$E_6$ is an attractive group for unification model building. However, the complexity of a rank 6 group makes it non-trivial to write down the structure of higher dimensional operators in an $E_6$ theory in terms of the states labeled by quantum numbers of the Standard Model gauge group. In this paper, we show the results of our computation of the Clebsch-Gordan coefficients for the products of the {27} with irreducible representations of higher dimensionality: ${78}$, ${351}$, ${351^\prime}$, ${\ol{351}}$, and ${\ol{351^\prime}}$. Application of these results to $E_6$ model building involving higher dimensional operators is straightforward.
hep-ph0101349.tex
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\def\NC{$\mid\! 0_{3}\rangle$}
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\begin{document}


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\begin{tabbing}
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxx\kill
 \> NUHEP-TH-00-81 \\ 
 \> SHEP-01-05 \\ 
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%
\vskip 0.4in
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\begin{center}


{\large \bf 
$E_6$ unification model building III.\\ 
Clebsch-Gordan coefficients in $E_6$ tensor products \\
of the ${\bf 27}$ with higher dimensional representations
}

\vskip 0.4in

Gregory W.~Anderson 
                   $^{a}$ 
and
Tom\'{a}\v{s} Bla\v{z}ek$^{*}$ 
                   $^{b}$ 




\vskip 0.2in

$^a$
    {\em Department of Physics and Astronomy, Northwestern University,}\\
    {\em 2145 Sheridan Road, Evanston, IL 60208, USA}\\
\vskip 0.05in

$^b$
    {\em Department of Physics and Astronomy, University of Southampton,}\\
    {\em Highfield, Southampton SO17 1BJ, UK}\\
\vskip 0.05in

e-mail: {\em anderson@susy.phys.nwu.edu, blazek@soton.ac.uk}

\vskip 0.2in

January, 2001 \\

\end{center}

\vskip 0.7in

%********1*********2*********3*********4*********5*********6*********7*********8
\begin{abstract}
$E_6$ is an attractive group for unification model building. However, the 
complexity of a rank 6 group makes it non-trivial 
to write down the structure of higher dimensional 
operators in an $E_6$ theory in terms of the states
labeled by quantum numbers of the Standard Model gauge group. 
In this paper, we show the results of our
computation of the Clebsch-Gordan coefficients
for the products of the {\bf 27} with 
irreducible representations of higher dimensionality:
${\bf 78}$, ${\bf 351}$, ${\bf 351^\prime}$, ${\bf \ol{351}}$, 
and ${\bf \ol{351^\prime}}$. Application of
these results to $E_6$ model building involving higher dimensional
operators is straightforward.


\end{abstract}


\vskip 0.5in

%PACS numbers:
%              02.20.Hj  Classical linear algebraic groups   ........... 7878 paper
%              03.65.Fd  Algebraic methods (in Quantum Mechanics)
%              11.30.Ly  Other internal and higher symmetries  ......... 7878 paper
%                                  (in Symmetry and Conservation Laws)
%              12.10.Dm  Unified theories and models of strong and 
%                        electroweak interactions
%                                  (in Unified Field Theories and Models)
%              12.60.Jv  Supersymmetric models
%                                  (in Models beyond the Standard Model)


\vskip 0.1in

$^*${\footnotesize On leave of absence from 
the Dept. of Theoretical Physics, Comenius Univ., Bratislava, Slovakia.}



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\section{Introduction}

$E_6$ is the minimal simple gauge group which could accommodate
one family of the observed fermions, {\it and} a family of Higgs states, into a
single gauge multiplet.\cite{grs} Therefore, unification models based on $E_6$ 
can provide relationships for the measured charged fermion masses 
and quark mixing angles: thirteen unrelated independent parameters of 
the Standard Model of elementary particles, and at the same time 
a small set of $E_6$ symmetric operators may relate the charged 
fermion data both to the masses and mixings in the neutrino sector 
and to the parameters of the Higgs sector.
In this respect, $E_6$ provides a framework for the
most economic unified supersymmetric theories. 

As is well known the key feature among the observed masses of the three generations
of fermions is the inter-generational hierarchy. 
Any unified model has to explain the origin of the hierarchy in terms of the dynamics
of the underlying theory. In $E_6$ models, the hierarchy can follow from
the pattern of the symmetry breaking as the rank 6 group is broken down to
the Standard Model gauge group, possibly in a succession of steps.
The hierarchy may be explicitly realized in terms of higher dimensional 
operators containing the light states after the superheavy degrees of freedom 
are integrated out of the effective theory generated below the $E_6$ breaking 
scale $M_6$. 
\cite{Froggatt_Nielsen}
From the technical point of view, the construction 
of higher dimensional $E_6$ symmetric operators and their structure
in terms of the Standard Model states is a non-trivial task. 
Assuming that the light states occupy the fundamental 27-dimensional irreducible 
representation (irrep), a complete knowledge of the tensor products 
of the {\bf 27} irrep with larger irreps is required. 
For instance,
the $E_6$ symmetry allows for a higher dimensional operator containing 
the product of three ${\bf 27}$s and a ${\bf 78}$, suppressed by some
heavy scale $M_H\geq M_6$. If the ${\bf 78}$ acquires a vacuum expectation
value (vev) $v_6\approx M_6$ and all three ${\bf 27}$s contain light states, 
such an operator contributes to the generation of fermion mass matrices. In particular, for
the first two families it may generate entries suppressed by $v_6/M_H$. 
Yet, the predictivity of $E_6$ can only be utilized if the exact
form of the singlet in 
${\bf 27} \otimes {\bf 27} \otimes {\bf 27} \otimes {\bf 78}$ 
in terms of the Standard Model states is known. If two of the ${\bf 27}$s are 
contracted antisymmetrically, one needs to know the 
Clebsch-Gordan decomposition of the ${\bf 351}$ in the tensor product
${\bf 27}\otimes {\bf 78}$, then the decomposition 
of the ${\bf \ol{27}}$ in the product ${\bf 27}\otimes {\bf 351}$,
and, lastly, the decomposition of the singlet in the product 
${\bf 27}\otimes {\bf \ol{27}}$.
However, a complete information on general 
tensor products of exceptional group $E_6$ is difficult to obtain. \cite{pat_Verma} 
As for particular computations, to our knowledge 
only the Clebsch-Gordan decompositions of ${\bf 27}\otimes {\bf 27}$, 
${\bf 27}\otimes {\bf \ol{27}}$, and ${\bf 78}\otimes {\bf 78}$
are presently available in the literature. \cite{pat_E6,abI,abII}.
(We note in passing that a separate paper \cite{bky} obtains a subset 
of the results needed for the operator ${\bf 27^3 78}$    
% ${\bf 27}^3{\bf 78}$
by studying a branching chain of $E_6$. The results presented in this paper
are relevant for the case when the vev is acquired by a
zero weight state of the ${\bf 78}$.) 

In this paper, we continue in our earlier work \cite{abI,abII}  and
provide basic group-theoretical tools for a 
construction of higher-dimensional $E_6$ symmetric operators. 
In particular, we present the results of our computation of 
the Clebsch-Gordan coefficients (CGCs) for the tensor products
involving 27, 78, and 351-dimensional representations, the lowest 
dimensional irreps in $E_6$.
%
%${\bf 27}\otimes {\bf 78}$,
%${\bf 27}\otimes {\bf 351}$,
%${\bf 27}\otimes {\bf \ol{351}}$,
%${\bf 27}\otimes {\bf 351^{\prime}}$,  and
%${\bf 27}\otimes {\bf \ol{351^{\prime}}}$.
%
Section 2 contains some necessary mathematical background for our study,
mostly concerned with lowering in the weight system of these irreps.
Our main results can be found in section 3, where we also comment 
on the construction and properties of the weight systems of the resulting 
irreps. Section 4 contains the summary, while one appendix provides details
on the lowering relations in the presence of degenerate weights.



\section{Mathematical Preliminaries}
\label{sc:MathPrelim}

In this work we consider tensor products of the fundamental 27-dimensional irrep
with higher dimensional 78 and 351-dimensional representations of $E_6$.
In particular, we compute the Clebsch-Gordan coefficients for the products\\
\parbox{12.7cm}
{\begin{eqnarray*}
{\bf 27}\otimes {\bf \:   \;\;78\:      } &=& {\bf     1728 } \oplus  {\bf \;\; 351 } \oplus  
                                            {\bf   \;  27 },                                 \\
{\bf 27}\otimes {\bf \:    351\:        } &=& {\bf \ol{7371}} \oplus  {\bf \ol{1728}} \oplus  
                                            {\bf \ol{ 351}} \oplus  {\bf \ol{  27}},         \\ 
{\bf 27}\otimes {\bf \:    351^{\prime} } &=& {\bf \ol{7722}} \oplus  {\bf \ol{1728}} \oplus  
                                            {\bf \;\ol{27}}                        ,         \\
{\bf 27}\otimes {\bf \:\ol{351}\:       } &=& {\bf \ol{5824}} \oplus  {\bf     2925 } \oplus  
                                            {\bf      650 } \oplus  {\bf       78 },         \\ 
{\bf 27}\otimes {\bf \:\ol{351^{\prime}}} &=& {\bf     3003 } \oplus  {\bf \ol{5824}} \oplus  
                                            {\bf      650 }.                                 
\end{eqnarray*}}
\hfill 
\parbox{0.7cm}
{\begin{eqnarray*}
   (\arabic{equation}{\mbox{a}}) \\ 
   (\arabic{equation}{\mbox{b}}) \\ 
   (\arabic{equation}{\mbox{c}}) \\ 
   (\arabic{equation}{\mbox{d}}) \\ 
   (\arabic{equation}{\mbox{e}}) 
                   \label{eq:dims_in_products}
\end{eqnarray*}}\\ 
\stepcounter{equation}
Before we discuss the construction of the weight systems of the irreps 
on the right hand side of these relations let us start first with 
the rules for the construction of the irreps on the left.

The key ingredient of our procedure is the lowering operation which is used to construct
a complete weight system by successive application of generators $E_{-\alpha_1},\ldots E_{-\alpha_6}$.
These are the generators which lie
outside of the diagonal Cartan subalgebra and correspond to the six simple roots of $E_6$. 
(Our choice of generators is described in more detail in \cite{abII}, 
and basically follows the standard conventions of \cite{Slansky_PhysRep}.)
The six generators act as ladder operators --- 
at each level the weight of the new state is obtained from the weight at the previous level
by subtracting (in the weight space) the respective simple root:
\be
   E_{-\alpha_i}|\, w\rangle\, = N_{-\alpha_i,\,w}|\, w-\alpha_i\rangle.
\label{eq:dfnN}
\ee
For the weight systems of the ${\bf 27}$ and ${\bf 78}$ constants $N_{-\alpha_i,w}$ satisfy
(see eq.(12) in \cite{abII})
\be
|N_{-\alpha_i,(w)_j}|^2 =          \langle \alpha_i| \,w\,\rangle\, 
                           +\, |\, \langle (w)_j\, |\, (w)_i\rangle|^2\;\; 
                               |N_{-\alpha_i,\,w+\alpha_i}|^2.
\label{eq:N-}
\ee
It is understood that the new state $|\, w-\alpha_i\rangle$ does not exist
if $N_{-\alpha_i,w}=0$  or the r.h.s. of (\ref{eq:N-}) turns out to be negative. 
The subscript on the weight $(w)$
%
% , not to be mistaken for a weight coordinate 
%   $w_i = \,\langle \alpha_i| w\rangle$, 
is only relevant for the six degenerate zero weight states
of the ${\bf 78}$ and is to be ignored for non-degenerate weights. 
In fact, the second term on the r.h.s. of (\ref{eq:N-}) never contributes when one constructs
the weight system of the ${\bf 27}$ as there are no higher multiplets than doublets
for any $SU(2)$ subgroup. 

We remark that throughout this work, and consistent with our previous studies \cite{abI,abII}, 
the lowering phase convention which always fixes constants $N$ to be real and non-negative
\be
    N_{-\alpha_i,\, w} \ge 0
\label{eq:Npositive}
\ee
is adopted for any simple root $\alpha_i$ and any weight system. Then
for the zero weight states of the ${\bf 78}$ the inner product in (\ref{eq:N-}) can
be expressed as
\be
\langle (w)_i\, |\, (w)_j\rangle = |A_{ij}| / 2,
\label{eq:ww_in_78}
\ee
where $A_{ij}\equiv \langle \alpha_i\, |\, \alpha_j\rangle$ are the elements 
of the Cartan matrix of $E_6$. \cite{Slansky_PhysRep,abII} 
This result follows from the decomposition of the ${\bf 78}$ weight states 
into the states of the fundamental representations in the tensor product 
${\bf 27}\otimes {\bf \ol{27}}$. \cite{abI}

In the appendix, we derive a generalized relation for $N_{-\alpha_i,w}$ for a weight system 
with multiple degenerate weights at different levels. We now discuss how to apply general 
formula (\ref{eq:app3}) 
to the weight systems of the 351-dimensional representations which appear on the left in 
eq.(\ref{eq:dims_in_products}b--e). These irreps, although already rather large, are still special 
because for each weight subspace a basis can be defined such that the application 
of a lowering ladder operator results in a single basis state, as indicated in eq.(\ref{eq:dfnN}). 
(For larger irreps, there are lowerings which lead to a linear combination 
of the basis states regardless of the basis definition.)
Moreover, if weight $(w)$ is degenerate and a state with weight $(w-\alpha)$ exists,
then the $(w+\alpha)$ weight state is either non-degenerate or does not exist at all.
Thus for the weight system of the ${\bf 351^\prime}$ or ${\bf 351}$ 
eq.(\ref{eq:app3}) reduces to a simple form
\be
|N_{-\alpha,\,(w)_a\rarrow (w-\alpha)_{A_a}}|^2 =  \langle \alpha|\, w\rangle\, 
                                        +\, | \langle (w)_a\, |\, (w)_c\rangle|^2\;\; 
                                          |N_{-\alpha,\, (w+\alpha)\rarrow (w)_c}|^2,
\label{eq:N-deg}
\ee
where, formally, the summation over $c$ is assumed in the last term, but no more than one state actually
contributes. Concrete applications of this formula are provided at the end of the section.

Compared to eq.(\ref{eq:N-}) both weights $(w)$ and $(w-\alpha)$ can now be degenerate. 
We find, however, that in the ${\bf 351^\prime}$ or ${\bf 351}$ the $(w+\alpha)$
weight state does not exist if $(w-\alpha)$ is a degenerate weight.
Hence if both $(w)$ and $(w-\alpha)$ are degenerate, 
$A_a$ can be set to $a$ by definition and (\ref{eq:N-deg}) can be
simplified even further:
\be
|N_{-\alpha,\,(w)_a\rarrow (w-\alpha)_{a}}|^2 =  \langle \alpha|\, w\rangle, 
\qquad\mbox{\rm for both }\:(w)\:\: \mbox{\rm and }\:\:(w-\alpha)\:\: \mbox{\rm degenerate.}
\label{eq:N-deg2}
\ee
This shows that the definition of the basis states (and their subscript labeling) 
in the degenerate subspaces of the ${\bf 351^\prime}$ or ${\bf 351}$ 
can be induced from the basis states at the previous level.
However, once $(w)$ is found degenerate, how do we know if $(w-\alpha)$
is going to be degenerate and what the dimensionality of this subspace is going to be?
Similarly to the  case of the ${\bf 78}$ weight system \cite{abII} the decomposition of the 
351-dimensional irreps into the states of the fundamental representations can be recalled.
The product 
${\bf \ol{27}}\otimes {\bf \ol{27}} = {\bf 351^\prime} \oplus  {\bf 351} \oplus  {\bf 27 }$
is conjugated to the product studied in ref.\cite{pat_E6}.
We refer to this work to claim that all degenerate weight subspaces in the 
${\bf 351^\prime}$ (or ${\bf 351}$) are of the same dimensionality and that
the degenerate weights follow the weight system of the ${\bf 27}$.
In the end it thus turns out that 
complete bases in the degenerate weight subspaces of the ${\bf 351^\prime}$ 
can be obtained
%
%   by following the lowering paths in the weight system of the ${\bf 27}$, 
%
starting from the four $(100000)$ weight states at level 8 of the ${\bf 351^\prime}$
\cite{footnote_1} :\\
\parbox{12cm}
{
  \beannmb
      |\,(100000)_3\rangle &=& E_{-\alpha_3} \: |\,(1\ol{1}2\ol{1}0\ol{1})\rangle \,/\sqrt{2},\\
      |\,(100000)_4\rangle &=& E_{-\alpha_4} \: |\,(10\ol{1}2\ol{1}0)\rangle      \,/\sqrt{2},     \\
      |\,(100000)_5\rangle &=& E_{-\alpha_5} \: |\,(100\ol{1}20)\rangle \,/\sqrt{2},     \\
      |\,(100000)_6\rangle &=& E_{-\alpha_6} \: |\,(10\ol{1}002)\rangle     \,/\sqrt{2}.      
  \eeannmb
}
\hfill
\parbox{3cm}
{
  \bea
    { }& &{ }\nnmb\\
    { }& &{ }      \label{eq:100000in351P}  \\
    { }& &{ }\nnmb
  \eea
}\\
With lowering convention (\ref{eq:Npositive}) 
the remaining 26 degenerate weight subspaces at lower levels can be specified as
$\,
|\,(w)_a\rangle = E_{-\alpha_A}\ldots E_{-\alpha_B}$ 
$|\,(100000)_a\rangle
$, 
$
a=3,4,5,6, 
$
%
% where $A\ldots B$ is the lowering path (symbolic notation for the sequence of lowering operators
% $E_{-\alpha_A}\ldots E_{-\alpha_B}$) leading to state $|\,(w)\rangle $ in the ${\bf 27}$.
%
where $E_{-\alpha_A}\ldots E_{-\alpha_B}$ is the lowering path 
leading to state $|(w)\,\rangle $ in the ${\bf 27}$.
For the ${\bf 351}$ the only difference is that the degenerate weight subspaces are five-dimensional
and the relations analogous to (\ref{eq:100000in351P}) also include 
\be
   |\,(100000)_2\rangle = E_{-\alpha_2} \:  |\,(02\ol{1}000)\rangle \,/\sqrt{2} 
   \label{eq:100000in351}        
\ee
when computing the $(100000)$ states at level 7 of this irrep.
We note that with this notation 
%
%[and assuming the lowering phase convention (\ref{eq:Npositive})]
%
the inner product in any degenerate weight subspace of both
the ${\bf 351^\prime}$ and ${\bf 351}$ satisfies 
\be
   \langle (w)_a\, |\, (w)_b\rangle = |A_{ab}| / 2
\label{eq:ww_in_351s}
\ee
(where $a,b = 3,4,5,6$ for the ${\bf 351^\prime}$, and 
       $a,b = 2,3,4,5,6$ for the ${\bf 351}$), 
in a close similarity to the degenerate zero weight 
subspace of the ${\bf 78}$, eq.(\ref{eq:ww_in_78}). 

As an example of the application of formula (\ref{eq:N-deg}) consider all possible
lowerings of the state $|\,F_5\rangle$ in the ${\bf 351^\prime}$ where, for brevity, 
$F$ stands for the $(100000)$ weight. Three different states at the next level can be obtained:
$  E_{-\alpha_1}\, |\,F_5\rangle = N_1\: |\,(\ol{1}10000)_5\rangle $,
$  E_{-\alpha_4}\, |\,F_5\rangle = N_4\: |\,101\ol{2}10\rangle $, and
$  E_{-\alpha_5}\, |\,F_5\rangle = N_5\: |\,1001\ol{2}0\rangle$.
Based on (\ref{eq:N-deg}) the constants are
\[
 \begin{array}{ccccccc}
  |N_1|^2 &=& 1 &+&                              0  &=& 1          \, , \\
  |N_4|^2 &=& 0 &+& (\frac{1}{2})^2   (\sqrt{2})^2  &=& \frac{1}{2}\, , \\  
  |N_5|^2 &=& 0 &+&             1^2 \;(\sqrt{2})^2  &=& 2          \, ,
 \end{array}
\]
as we have already showed in the second and third equation (\ref{eq:100000in351P}) that 
\[
  N_{-\alpha_4,\,(F+\alpha_4)\rightarrow (F)_4} \, = \,
  N_{-\alpha_5,\,(F+\alpha_5)\rightarrow (F)_5} \, = \, \sqrt{2}.
\nonumber
\]
Implicitly, we also used the fact that (\ref{eq:100000in351P}) 
represents the only way how the $(100000)$ weight states can be obtained
from the states at the previous level.
Note that (\ref{eq:N-deg2}) could be used to compute $N_1$ since both $(100000)$
and $(\ol{1}10000)$ are degenerate weights.

Finally, we remark that the properties of the ${\bf \ol{351^\prime}}$ and ${\bf \ol{351}}$ are easily
derived from the properties of the ${\bf 351^\prime}$ and ${\bf 351}$ after the Dynkin coordinates 
[and any other indices in Dynkin formalism, like e.g., the labeling of states in 
eq.(\ref{eq:100000in351P})]
$1$ and $2$ are exchanged with $5$ and $4$, respectively.



\section{Construction of Clebsch-Gordan Coefficients}

Tensor products in eq.(\ref{eq:dims_in_products})
can be expressed in terms of the highest weights as\\
\parbox{14cm}
{\begin{eqnarray*}
(100000)\otimes(000001) &=& (100001)\oplus (000100) \oplus (100000)                , \\ 
(100000)\otimes(000100) &=& (100100)\oplus (000011) \oplus (010000) \oplus (000010), \\ 
(100000)\otimes(000020) &=& (100020)\oplus (000011)                 \oplus (000010), \\     
(100000)\otimes(010000) &=& (110000)\oplus (001000) \oplus (100010) \oplus (000001), \\ 
(100000)\otimes(200000) &=& (300000)\oplus (110000) \oplus (100010)                . 
\end{eqnarray*}}
\hfill 
\parbox{1cm}
{\begin{eqnarray*}
   (\arabic{equation}{\mbox{a}}) \\ 
   (\arabic{equation}{\mbox{b}}) \\ 
   (\arabic{equation}{\mbox{c}}) \\ 
   (\arabic{equation}{\mbox{d}}) \\ 
   (\arabic{equation}{\mbox{e}}) 
                   \label{eq:weights_in_products}
\end{eqnarray*}}\\ 
\stepcounter{equation}
For each product we start with the construction of the weight system of the first irrep on 
the r.h.s..
% of (\ref{eq:weights_in_products}). 
The highest weight state of this irrep is non-degenerate and can always be expressed as 
a trivial combination of the highest weight states of the two irreps on the left-hand side,
with the CGC being equal to $+1$. 
In the absence of a simple method how to determine the bases in the degenerate weight subspaces
which follow at lower levels for each of these irreps, we compute directly the complete weight
system in each case. However, note that simple lowering (\ref{eq:dfnN}) does not necessarily
hold for weights with multiple degeneracies, as is discussed in the appendix. States at lower 
levels are then computed by successive lowerings 
%with generators $E_{-\alpha_1}$, ..$E_{-\alpha_6}$ 
applied to the states of the 
${\bf 27}$, and ${\bf 78}$ in case $(a)$ or 
one of the 351-dimensional irreps in cases $(b)$--$(e)$. These lowerings were described in detail
in the previous section. The computed state is accepted and kept as a new basis state if it
cannot be expressed as a linear combination of the previously obtained basis states with the same weight.

It is not necessary to show the Clebsch-Gordan coefficients for every linearly independent state, 
since there are many states with the same CGCs. Instead, we
present the results just for the dominant weight states. Dominant weights are weights
with all Dynkin coordinates non-negative. The CGCs for the remaining states can then be 
determined using the charge conjugation operators \cite{pat_CCO, footnote_2}, or in a straightforward way by
direct lowering.
In tables \ref{t:paths_1728}--\ref{t:paths_3003} we present lowering paths for the
dominant weight states of the ${\bf 1728}$, ${\bf \ol{7371}}$, 
${\bf \ol{7722}}$, ${\bf \ol{5824}}$, and ${\bf 3003}$ irreps. In our abbreviated notation,
lowering path, let's say, 3421 stands for $E_{-\alpha_3} E_{-\alpha_4} E_{-\alpha_2} E_{-\alpha_1}$
applied (from the right) to the highest weight state.  
The lowering paths in tables \ref{t:paths_1728}--\ref{t:paths_3003} actually specify 
our choice of bases for particular dominant weight subspaces.
Explicit Clebsch-Gordan decomposition of the dominant weight states 
is important because, typically, the multiplicity of degeneracy (i.e., the dimensionality
of the weight subspace)
changes compared to the degeneracy at the previous level. Clearly, that is why these states cannot be 
obtained by generalized charge conjugation from the states at the previous levels.
Moreover, it is important to check the completeness of a reducible dominant weight subspace.
If it is impossible to complete its basis by lowering the states at the previous level, 
new weight systems open up and the remaining basis vectors are their highest weight states.
This is what happens for every dominant weight in the tensor products 
${\bf 78}\otimes {\bf 78}$ \cite{abII}, ${\bf 27}\otimes {\bf \ol{27}}$ \cite{abI},
or ${\bf 27}\otimes {\bf 27}$ \cite{abI} studied in the earlier work.
However, this property of the dominant weights is no longer true for the products studied here.
We now discuss shortly the dominant weights in each of the products in (\ref{eq:weights_in_products}).

\vspace{3mm}
\no
{\itshape
(a) {\bfseries 
      (100000)$\,\otimes\,$(000001) $=$ (100001)$\,\oplus\,$(000100)$\,\oplus\,$(100000)
    }
}

             At level 4 of the 1728-dimensional {\bf (100001)} irrep
             we find four states with weight $(000100)$. This weight space,
             however, is five-dimensional, and the computation of the
             state orthogonal to the previous four yields the highest
             weight state of the ${\bf 351}$ irrep. 
             (See table \ref{t:78cgc000100}.)
             Similarly, at level 11 we find 16-fold degenerate weight $(100000)$,
             while this reducible subspace unfolds to be 22-dimensional. Since there are
             five distinct states of the same weight in the ${\bf 351}$, there is
             room for one extra state. Once computed as orthogonal to all the
             other 21 states it becomes the highest weight state of the
             fundamental 27-dimensional {\bf (100000)} irrep. 
             Note that in table \ref{t:78cgc100000} we keep the labeling 
             of the five $(100000)$ states of the ${\bf 351}$ 
             consistent with the notation introduced in equations 
             (\ref{eq:100000in351P}) and (\ref{eq:100000in351}).

\vspace{3mm}
\no
{\itshape
(b) {\bfseries 
      (100000)$\,\otimes\,$(000100) $=$ (100100)$\,\oplus\,$(000011)$\,\oplus\,$(010000)$\,\oplus\,$(000010)
    }
}

             Lowering down to level 4 of the 7371-dimensional {\bf (100100)} irrep       
             we obtain four distinct $(000011)$ weight states spanned over 
             a five-dimensional reducible subspace. The last basis state in this subspace,
             orthogonal to the four from the ${\bf \ol{7371}}$, becomes the highest
             weight state of the ${\bf \ol{1728}}$. 
             (See table \ref{t:351cgc000011}.)
             This is a conjugate irrep to the ${\bf 1728}$ described in {\it (a)}.
             The lowering paths to the dominant weights in its weight system can be
             obtained from table \ref{t:paths_1728} (replacing 1 and 2 with 5 and 4, 
             and {\it vice versa}. Proceeding to level 7 
             a five-fold degenerate $(200000)$ dominant weight is found:
             \be
                 |200000_a\rangle = |100000\rangle\,|100000_a\rangle, 
                 \quad\quad a=2,\ldots 6
             \label{eq:200000b}
             \ee
             which, obviously, does not leave any extra space 
             for states outside of the ${\bf \ol{7371}}$. This is consistent
             with no observation in {\it (a)} of a dominant weight $(000020)$
             in the weight system of the ${\bf 1728}$, and also with the fact that
             there is no ${\bf (200000)}$ irrep on the right side of 
             (\ref{eq:weights_in_products}b). The charge conjugation operators 
             can be used to show that a five-fold degenerate weight subspace
             with CGCs equal to 1
             is then present at odd levels of the ${\bf \ol{7371}}$ from this level down
             until 
             subspace $(0000\ol{2}0)$ emerges at level 39.
             Next, at level 8 fifteen linearly independent $(010000)$ weight states
             are present, while the weight subspace turns out to be 20-dimensional.
             Not surprisingly there are four states which belong to the 
             weight system of the ${\bf \ol{1728}}$ (compare with {\it (a)} above),
             and the remaining basis state, orthogonal to the previous nineteen,
             represents the highest weight state of the ${\bf \ol{351}}$.
             Lastly, at level 15 we get the reducible $(000010)$ weight subspace,
             which is 66-dimensional. That makes room for the highest weight of the
             ${\bf \ol{27}}$, since there are 44 basis states present 
             in the ${\bf \ol{7371}}$
             together with 16  states of the ${\bf \ol{1728}}$. Additional five states
             of the ${\bf \ol{351}}$ should be expected based on 
             eqs.(\ref{eq:100000in351P}, \ref{eq:100000in351}).
             The CGCs for this subspace are presented in 
             tables \ref{t:351cgc000010} and  \ref{t:351cgc000010part2}.

\vspace{3mm}
\no
{\itshape
(c) {\bfseries 
       (100000)$\,\otimes\,$(000020) $=$ (100020)$\,\oplus\,$(000011)$\,\oplus\,$(000010)
    }
}

             In the construction of the 7722-dimensional ${\bf (100020)}$ irrep one finds 
             a dominant weight already at level 1, 
             \be 
                 |100100\rangle = |100000\rangle\,|000100\rangle.
             \ee
             It occupies a one-dimensional subspace, which
             is consistent with the absence of the ${\bf (100100)}$ irrep
             in product (\ref{eq:weights_in_products}c). 
%             and shows how elegant and efficient the Dynkin formalism is.
             The first degenerate dominant weight is obtained at level 5. 
             There, a six-dimensional $(000011)$ weight subspace contains five
             linearly independent states of the ${\bf \ol{7722}}$. The basis in
             this reducible subspace is completed by the highest weight state 
             of the ${\bf \ol{1728}}$ (table \ref{t:351primecgc000011}).
             Proceeding further, there is no room
             for the highest weight state of a new irrep when dominant
             weights $(200000)$ and $(010000)$ are encountered at levels 8 and 9,
             respectively. The $(200000)$ subspace is four-dimensional and its basis 
             can be specified as in eq.(\ref{eq:200000b}). (The states are 
             now numbered as $a=3,4,5,6$.) The CGC decomposition of the
             $(010000)$ subspace can be found in table \ref{t:351primecgc010000}.
             Finally, at level 16 the last dominant weight in this product is
             unveiled. The reducible $(000010)$ weight subspace turns out to 
             be 57-dimensional,
             with 40 basis states coming from the ${\bf \ol{7722}}$ and 16 states 
             from the ${\bf \ol{1728}}$. The remaining state, orthogonal to them,
             becomes the highest weight state of the ${\bf \ol{27}}$ 
             (see tables \ref{t:351primecgc000010} and \ref{t:351primecgc000010part2}).

\vspace{3mm}
\no
{\itshape
(d) {\bfseries 
       (100000)$\,\otimes\,$(010000) $=$ (110000)$\,\oplus\,$(001000)$\,\oplus\,$(100010)$\,\oplus\,$(000001)
    }
}

             In this product, we find the dominant weight states encountered
             already in the decomposition of ${\bf 78}\otimes {\bf 78}$ and 
             ${\bf 27}\otimes {\bf \ol{27}}$. At level 2 of the
             $\bf (110000)$ weight system ({\it i.e.}, the ${\bf \ol{5824}}$ irrep) 
             we reach the 3-dimensional $(001000)$ subspace, with two states in the ${\bf \ol{5824}}$ 
             and the third one being the highest weight state of the ${\bf 2925}$,
             as shown in table  \ref{t:351barcgc001000}.
             Then following the lowering paths in table \ref{t:paths_5824bar}, 
             table II in Ref.\cite{abII}, and table I in Ref.\cite{abI} the dominant
             weights $(100010)$, $(000001)$,  and $(000000)$ follow at levels
             7, 12, and 23, respectively. The CGCs for these dominant weights
             can be found in tables \ref{t:351barcgc100010}--\ref{t:351barcgc000000part4}.
             The reducible $(000000)$ subspace is 135-dimensional and represents
             the most (technically) involved computation in this study. Obviously,
             it cannot (and does not) leave any room for the singlet since the two
             representations in the product are not conjugate to each other.

\vspace{3mm}
\no
{\itshape
(e) {\bfseries 
      (100000)$\,\otimes\,$(200000) $=$ (300000)$\,\oplus\,$(110000)$\,\oplus\,$(100010)
    }
}

             The 3003-dimensional ${\bf (300000)}$ irrep contains a dominant weight already at level 1:
             \be
                 |110000\rangle = (\,|\ol{1}10000\rangle\,|200000\rangle
                                  + \sqrt{2}\,|100000\rangle\,|010000\rangle)/\sqrt{3}.
             \ee
             The orthogonal combination
             \be
                 |110000\rangle = (\,\sqrt{2} |\ol{1}10000\rangle\,|200000\rangle
                                  -           |100000\rangle\,|010000\rangle)/\sqrt{3}
             \ee
             forms the highest weight state of the ${\bf \ol{5824}}$.
             Since then, the same dominant weights occur as in the weight system of the
             ${\bf \ol{5824}}$ described under {\it (d)} above. There are, however, no ${\bf 2925}$
             and ${\bf 78}$ irreps in this product (see tables \ref{t:351prime_barcgc001000},
             \ref{t:351prime_barcgc000001}, and \ref{t:351prime_barcgc000001part2}), 
             just the highest weight state of the ${\bf 650}$ completes the 13-dimensional $(100010)$ subspace 
             at level 8 (table \ref{t:351prime_barcgc100010}). The reducible $(000000)$ weight subspace 
             is 108-dimensional and its decomposition can be found 
             in tables \ref{t:351prime_barcgc000000part1}--\ref{t:351prime_barcgc000000part4}.




\section{Summary}

We have presented the Clebsch-Gordan decomposition of the $E_6$ tensor
products of the fundamental ${\bf 27}$ irrep 
with the 78- and 351-dimensional irreps. Analogous products involving
the ${\bf \ol{27}}$ instead of the ${\bf 27}$ can now be obtained trivially
by charge conjugation. It is straightforward to apply these results to
the construction of higher dimension operators in $E_6$ model building
\cite{ab_models}.



\section*{Appendix: The problem of Degenerate Weights}

\appendix



\def\sp#1#2{\langle\, #1 , #2\, \rangle}


Rules (\ref{eq:dfnN},\ref{eq:N-}) are insufficient for representations 
with degenerate weights at successive levels. 
For degenerate weights we must first identify a particular 
basis. Label  the degenerate basis states of weights
$(w+\alpha)$, $(w)$, and $(w-\alpha)$ as
\begin{eqnarray*}
&\mid&\!\! (w+\alpha)_{\Gamma} \rangle, \quad\mbox{where}\quad \Gamma = 1,\ldots D_{w+\alpha}, \\
&\mid&\!\! \;\; w_{c}\qquad\:  \rangle, \quad\mbox{where}\quad      c = 1,\ldots D_{w}, \qquad \mbox{and} \\
&\mid&\!\! (w-\alpha)_{C}      \rangle, \quad\mbox{where}\quad      C = 1,\ldots D_{w-\alpha}. 
\end{eqnarray*}
$D_w$ stands for the degeneracy of $(w)$.  
The basis states are in general non-orthogonal.
In our notation, they are always normalized to unity:
$\langle w_{c} \mid w_{c} \rangle = 1$.
The identity operator in the degenerate subspace is
\begin{eqnarray}
I\; &=& G_{ab} \mid w_a\, \rangle \langle\, w_b \mid, 
\cr
G_{ab} &=& (M^{-1})_{ab}, \quad \mbox{where}\quad M_{ab} = \langle\, w_a \mid w_b\, \rangle.
\end{eqnarray}

Although the basis is non-orthogonal, we can construct state
vectors which are orthogonal to any state except the state
we are interested in
\begin{eqnarray}
\mid      \hat{w_b}\, \rangle &=&  \mid w_a\, \rangle\: G_{ab}, \cr
\langle \,\hat{w_a}   \mid    &=&  G_{ab}  \: \langle\, w_b \mid,
\end{eqnarray}
which satisfy
\begin{equation}
\langle \,w_c \mid \hat{w_a}\, \rangle = \langle \,\hat{w_a} \mid w_c\, \rangle = \delta_{ac}.
\end{equation}
\noindent
A general raising or lowering of a degenerate weight state can be written as:
\begin{eqnarray}
E_{\:\alpha_i} \mid w_c \rangle &=&
    N_{\:\alpha_i,\,w_c \rightarrow (w+\alpha_i)_\Gamma} \;\,
                               \mid (w+\alpha_i)_\Gamma \rangle \\
E_{ -\alpha_i} \mid w_c \rangle &=&
    N_{ -\alpha_i,\,w_c \rightarrow (w-\alpha_i)_C} 
                               \mid (w-\alpha_i)_C \rangle
\label{eq:general_lowering_relation} 
\end{eqnarray}
where there is a possible sum over the states on the right hand side 
[compare (\ref{eq:general_lowering_relation})  with (\ref{eq:dfnN})]. 
The lowering normalization constant can then be expressed only as a sum of matrix elements
$N_{-\alpha_i,\,w_{a} \rightarrow (w-\alpha_i)_A}
= G^{(w-\alpha_i)}_{AB}\,
\langle (w-\alpha_i)_B \mid E_{-\alpha_i} \mid w_{a} \rangle $.\cite{footnote_3} 
Using $E_{\alpha} = E_{-\alpha}^{\dag}$
and the defining relation (\ref{eq:general_lowering_relation}) 
we derive
\begin{eqnarray}
& &
    N_{-\alpha,\, w_a \rightarrow (w-\alpha)_A}\, 
N^{*}_{-\alpha,\, w_b \rightarrow (w-\alpha)_B}\,
\langle \,(w-\alpha)_B \mid (w-\alpha)_A \rangle 
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\cr
& & \qquad
= 
\langle w_b \mid E_{\alpha} E_{-\alpha} \mid w_a \rangle
\cr
& & \qquad
= 
\langle w_b \mid [ E_{\alpha}, E_{-\alpha} ] 
                +  E_{-\alpha} E_{\alpha} 
  \mid  w_a \rangle 
\cr
& & \qquad
= 
\langle w_b \mid w_a \rangle \,\sp{\alpha}{w} + \,
G^{w+\alpha}_{\Gamma\Delta} \,
\langle w_b \mid E_{-\alpha} \mid (w+\alpha)_\Gamma \rangle\,
\langle (w+\alpha)_\Delta \mid E_{\alpha} \mid  w_a \rangle 
\cr
& & \qquad
= 
\langle w_b \mid w_a \rangle \,\sp{\alpha}{w} + \,
G^{w+\alpha}_{\Gamma\Delta} \,
N_{-\alpha,\,(w+\alpha)_\Gamma \rightarrow w_c}
\langle w_b \mid w_c \rangle\,
N^{*}_{-\alpha,\, (w+\alpha)_\Delta \rightarrow w_d}
\langle w_d \mid w_a \rangle.
\end{eqnarray}

\noindent
Hence 
\begin{eqnarray}
  \mbox{\rule[-0.40cm]{0mm}{0.5cm}}
  (G^{w-\alpha})^{-1}_{AB}&\!\!\!
      N_{-\alpha,\, w_a \rightarrow (w-\alpha)_A} \,
  N^{*}_{-\alpha,\, w_b \rightarrow (w-\alpha)_B} 
  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
 \cr
 =
 &\!
  (G^w)^{-1}_{ab} \sp{\alpha}{w} +\, 
  G^{w+\alpha}_{\Gamma\Delta} \,(G^w)^{-1}_{bc}(G^w)^{-1}_{da}\,
      N_{-\alpha,\, (w+\alpha)_\Gamma \rightarrow w_c}\,
  N^{*}_{-\alpha,\, (w+\alpha)_\Delta \rightarrow w_d}\,.
\label{eq:app2}
\end{eqnarray}
%
%
%
For $a=b$ we get:
%
\begin{eqnarray}
  \mbox{\rule[-0.40cm]{0mm}{0.5cm}}
  (G^{w-\alpha})^{-1}_{AB} &\!\!\!
      N_{-\alpha,\, w_a \rightarrow (w-\alpha)_A} \,
  N^{*}_{-\alpha,\, w_a \rightarrow (w-\alpha)_B} 
  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
 \cr
 =
 &\!
  \sp{\alpha}{w} +\, 
  G^{w+\alpha}_{\Gamma\Delta} \,(G^w)^{-1}_{ac}(G^w)^{-1}_{ad}\,
      N_{-\alpha,\, (w+\alpha)_\Gamma \rightarrow w_c}\,
  N^{*}_{-\alpha,\, (w+\alpha)_\Delta \rightarrow w_d}\,.
  \qquad\qquad
\label{eq:app3}
\end{eqnarray}


\noindent
Another useful expression can be found
by contracting relation (\ref{eq:app2}) with $G^{w}_{ab}\:$:
\begin{eqnarray}
\mbox{\rule[-0.40cm]{0mm}{0.5cm}}
(G^w)_{ab} \!\!\!\!\!\! & 
(G^{w-\alpha})^{-1}_{AB}\;
    N_{-\alpha,\, w_a \rightarrow (w-\alpha)_A} \,
N^{*}_{-\alpha,\, w_b \rightarrow (w-\alpha)_B} 
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\cr
=
&\!
\sp{\alpha}{w} \, D_w + \,
G^{w+\alpha}_{\Gamma\Delta}\, (G^w)^{-1}_{cd} \,
    N_{-\alpha,\, (w+\alpha)_\Gamma \rightarrow w_c} \,
N^{*}_{-\alpha,\, (w-\alpha)_\Delta \rightarrow w_d} \,.
\qquad\qquad\qquad\qquad\qquad
\end{eqnarray}
This expression is easily iterated along a sequence
of lowerings with the same ladder operator:
\begin{eqnarray}
\mbox{\rule[-0.40cm]{0mm}{0.5cm}}
(G^w)_{ab} \!\!\!\!\!\! & 
(G^{w-\alpha})^{-1}_{AB}\;  
    N_{-\alpha,\, w_a \rightarrow (w-\alpha)_A} \, 
N^{*}_{-\alpha,\, w_b \rightarrow (w-\alpha)_B} 
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\cr
=
&\!
\sp{\alpha}{w} \, D_w + \sp{\alpha}{w+\alpha} \, D_{w+\alpha} +\,
G^{w+2\alpha}_{\gamma\delta}\, (G^{w+\alpha})^{-1}_{\Gamma\Delta} \,
    N_{-\alpha,\, (w+2\alpha)_\gamma \rightarrow (w+\alpha)_\Gamma}\,
N^{*}_{-\alpha,\, (w+2\alpha)_\delta \rightarrow (w+\alpha)_\Delta}
\cr
=
&\!
\sp{\alpha}{w}\,D_w  +          \sp{\alpha}{w+ \alpha}\, D_{w+ \alpha}
                     + \ldots + \sp{\alpha}{w+k\alpha}\, D_{w+k\alpha}\, , 
\qquad\qquad\qquad\qquad\qquad\qquad\quad
\end{eqnarray}
where $(w+k\alpha)$ is the highest weight in the $SU(2)$ subgroup
chain $(w)$, $(w+\alpha)$, $(w+2\alpha)$, $\ldots$ present in the weight system.

Finally, for completeness, when raising operators are applied, eq.(\ref{eq:app3})
can be written as
\begin{eqnarray}
  \mbox{\rule[-0.40cm]{0mm}{0.5cm}}
  (G^{w+\alpha})^{-1}_{\Gamma\Delta}&\!\!\!
      N_{\alpha,\, w_a \rightarrow (w+\alpha)_\Gamma} \,
  N^{*}_{\alpha,\, w_a \rightarrow (w+\alpha)_\Delta} 
  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
 \cr
 =
 &\!
  - \sp{\alpha}{w} +\, 
  G^{w-\alpha}_{CD} \,(G^w)^{-1}_{ac}(G^w)^{-1}_{ad}\,
      N_{\alpha,\, (w-\alpha)_C \rightarrow w_c}\,
  N^{*}_{\alpha,\, (w-\alpha)_D \rightarrow w_d}.
  \qquad\qquad
\end{eqnarray}




\subsubsection*{Special Cases: Lowering within basis states}
 
Consider a series of states connected by repeated application
of the same lowering operator $E_{-\alpha}$. Choose any 
states with degenerate weights obtained in this series as part 
of the basis for the degenerate weights and label these states by $i$.
For the sequence: 
$\ldots, (w + \alpha)_i, w_i, (w-\alpha)_i, \ldots$ the
generalized recursion relation (\ref{eq:app3}) reduces to:
\be
  \mbox{\rule[-0.40cm]{0mm}{0.5cm}}
  \mid N_{-\alpha,\, w_i \rightarrow (w -\alpha)_i} \mid^2 
  \, = \, \sp{\alpha}{w} + \:
  G^{w+\alpha}_{\Gamma\Delta} (G^w)^{-1}_{ic}
                              (G^w)^{-1}_{id}
      N_{-\alpha,\, (w +\alpha)_\Gamma \rightarrow w_c}
  N^{*}_{-\alpha,\, (w +\alpha)_\Delta \rightarrow w_d}\, .
\ee
When $(w + \alpha)_i$ is the only state which can be lowered by $E_{-\alpha}$ 
to obtain a state of weight $w$ we get 
\be
  \mbox{\rule[-0.40cm]{0mm}{0.5cm}}
  \mid N_{-\alpha,\, w_i \rightarrow (w -\alpha)_i} \mid^2 
  = \sp{\alpha}{w} + \:
  G^{(w+\alpha)}_{ii} 
  |N_{-\alpha,\, (w +\alpha)_i \rightarrow w_i}|^2.
\ee
This includes the case of a non-degenerate $(w + \alpha)$ weight subspace.
When $(w + \alpha)$ is non-degenerate $G^{(w+\alpha)}_{ii} = 1$, which
further simplifies the above relation.



A special case of interest is lowering the degenerate
zero weight states of the adjoint representation which
correspond to the Cartan sub-algebra.  These
degenerate weight states can be labeled $\mid (0)_i\, \rangle$ where 
the $i$-th degenerate weight is obtained by 
$E_{-\alpha_i} \mid \alpha_i\, \rangle \propto\:
\mid (0)_i\, \rangle$. This basis, however, is not orthogonal.
When lowering such a basis state the general formula 
(\ref{eq:app3}) reduces to:
\be
    \mid N_{-\alpha_i, (0)_j \rightarrow (-\alpha_i)} \mid^2
 \: = \:
    \left[ (G^{(0)})^{-1}_{ji} \right]^2
    \mid N_{-\alpha_i,\, (\alpha_i) \rightarrow (0)_i }\mid^2 
 \: = \: 
    \langle (0)_j\, |\, (0)_i\rangle^2\;\; 
    \mid N_{-\alpha_i,\, (\alpha_i) \rightarrow (0)_i }\mid^2 .
\ee
This result 
is consistent with formula (\ref{eq:N-}) in section \ref{sc:MathPrelim}
when applied to the zero weight states of the ${\bf 78}$ in $E_6$.


\newpage

\begin{thebibliography}{99}

\bibitem{grs} F.~Gursey, P.~Ramond, and P.~Sikivie,
                             Phys. Lett. B{\bf 60} (1976) 177.

\bibitem{Froggatt_Nielsen}  C.D.~Froggatt and H.B.~Nielsen,
                             Nucl. Phys. {\bf B147} (1979) 277.


\bibitem{pat_Verma} Sai-Ping Li, R.V.~Moody, M.~Nicolescu, and J.~Patera,
                              J. Math. Phys. {\bf 27} (1986) 668.

\bibitem{pat_E6} In-Guy Koh, J.~Patera, and C.~Rousseau, 
                              J. Math. Phys. {\bf 25} (1984) 2863.

\bibitem{abI} G.~Anderson and T.~Bla\v{z}ek, 
                              J. Math. Phys. {\bf 41} (2000) 4808.

\bibitem{abII} G.~Anderson and T.~Bla\v{z}ek, 
                              J. Math. Phys. {\bf 41} (2000) 8170.

\bibitem{bky} M.~Bando, T.~Kugo, and K.~Yoshioka,
%                    hep-ph/0003220,
                     Prog. Theor. Phys. {\bf 104} (2000) 211. 

\bibitem{Slansky_PhysRep} R.~Slansky, Phys. Rep. {\bf 79} (1981) 1.


\bibitem{footnote_1} For roots and weights we use the notation where $\ol{x}\equiv -x$.

\bibitem{pat_CCO} R.~V.~Moody and J.~Patera,
                                J. Math. Phys. {\bf 25} (1984) 2838.

\bibitem{footnote_2} Some applications of the charge conjugation operators 
                     can be found in ref.\cite{abI}.


\bibitem{ab_models} G.~Anderson and T.~Bla\v{z}ek, in preparation.


\bibitem{footnote_3} It is not always possible to choose a single basis
                     for all of the states of degenerate weights in a representation 
                     such that lowering a particular state results
                     in a particluar state of lower weight (as opposed to a linear combination
                     of weights) for all possible lowerings.  The simplest example of 
                     this is the 27 dimensional representation of $SU(3)$.


\end{thebibliography}

\newpage




\protect
\begin{table}
\caption{ 
{\bf Bases in the dominant weight subspaces 
of the 1728-dimensional $(100001)$ irrep.} 
}
\label{t:paths_1728}
\begin{tabular}{|l|c||l|c|}
\hline
                                   &                &                                   &              \\
  Weight state                     &  Lowering path &  Weight state                     & Lowering path\\
                                   &                &                                   &              \\
\hline 						    						       
                                   &                &                                   &              \\
$\;\;\mid\!\! 000100_6     \rangle$&  6321          &$\;\;\mid\!\! 100000_{\: 6}\rangle$& 65324436321  \\
$\;\;\mid\!\! 000100_3     \rangle$&  3621          &$\;\;\mid\!\! 100000_{\: 7}\rangle$& 64534236321  \\
$\;\;\mid\!\! 000100_2     \rangle$&  2361          &$\;\;\mid\!\! 100000_{\: 8}\rangle$& 63214534236  \\
$\;\;\mid\!\! 000100_1     \rangle$&  1236          &$\;\;\mid\!\! 100000_{\: 9}\rangle$& 63243654321  \\
                                   &                &$\;\;\mid\!\! 100000_{10}  \rangle$& 53624436321  \\
 \cline{1-2}                       &                &$\;\;\mid\!\! 100000_{11}  \rangle$& 54321634236  \\
$\;\;\mid\!\! 100000_{\: 1}\rangle$& 12364534236    &$\;\;\mid\!\! 100000_{12}  \rangle$& 32164534236  \\
$\;\;\mid\!\! 100000_{\: 2}\rangle$& 21364534236    &$\;\;\mid\!\! 100000_{13}  \rangle$& 32643654321  \\
$\;\;\mid\!\! 100000_{\: 3}\rangle$& 25364436321    &$\;\;\mid\!\! 100000_{14}  \rangle$& 32643254361  \\
$\;\;\mid\!\! 100000_{\: 4}\rangle$& 24534636321    &$\;\;\mid\!\! 100000_{15}  \rangle$& 45346236321  \\
$\;\;\mid\!\! 100000_{\: 5}\rangle$& 23643254361    &$\;\;\mid\!\! 100000_{16}  \rangle$& 45321634236  \\
                                   &                &                                   &              \\
\hline					                    
\end{tabular}				                    
\end{table}				                    



\protect
\begin{table}
\caption{ 
{\bf Bases in the dominant weight subspaces 
of the $(100100)$ irrep, the ${\bf \ol{7371}}$.} 
$\mid\!\! 000010_n \rangle$ states are marked $\mid\!\! \ol{F}_n \rangle$ for brevity.
}
\label{t:paths_7371bar}
\small
\begin{tabular}{|l|c||l|c||l|c|}
\hline
                            &              & \multicolumn{4}{c|}{ }                                                    \\
 Weight state               & Lowering path& \multicolumn{4}{c|}{Lowering paths to $(000010)$ weight states }          \\
                            &              & \multicolumn{4}{c|}{ }                                                    \\
\hline 
                            &              &                             &                  &                             &                  \\
$\gapA 000011_4     \rangle$&  4321        & $\gapA \ol{F}_{\: 1}\rangle$& 514362236434321  & $\gapA \ol{F}_{23}  \rangle$& 645342136234321  \\
$\gapA 000011_3     \rangle$&  3421        & $\gapA \ol{F}_{\: 2}\rangle$& 563214436234321  & $\gapA \ol{F}_{24}  \rangle$& 643452136234321  \\
$\gapA 000011_2     \rangle$&  2341        & $\gapA \ol{F}_{\: 3}\rangle$& 536214436234321  & $\gapA \ol{F}_{25}  \rangle$& 643621345234321  \\
$\gapA 000011_1     \rangle$&  1234        & $\gapA \ol{F}_{\: 4}\rangle$& 523614436234321  & $\gapA \ol{F}_{26}  \rangle$& 636231245434321  \\
                            &              & $\gapA \ol{F}_{\: 5}\rangle$& 145362236434321  & $\gapA \ol{F}_{27}  \rangle$& 633221143645234  \\
 \cline{1-2}                &              & $\gapA \ol{F}_{\: 6}\rangle$& 146234536234321  & $\gapA \ol{F}_{28}  \rangle$& 632145364234321  \\
$\gapA 200000_6     \rangle$& 6345234      & $\gapA \ol{F}_{\: 7}\rangle$& 143621236345234  & $\gapA \ol{F}_{29}  \rangle$& 344523126634321  \\
$\gapA 200000_3     \rangle$& 3645234      & $\gapA \ol{F}_{\: 8}\rangle$& 143622336435421  & $\gapA \ol{F}_{30}  \rangle$& 343221166345234  \\
$\gapA 200000_4     \rangle$& 4365234      & $\gapA \ol{F}_{\: 9}\rangle$& 162363245434321  & $\gapA \ol{F}_{31}  \rangle$& 345234126634321  \\
$\gapA 200000_5     \rangle$& 5436234      & $\gapA \ol{F}_{10}  \rangle$& 162332143645234  & $\gapA \ol{F}_{32}  \rangle$& 345216321436234  \\
$\gapA 200000_2     \rangle$& 2364534      & $\gapA \ol{F}_{11}  \rangle$& 162332435644321  & $\gapA \ol{F}_{33}  \rangle$& 342231166345234  \\
                            &              & $\gapA \ol{F}_{12}  \rangle$& 134562236434321  & $\gapA \ol{F}_{34}  \rangle$& 342345126634321  \\
 \cline{1-2}                &              & $\gapA \ol{F}_{13}  \rangle$& 134632364523421  & $\gapA \ol{F}_{35}  \rangle$& 342312632645341  \\
$\gapA 010000_{\: 1}\rangle$& 23645341     & $\gapA \ol{F}_{14}  \rangle$& 134621236345234  & $\gapA \ol{F}_{36}  \rangle$& 342163423546321  \\
$\gapA 010000_{\: 2}\rangle$& 32645341     & $\gapA \ol{F}_{15}  \rangle$& 134622336435421  & $\gapA \ol{F}_{37}  \rangle$& 245134326634321  \\
$\gapA 010000_{\: 3}\rangle$& 31645234     & $\gapA \ol{F}_{16}  \rangle$& 121364236345234  & $\gapA \ol{F}_{38}  \rangle$& 241345326634321  \\
$\gapA 010000_{\: 4}\rangle$& 35644321     & $\gapA \ol{F}_{17}  \rangle$& 122334546634321  & $\gapA \ol{F}_{39}  \rangle$& 241332166345234  \\
$\gapA 010000_{\: 5}\rangle$& 36435421     & $\gapA \ol{F}_{18}  \rangle$& 123456321436234  & $\gapA \ol{F}_{40}  \rangle$& 213245346634321  \\
$\gapA 010000_{\: 6}\rangle$& 61345234     & $\gapA \ol{F}_{19}  \rangle$& 624134536234321  & $\gapA \ol{F}_{41}  \rangle$& 213243632645341  \\
$\gapA 010000_{\: 7}\rangle$& 65434321     & $\gapA \ol{F}_{20}  \rangle$& 621363245434321  & $\gapA \ol{F}_{42}  \rangle$& 213456321436234  \\
$\gapA 010000_{\: 8}\rangle$& 64534321     & $\gapA \ol{F}_{21}  \rangle$& 621332143645234  & $\gapA \ol{F}_{43}  \rangle$& 453423126634321  \\
$\gapA 010000_{\: 9}\rangle$& 63245341     & $\gapA \ol{F}_{22}  \rangle$& 621332435644321  & $\gapA \ol{F}_{44}  \rangle$& 432163423546321  \\
$\gapA 010000_{10}  \rangle$& 41365234     &                             &                  &                             &                  \\
$\gapA 010000_{11}  \rangle$& 43265341     &                             &                  &                             &                  \\
$\gapA 010000_{12}  \rangle$& 43546321     &                             &                  &                             &                  \\
$\gapA 010000_{13}  \rangle$& 51436234     &                             &                  &                             &                  \\
$\gapA 010000_{14}  \rangle$& 54326341     &                             &                  &                             &                  \\
$\gapA 010000_{15}  \rangle$& 12364534     &                             &                  &                             &                  \\
                            &              &                             &                  &                             &                  \\
\hline					                    
%\normalsize
\end{tabular}				                    
\end{table}				                    
\normalsize




\protect
\begin{table}
\caption{ 
{\bf Bases in the dominant weight subspaces 
of the $(100020)$ irrep, the ${\bf \ol{7722}}$.} 
(100100) weight is left out as trivial.
$\mid\!\! 000010_n \rangle$ states are marked $\mid\!\! \ol{F}_n \rangle$ for brevity.
}
\label{t:paths_7722bar}
\small
\begin{tabular}{|l|c||l|c||l|c|}
\hline
                            &              & \multicolumn{4}{c|}{ }                                                    \\
 Weight state               & Lowering path& \multicolumn{4}{c|}{Lowering paths to $(000010)$ weight states }          \\
                            &              & \multicolumn{4}{c|}{ }                                                    \\
\hline 
                            &              &                             &                  &                             &                  \\
$\gapA 000011_5     \rangle$& 54321        & $\gapA \ol{F}_{\: 1}\rangle$& 5632144362345321 & $\gapA \ol{F}_{21}  \rangle$& 3164213623452345 \\
$\gapA 000011_4     \rangle$& 45321        & $\gapA \ol{F}_{\: 2}\rangle$& 5362144362345321 & $\gapA \ol{F}_{22}  \rangle$& 3164223645345321 \\
$\gapA 000011_3     \rangle$& 34521        & $\gapA \ol{F}_{\: 3}\rangle$& 5236144362345321 & $\gapA \ol{F}_{23}  \rangle$& 3445231266345321 \\
$\gapA 000011_2     \rangle$& 23451        & $\gapA \ol{F}_{\: 4}\rangle$& 5123644362345321 & $\gapA \ol{F}_{24}  \rangle$& 3452341266345321 \\
$\gapA 000011_1     \rangle$& 12345        & $\gapA \ol{F}_{\: 5}\rangle$& 5432163452364321 & $\gapA \ol{F}_{25}  \rangle$& 3452163452364321 \\
                            &              & $\gapA \ol{F}_{\: 6}\rangle$& 6421345362345321 & $\gapA \ol{F}_{26}  \rangle$& 3452163421362345 \\
 \cline{1-2}                &              & $\gapA \ol{F}_{\: 7}\rangle$& 6412345362345321 & $\gapA \ol{F}_{27}  \rangle$& 3422636345123451 \\
$\gapA 200000_6     \rangle$& 63452345     & $\gapA \ol{F}_{\: 8}\rangle$& 6453421362345321 & $\gapA \ol{F}_{28}  \rangle$& 3423451266345321 \\
$\gapA 200000_3     \rangle$& 36452345     & $\gapA \ol{F}_{\: 9}\rangle$& 6434521362345321 & $\gapA \ol{F}_{29}  \rangle$& 3423126633454521 \\
$\gapA 200000_4     \rangle$& 43652345     & $\gapA \ol{F}_{10}  \rangle$& 6212363343454521 & $\gapA \ol{F}_{30}  \rangle$& 2451343266345321 \\
$\gapA 200000_5     \rangle$& 54362345     & $\gapA \ol{F}_{11}  \rangle$& 6213632343454521 & $\gapA \ol{F}_{31}  \rangle$& 2411363623452345 \\
                            &              & $\gapA \ol{F}_{12}  \rangle$& 6213321436452345 & $\gapA \ol{F}_{32}  \rangle$& 2413453266345321 \\
 \cline{1-2}                &              & $\gapA \ol{F}_{13}  \rangle$& 6213324356454321 & $\gapA \ol{F}_{33}  \rangle$& 2413322663453451 \\
$\gapA 010000_{\: 1}\rangle$& 263453451    & $\gapA \ol{F}_{14}  \rangle$& 6136232343454521 & $\gapA \ol{F}_{34}  \rangle$& 2132453466345321 \\
$\gapA 010000_{\: 2}\rangle$& 163452345    & $\gapA \ol{F}_{15}  \rangle$& 6134522363454321 & $\gapA \ol{F}_{35}  \rangle$& 2136453421362345 \\
$\gapA 010000_{\: 3}\rangle$& 136452345    & $\gapA \ol{F}_{16}  \rangle$& 6134213623452345 & $\gapA \ol{F}_{36}  \rangle$& 1453622363454321 \\
$\gapA 010000_{\: 4}\rangle$& 143652345    & $\gapA \ol{F}_{17}  \rangle$& 6334542361234521 & $\gapA \ol{F}_{37}  \rangle$& 1423453266345321 \\
$\gapA 010000_{\: 5}\rangle$& 154362345    & $\gapA \ol{F}_{18}  \rangle$& 6321453623454321 & $\gapA \ol{F}_{38}  \rangle$& 1236453421362345 \\
$\gapA 010000_{\: 6}\rangle$& 653454321    & $\gapA \ol{F}_{19}  \rangle$& 3164362123452345 & $\gapA \ol{F}_{39}  \rangle$& 4534231266345321 \\
$\gapA 010000_{\: 7}\rangle$& 645345321    & $\gapA \ol{F}_{20}  \rangle$& 3164522363454321 & $\gapA \ol{F}_{40}  \rangle$& 4532163452364321 \\
$\gapA 010000_{\: 8}\rangle$& 633454521    &                             &                  &                             &                  \\
$\gapA 010000_{\: 9}\rangle$& 326453451    &                             &                  &                             &                  \\
$\gapA 010000_{10}  \rangle$& 356454321    &                             &                  &                             &                  \\
$\gapA 010000_{11}  \rangle$& 344655321    &                             &                  &                             &                  \\
$\gapA 010000_{12}  \rangle$& 455364321    &                             &                  &                             &                  \\
$\gapA 010000_{13}  \rangle$& 453263451    &                             &                  &                             &                  \\
$\gapA 010000_{14}  \rangle$& 543263451    &                             &                  &                             &                  \\
                            &              &                             &                  &                             &                  \\
\hline					                    
%\normalsize
\end{tabular}				                    
\end{table}				                    
\normalsize




\protect
\begin{table}
\caption{ 
{\bf Bases in the dominant weight subspaces 
of the $(110000)$ irrep, the ${\bf \ol{5824}}$.} 
}
\label{t:paths_5824bar}
\small
\begin{tabular}{|l|c||l|c||l|c|}
\hline
                            &              & \multicolumn{4}{c|}{ }                                              \\
 Weight state               & Lowering path& \multicolumn{4}{c|}{Lowering paths to zero weight states }          \\
                            &              & \multicolumn{4}{c|}{ }                                              \\
\hline 
                            &              &                        &                         &                        &                         \\
$\gapA 001000_2     \rangle$&  21          & $\gapA 0_{\: 1}\rangle$& 65241364345236342133221 & $\gapA 0_{33}  \rangle$& 51453643212233664433221 \\
$\gapA 001000_1     \rangle$&  12          & $\gapA 0_{\: 2}\rangle$& 65241363344521322364321 & $\gapA 0_{34}  \rangle$& 51436213452233664433221 \\
                            &              & $\gapA 0_{\: 3}\rangle$& 65142364345236342133221 & $\gapA 0_{35}  \rangle$& 51436213422334566321432 \\
 \cline{1-2}                &              & $\gapA 0_{\: 4}\rangle$& 65142363344521322364321 & $\gapA 0_{36}  \rangle$& 53623124536436214433221 \\
$\gapA 100010_1     \rangle$& 1236432      & $\gapA 0_{\: 5}\rangle$& 65362312453436214433221 & $\gapA 0_{37}  \rangle$& 53621453623436214433221 \\
$\gapA 100010_2     \rangle$& 2136432      & $\gapA 0_{\: 6}\rangle$& 65321453623436214433221 & $\gapA 0_{38}  \rangle$& 53621443632314521236432 \\
$\gapA 100010_3     \rangle$& 2364321      & $\gapA 0_{\: 7}\rangle$& 65321443632314521236432 & $\gapA 0_{39}  \rangle$& 53621443632314522364321 \\
$\gapA 100010_4     \rangle$& 6433221      & $\gapA 0_{\: 8}\rangle$& 65321443632314522364321 & $\gapA 0_{40}  \rangle$& 54433221166345322364321 \\
$\gapA 100010_5     \rangle$& 6321432      & $\gapA 0_{\: 9}\rangle$& 62451364345236342133221 & $\gapA 0_{41}  \rangle$& 24512345632133664433221 \\
$\gapA 100010_6     \rangle$& 4321632      & $\gapA 0_{10}  \rangle$& 62451363344521322364321 & $\gapA 0_{42}  \rangle$& 24513245632133664433221 \\
$\gapA 100010_7     \rangle$& 3216432      & $\gapA 0_{11}  \rangle$& 62113344223366554433221 & $\gapA 0_{43}  \rangle$& 24513246321334566321432 \\
$\gapA 100010_8     \rangle$& 3264321      & $\gapA 0_{12}  \rangle$& 62136324436321554433221 & $\gapA 0_{44}  \rangle$& 24513213466345321236432 \\
                            &              & $\gapA 0_{13}  \rangle$& 62133221443366554433221 & $\gapA 0_{45}  \rangle$& 24513213466345322364321 \\
 \cline{1-2}                &              & $\gapA 0_{14}  \rangle$& 62133214432635544321632 & $\gapA 0_{46}  \rangle$& 23312435454634216633221 \\
$\gapA 000001_{\: 1}\rangle$& 653214433221 & $\gapA 0_{15}  \rangle$& 64152364345236342133221 & $\gapA 0_{47}  \rangle$& 23245341166345321236432 \\
$\gapA 000001_{\: 2}\rangle$& 645342133221 & $\gapA 0_{16}  \rangle$& 64152363344521322364321 & $\gapA 0_{48}  \rangle$& 23245341166345322364321 \\
$\gapA 000001_{\: 3}\rangle$& 536214433221 & $\gapA 0_{17}  \rangle$& 64536421345236342133221 & $\gapA 0_{49}  \rangle$& 23611435422334566321432 \\
$\gapA 000001_{\: 4}\rangle$& 532144321632 & $\gapA 0_{18}  \rangle$& 64536213344521322364321 & $\gapA 0_{50}  \rangle$& 23612344321635544321632 \\
$\gapA 000001_{\: 5}\rangle$& 523614433221 & $\gapA 0_{19}  \rangle$& 64534532364312364232121 & $\gapA 0_{51}  \rangle$& 41534563212233664433221 \\
$\gapA 000001_{\: 6}\rangle$& 523144321632 & $\gapA 0_{20}  \rangle$& 61236324436321554433221 & $\gapA 0_{52}  \rangle$& 41534632122334566321432 \\
$\gapA 000001_{\: 7}\rangle$& 512364433221 & $\gapA 0_{21}  \rangle$& 61233221443366554433221 & $\gapA 0_{53}  \rangle$& 41536213452233664433221 \\
$\gapA 000001_{\: 8}\rangle$& 512344321632 & $\gapA 0_{22}  \rangle$& 61233214432635544321632 & $\gapA 0_{54}  \rangle$& 41536213422334566321432 \\
$\gapA 000001_{\: 9}\rangle$& 544362133221 & $\gapA 0_{23}  \rangle$& 63623124436321554433221 & $\gapA 0_{55}  \rangle$& 45334221166345321236432 \\
$\gapA 000001_{10}  \rangle$& 415321236432 & $\gapA 0_{24}  \rangle$& 63322114432635544321632 & $\gapA 0_{56}  \rangle$& 45334221166345322364321 \\
$\gapA 000001_{11}  \rangle$& 415322364321 & $\gapA 0_{25}  \rangle$& 52331266453436214433221 & $\gapA 0_{57}  \rangle$& 45345632364312364232121 \\
$\gapA 000001_{12}  \rangle$& 425132136432 & $\gapA 0_{26}  \rangle$& 52361236453436214433221 & $\gapA 0_{58}  \rangle$& 45346323465121322364321 \\
$\gapA 000001_{13}  \rangle$& 425132364321 & $\gapA 0_{27}  \rangle$& 52361234536436214433221 & $\gapA 0_{59}  \rangle$& 13456342122334566321432 \\
$\gapA 000001_{14}  \rangle$& 433654232121 & $\gapA 0_{28}  \rangle$& 52361453623436214433221 & $\gapA 0_{60}  \rangle$& 13456213452233664433221 \\
$\gapA 000001_{15}  \rangle$& 436542133221 & $\gapA 0_{29}  \rangle$& 52361443632314521236432 & $\gapA 0_{61}  \rangle$& 13456213422334566321432 \\
$\gapA 000001_{16}  \rangle$& 364354232121 & $\gapA 0_{30}  \rangle$& 52361443632314522364321 & $\gapA 0_{62}  \rangle$& 13456223645363214433221 \\
$\gapA 000001_{17}  \rangle$& 314521236432 & $\gapA 0_{31}  \rangle$& 51362362453436214433221 & $\gapA 0_{63}  \rangle$& 36231244332166554433221 \\
$\gapA 000001_{18}  \rangle$& 314522364321 & $\gapA 0_{32}  \rangle$& 51362324536436214433221 & $\gapA 0_{64}  \rangle$& 36231244321635544321632 \\
$\gapA 000001_{19}  \rangle$& 322436543121 &                        &                         &                        &                         \\
$\gapA 000001_{20}  \rangle$& 321432654321 &                        &                         &                        &                         \\
$\gapA 000001_{21}  \rangle$& 211342365432 &                        &                         &                        &                         \\
$\gapA 000001_{22}  \rangle$& 213243654321 &                        &                         &                        &                         \\
$\gapA 000001_{23}  \rangle$& 213645321432 &                        &                         &                        &                         \\
$\gapA 000001_{24}  \rangle$& 123645321432 &                        &                         &                        &                         \\
                            &              &                        &                         &                        &                         \\
\hline					                    
%\normalsize
\end{tabular}				                    
\end{table}				                    
\normalsize



\protect
\begin{table}
\caption{ 
{\bf Bases in the dominant weight subspaces 
of the 3003-dimensional $(300000)$ irrep.} 
(110000) weight is left out as trivial.
}
\label{t:paths_3003}
\begin{tabular}{|l|c||l|c|}
\hline
                                   &                 &                                   &                          \\
  Weight state                     &  Lowering path  &  Weight state                     &  Lowering path           \\
                                   &                 &                                   &                          \\
\hline 						    
                                   &                 &                                   &                          \\
$\;\;\mid\!\! 001000       \rangle$&  211            &$\;\;\mid\!\! 000000_{\: 1}\rangle$& 652413643453323643222111 \\
                                   &                 &$\;\;\mid\!\! 000000_{\: 2}\rangle$& 651423643453323643222111 \\
 \cline{1-2}                       &                 &$\;\;\mid\!\! 000000_{\: 3}\rangle$& 653623124536444333222111 \\
$\;\;\mid\!\! 100010_6     \rangle$&  64332211       &$\;\;\mid\!\! 000000_{\: 4}\rangle$& 653214436323145236432211 \\
$\;\;\mid\!\! 100010_3     \rangle$&  36432211       &$\;\;\mid\!\! 000000_{\: 5}\rangle$& 624513643453323643222111 \\
$\;\;\mid\!\! 100010_2     \rangle$&  23643211       &$\;\;\mid\!\! 000000_{\: 6}\rangle$& 621134342362365544332211 \\
$\;\;\mid\!\! 100010_1     \rangle$&  12364321       &$\;\;\mid\!\! 000000_{\: 7}\rangle$& 621332243643655443322111 \\
                                   &                 &$\;\;\mid\!\! 000000_{\: 8}\rangle$& 641523643453323643222111 \\
 \cline{1-2}                       &                 &$\;\;\mid\!\! 000000_{\: 9}\rangle$& 645364332345221236432111 \\
$\;\;\mid\!\! 000001_{\: 1}\rangle$&  6544333222111  &$\;\;\mid\!\! 000000_{10}  \rangle$& 612332243643655443322111 \\
$\;\;\mid\!\! 000001_{\: 2}\rangle$&  5364433222111  &$\;\;\mid\!\! 000000_{11}  \rangle$& 524536361236444333222111 \\
$\;\;\mid\!\! 000001_{\: 3}\rangle$&  5236443322111  &$\;\;\mid\!\! 000000_{12}  \rangle$& 524133214663452364332211 \\
$\;\;\mid\!\! 000001_{\: 4}\rangle$&  5123644332211  &$\;\;\mid\!\! 000000_{13}  \rangle$& 524133245434666333222111 \\
$\;\;\mid\!\! 000001_{\: 5}\rangle$&  4251364332211  &$\;\;\mid\!\! 000000_{14}  \rangle$& 513645236236444333222111 \\
$\;\;\mid\!\! 000001_{\: 6}\rangle$&  4152364332211  &$\;\;\mid\!\! 000000_{15}  \rangle$& 514362134523623644332211 \\
$\;\;\mid\!\! 000001_{\: 7}\rangle$&  4536433222111  &$\;\;\mid\!\! 000000_{16}  \rangle$& 536214436323145236432211 \\
$\;\;\mid\!\! 000001_{\: 8}\rangle$&  3145236432211  &$\;\;\mid\!\! 000000_{17}  \rangle$& 245133214663452364332211 \\
$\;\;\mid\!\! 000001_{\: 9}\rangle$&  3452364322111  &$\;\;\mid\!\! 000000_{18}  \rangle$& 245133245434666333222111 \\
$\;\;\mid\!\! 000001_{10}  \rangle$&  2345123643211  &$\;\;\mid\!\! 000000_{19}  \rangle$& 213216324364365544332211 \\
                                   &                 &$\;\;\mid\!\! 000000_{20}  \rangle$& 415345234234666333222111 \\
                                   &                 &$\;\;\mid\!\! 000000_{21}  \rangle$& 415362134523623644332211 \\
                                   &                 &$\;\;\mid\!\! 000000_{22}  \rangle$& 453342211663452364332211 \\
                                   &                 &$\;\;\mid\!\! 000000_{23}  \rangle$& 134562134523623644332211 \\
                                   &                 &$\;\;\mid\!\! 000000_{24}  \rangle$& 345634234653221236432111 \\
                                   &                 &                                   &                          \\
\hline					                    
\end{tabular}				                    
\end{table}				                    





%******************************************************************************************************%
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%                                                                                                      %
%                                     CGG  for   27 x 78                                               %
%                                                                                                      %
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%******************************************************************************************************%


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000100) dominant weight in (100000)$\otimes$(000001).}
Each entry should be divided by the respective number in the last row to keep
the states normalized to 1.
}
\label{t:78cgc000100}
\footnotesize
%\tiny
%                 1  2345  6
%\begin{tabular}{|c||cccc||c|}
\begin{tabular}{|@{\hspace{0.5mm}}c
                |c@{$\,$}c@{$\,$}c@{$\,$}c@{$\,$}
                |c@{$\,$}|}
\hline
\mbox{ } & \multicolumn{4}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\mbox{ } & \multicolumn{4}{ c|}{\normalsize \em (100001) } & {\normalsize \em (000100) } \\
\mbox{ } & \multicolumn{4}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\cline{2-6}
 & 
   $\,\mid$000100$_{6}\rangle$ & 
   $\,\mid$000100$_{3}\rangle$ & 
   $\,\mid$000100$_{2}\rangle$ & 
   $\,\mid$000100$_{1}\rangle$ & 
   $\,\mid$000100$        \rangle$ \\ 
\hline
%                         &  1      2      3      4      1           
%                         ----------------------------------------------------------
\AA00010\1\MM000001\ZZ    &  1  &      &      &      & $\;\,$1  \\
\AA00\1101\MM00100\1\ZZ   &  1  &   1  &      &      &      -1  \\
\AA0\11000\MM01\1100\ZZ   &     &   1  &   1  &      & $\;\,$1  \\
\AA\110000\MM1\10100\ZZ   &     &      &   1  &   1  &      -1  \\
\AA100000\MM\100100\ZZ    &     &      &      &   1  & $\;\,$1  \\
\hline
                          & $\sqrt{2}$ & $\sqrt{2}$ & $\sqrt{2}$ & $\sqrt{2}$ &$\sqrt{5}$ \\
\hline
\end{tabular}
\end{table}





\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (100000) dominant weight in (100000)$\otimes$(000001).}
The fundamental $(100000)$ irrep is marked as $F$, and its highest weight state as \AA F\ZZ\.
\AA n\ZZ\ is an abbreviation for \AA100000$_n$\ZZ. Numbering of the degenerate states is 
consistent with table \ref{t:paths_1728} and eqs.(\ref{eq:100000in351P}--\ref{eq:100000in351}).
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:78cgc100000}
%\small
\footnotesize
%\tiny
%                 12  3456789012345678  90123  4
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrr||rrrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                                                        r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{16}{|c|}{\mbox{ }} & 
      \multicolumn{ 5}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{16}{|c|}{\normalsize \em (100001) } &
      \multicolumn{ 5}{|c|}{\normalsize \em (000100) } &
                           {\normalsize \em F $\;$} \\
 & &  \multicolumn{16}{|c|}{\mbox{ }} & 
      \multicolumn{ 5}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-24}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
                                                            \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& 
$\!\!$\BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& 
$\!\!$\AA F\ZZ$\;$ \\ 
\hline
%                         &  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6*   2   3   4   5   6*         1 
%                          -----------------------------------------------------------------------------------------------------
\AA100000\ZZ&\NA          &$\sqrt{2}$&&&   &   &   &   &   &   &   &   &   &   &   &   &   &    &   &   &   &   &  $\sqrt{32}$\\
\AA100000\ZZ&\NB          &&$\sqrt{2}$&&   &   &   &   &   &   &   &   &   &   &   &   &   & $\sqrt{2}$&&&  &   & -$\sqrt{50}$\\
\AA100000\ZZ&\NC          &    &   &   &   &   &   &   &   &   &   &&$\sqrt{2}$&&  &   &   &  &$\sqrt{2}$&& &   &  $\sqrt{72}$\\
\AA100000\ZZ&\ND          &    &   &   &   &   &   &   &   &   &   &   &   &   &&&$\sqrt{2}$&   &&$\sqrt{2}$&&  & -$\sqrt{32}$\\
\AA100000\ZZ&\NE          &    &   &   &   &   &   &   &   &   &&$\sqrt{2}$&&  &   &   &   &    &   &&$\sqrt{2}$&& $\sqrt{ 8}\;$\\
\AA100000\ZZ&\NF          &    &   &   &   &   &   &&$\sqrt{2}$&&  &   &   &   &   &   &   &    &   &&&$\sqrt{2}$&-$\sqrt{18}$\\
%
\AA\110000\ZA2\10000\ZZ   & \p1&\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\mwo&   &   &   &   &   \mC   \\ 
\AA0\11000\ZA11\1000\ZZ   &    &\p1&   &   &\p1&   &   &   &   &   &   &\p1&   &\p1&   &   &\mwo&\m1&   &   &   &   \pC   \\
\AA00\1101\ZA101\10\1\ZZ  &    &   &   &   &   &   &\p1&\p1&   &   &   &\p1&   &\p1&   &\p1&    &\m1&\m1&   &\m1&   \mC   \\
\AA000\111\ZA1001\1\1\ZZ  &    &   &   &   &   &\p1&\p1&   &   &   &\p1&   &   &   &   &\p1&    &   &\m1&\m1&\p1&   \pC   \\ 
\AA00010\1\ZA100\101\ZZ   &    &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &\p1&   &    &   &\p1&   &\m1&   \pC   \\ 
\AA0000\11\ZA10001\1\ZZ   &    &   &   &   &   &\p1&   &   &\p1&   &\p1&   &   &   &   &   &    &   &   &\m1&\m1&   \mC   \\
\AA001\11\1\ZA10\11\11\ZZ &    &   &   &   &   &\p1&\p1&   &   &\p1&   &   &   &\p1&\p1&   &    &\p1&\p1&\p1&\p1&   \mC   \\ 
\AA0010\1\1\ZA10\1011\ZZ  &    &   &   &   &   &\p1&   &   &\p1&\p1&   &   &\p1&   &   &   &    &\m1&   &\p1&\m1&   \pC   \\
\AA01\1010\ZA1\110\10\ZZ  &    &   &\p1&   &\p1&   &   &   &   &\p1&   &   &   &\p1&   &   & \p1&\p1&   &\m1&   &   \pC   \\ 
\AA01\11\10\ZA1\11\110\ZZ &    &   &\p1&\p1&   &   &   &   &   &\p1&   &   &\p1&   &\p1&   &\mwo&\m1&\m1&\m1&   &   \mC   \\
\AA1\10010\ZA0100\10\ZZ   &    &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   & \p1&   &   &\p1&   &   \mC   \\ 
\AA010\100\ZA1\10100\ZZ   &    &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   & \p1&   &\m1&   &   &   \pC   \\
\AA1\101\10\ZA010\110\ZZ  &    &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &\mwo&   &\p1&\p1&   &   \pC   \\ 
\AA1\11\100\ZA01\1100\ZZ  &    &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   & \p1&\p1&\p1&   &   &   \mC   \\
\AA10\1001\ZA00100\1\ZZ   &    &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   &    &\p1&   &   &\p1&   \pC   \\
\AA10000\1\ZA000001\ZZ    &    &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &    &   &   &   &\p1&   \mC   \\
\hline			    
       &        & $\sqrt{3}$& \pB & \pB & \pB & \pB & \pB & \pB & \pB & \pB & \pB &
                                                      \pB & \pB & \pB & \pB & \pB & \pB &
                              $\sqrt{10}$& $\sqrt{10}$& $\sqrt{10}$& $\sqrt{10}$& $\sqrt{10}$&
                              $\sqrt{156}\;$  \\
\hline
\end{tabular}
\end{table}


%******************************************************************************************************%
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%                                                                                                      %
%                                     CGG  for   27 x 351                                              %
%                                                                                                      %
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%******************************************************************************************************%


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000011) dominant weight in (100000)$\otimes$(000100).}
Each entry should be divided by the respective number in the last row to keep
the states normalized to 1.
}
\label{t:351cgc000011}
\footnotesize
%\tiny
%                 1  2345  6
%\begin{tabular}{|c||cccc||c|}
\begin{tabular}{|@{\hspace{0.5mm}}c
                |c@{$\,$}c@{$\,$}c@{$\,$}c@{$\,$}
                |c@{$\,$}|}
\hline
\mbox{ } & \multicolumn{4}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\mbox{ } & \multicolumn{4}{ c|}{\normalsize \em (100100) } & {\normalsize \em (000011) } \\
\mbox{ } & \multicolumn{4}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\cline{2-6}
 & 
   $\,\mid$000011$_{4}\rangle$ & 
   $\,\mid$000011$_{3}\rangle$ & 
   $\,\mid$000011$_{2}\rangle$ & 
   $\,\mid$000011$_{1}\rangle$ & 
   $\,\mid$000011$        \rangle$ \\ 
\hline
%                         &  4      3      2      1      S           
%                         ----------------------------------------------------------
\AA000\111\MM000100\ZZ    &  1  &      &      &      & $\;\,$1  \\
\AA00\1101\MM001\110\ZZ   &  1  &   1  &      &      &      -1  \\
\AA0\11000\MM01\1011\ZZ   &     &   1  &   1  &      & $\;\,$1  \\
\AA\110000\MM1\10011\ZZ   &     &      &   1  &   1  &      -1  \\
\AA100000\MM\100011\ZZ    &     &      &      &   1  & $\;\,$1  \\
\hline
                          & $\sqrt{2}$ & $\sqrt{2}$ & $\sqrt{2}$ & $\sqrt{2}$ &$\sqrt{5}$ \\
\hline
\end{tabular}
\end{table}







\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (010000) dominant weight in (100000)$\otimes$(000100).}
The $(010000)$ irrep is marked as $\ol{G}$, and its highest weight state as $\mid \ol{G}\rangle$.
%% In the last column, $\ol{G}$ stands for the $(010000)$ irrep. 
\AA n\ZZ\ is an abbreviation for \AA010000$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_7371bar} and \ref{t:paths_1728}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351cgc010000}
%\small
\footnotesize
%\tiny
%                 12  345678901234567  8901  2
%\begin{tabular}{|cc||rrrrrrrrrrrrrrr||rrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                                                        r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{15}{|c|}{\mbox{ }} & 
      \multicolumn{ 4}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{15}{|c|}{\normalsize \em (100100) } &
      \multicolumn{ 4}{|c|}{\normalsize \em (000011) } &
                           {\normalsize \em $\ol{G}$} \\
 & &  \multicolumn{15}{|c|}{\mbox{ }} & 
      \multicolumn{ 4}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-22}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
                                                            \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& 
$\!\!$\BG 6\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ$\:$ & 
\AA $\ol{G}$ \ZZ \\ 
\hline
%                               1   2   3   4   5   6   7   8   9   0   1   2   3   4   5*  1   2   3   4*   1
%                          ---------------------------------------------------------------------------------------
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &             \\
\AA100000\ZA\110000$_2$\ZZ &     &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   5$\;\;$   \\
\AA100000\ZA\110000$_3$\ZZ &     &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &  -6$\;\;$   \\
\AA100000\ZA\110000$_4$\ZZ &     &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\sQ&   &   4$\;\;$   \\
\AA100000\ZA\110000$_5$\ZZ &     &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\sQ&  -2$\;\;$   \\
\AA100000\ZA\110000$_6$\ZZ &     &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   3$\;\;$   \\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &             \\
\AA\110000\ZA100000$_2$\ZZ &  \sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &  -5$\;\;$   \\
\AA\110000\ZA100000$_3$\ZZ &     &\sQ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   6$\;\;$   \\
\AA\110000\ZA100000$_4$\ZZ &     &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &   &   &   &\qQ&   &  -4$\;\;$   \\
\AA\110000\ZA100000$_5$\ZZ &     &   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &   &\qQ&   2$\;\;$   \\
\AA\110000\ZA100000$_6$\ZZ &     &   &   &   &   &\p1&   &   &\sQ&   &   &   &   &   &   &\qQ&   &   &   &  -3$\;\;$   \\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &             \\
\AA0\11000\ZA02\1000\ZZ    &  \p1&\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   $\sqrt{8}$\\
\AA00\1101\ZA011\10\1\ZZ   &     &\p1&   &   &\p1&   &   &\p1&\p1&   &\p1&   &   &   &   &\p1&\p1&\p1&   &  -$\sqrt{8}$\\
\AA000\111\ZA0101\1\1\ZZ   &     &   &   &   &   &   &\p1&\p1&   &   &\p1&   &   &\p1&   &\m1&   &\p1&\p1&   $\sqrt{8}$\\
\AA00010\1\ZA010\101\ZZ    &     &   &   &   &   &   &   &\p1&\p1&   &   &\p1&   &   &   &\p1&   &\m1&   &   $\sqrt{8}$\\
\AA0000\11\ZA01001\1\ZZ    &     &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &\p1&   &   &\p1&  -$\sqrt{8}$\\
\AA001\11\1\ZA01\11\11\ZZ  &     &   &   &\p1&\p1&   &\p1&\p1&   &   &   &\p1&   &   &   &\m1&\m1&\m1&\m1&  -$\sqrt{8}$\\
\AA0010\1\1\ZA01\1011\ZZ   &     &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &\p1&\p1&   &\m1&   $\sqrt{8}$\\
\AA01\1010\ZA0010\10\ZZ    &     &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &\m1&   &\p1&   $\sqrt{8}$\\
\AA01\11\10\ZA001\110\ZZ   &     &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&\p1&\p1&  -$\sqrt{8}$\\
\AA010\100\ZA000100\ZZ     &     &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   $\sqrt{8}$\\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &             \\
\hline			    
       &       & $\sqrt{3}$& \pB & \sQ & \pB & \pB & 
                  \sQ      & \pB & \pB & \pB & \sQ &
                  \pB      & \pB & \sQ & \pB & \sQ &
                              $\sqrt{10}$& $\sqrt{10}$& $\sqrt{10}$& $\sqrt{10}$&
                              10$\;$  \\
\hline
\end{tabular}
\end{table}


\clearpage



\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000010) dominant weight in (100000)$\otimes$(000100).}
\AA n\ZZ\ is an abbreviation for \AA000010$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_7371bar}, and \ref{t:paths_1728}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351cgc000010}
%\small
%\footnotesize
\tiny
%                 12  345678901234567890123456789012345  (33 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{33}{|c|}{\mbox{ }} \\
 & &  \multicolumn{33}{|c|}{\normalsize \em (100100) } \\
 & &  \multicolumn{33}{|c|}{\mbox{ }} \\
\cline{3-35}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ& \AA 33\ZZ$\:$ \\
\hline
%
% 0 0 0 0 1 0                1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3 
%                         ------------------------------------------------------------------------------------------------------------------------------------   
\AA100000\ZA\100010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%36
\AA100000\ZA\100010$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%34
\AA100000\ZA\100010$_4$\ZZ &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%24
\AA100000\ZA\100010$_5$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%40
\AA100000\ZA\100010$_6$\ZZ &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%20
%                                                                                                                                                                   .
\AA\100010\ZA100000$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%37
\AA\100010\ZA100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%29
\AA\100010\ZA100000$_4$\ZZ &   &   &   &   &\sQ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%13
\AA\100010\ZA100000$_5$\ZZ &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\% 2
\AA\100010\ZA100000$_6$\ZZ &   &   &   &   &   &   &   &   &\p1&   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%21
%                                                                                                                                                                   .
\AA\110000\ZA1\10010$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%35
\AA\110000\ZA1\10010$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\%33
\AA\110000\ZA1\10010$_4$\ZZ&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\%23
\AA\110000\ZA1\10010$_5$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%39
\AA\110000\ZA1\10010$_6$\ZZ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%19
%                                                                                                                                                                   .
\AA1\10010\ZA\110000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%38
\AA1\10010\ZA\110000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   \\%30
\AA1\10010\ZA\110000$_4$\ZZ&   &   &   &   &\sQ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%14
\AA1\10010\ZA\110000$_5$\ZZ&\sQ&   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\% 4
\AA1\10010\ZA\110000$_6$\ZZ&   &   &   &   &   &   &   &   &\p1&   &\sQ&   &   &   &   &   &   &   &   &\p1&   &\sQ&   &   &   &   &   &\sQ&   &   &   &   &   \\%22
%                                                                                                                                                                   .
\AA0\11000\ZA01\1010$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%32
\AA0\11000\ZA01\1010$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\%62
\AA0\11000\ZA01\1010$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   \\%46
\AA0\11000\ZA01\1010$_5$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   \\%61
\AA0\11000\ZA01\1010$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\%60
%                                                                                                                                                                   .
\AA01\1010\ZA0\11000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%31
\AA01\1010\ZA0\11000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%63
\AA01\1010\ZA0\11000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\sQ&   &   \\%59
\AA01\1010\ZA0\11000$_5$\ZZ&   &   &\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   \\%10
\AA01\1010\ZA0\11000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\sQ&   &   &   &\p1&   &   &   &   &   &   &   \\%45
%                                                                                                                                                                   .
\AA00\1101\ZA001\11\1$_2$\ZZ&  &   &   &   &   &\p1&\p1&   &   &\p1&   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%18
\AA00\1101\ZA001\11\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1\\%44
\AA00\1101\ZA001\11\1$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   \\%54
\AA00\1101\ZA001\11\1$_5$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   \\%50
\AA00\1101\ZA001\11\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   \\%58
%                                                                                                                                                                   .
\AA001\11\1\ZA00\1101$_2$\ZZ&  &   &   &   &   &\p1&   &\p1&\p1&   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%15
\AA001\11\1\ZA00\1101$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\%41
\AA001\11\1\ZA00\1101$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\sQ&\p1&   &\sQ&   &   \\%51
\AA001\11\1\ZA00\1101$_5$\ZZ&  &\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   \\% 6
\AA001\11\1\ZA00\1101$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   \\%55
%                                                                                                                                                                   .
\AA000\111\ZA00010\1$_2$\ZZ&   &   &   &   &   &\p1&\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%17
\AA000\111\ZA00010\1$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\%42
\AA000\111\ZA00010\1$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\sQ&   &   &   &   &   \\%52
\AA000\111\ZA00010\1$_5$\ZZ&   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\% 8
\AA000\111\ZA00010\1$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &\p1&   &   &   \\%56
%                                                                                                                                                                   .
\AA00010\1\ZA000\111$_2$\ZZ&   &   &   &   &   &\p1&   &\p1&   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%16
\AA00010\1\ZA000\111$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\%43
\AA00010\1\ZA000\111$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &\p1&   &   &\p1&   \\%53
\AA00010\1\ZA000\111$_5$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\%49
\AA00010\1\ZA000\111$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\%57
%*                                                                                                                                                                  .
\AA\1001\10\ZA100\120\ZZ   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\% 1
\AA1\101\10\ZA\110\120\ZZ  &\p1&   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\% 3
\AA01\11\10\ZA0\11\120\ZZ  &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\% 9
\AA0010\1\1\ZA00\1021\ZZ   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   \\% 5
\AA0000\11\ZA00002\1\ZZ    &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\% 7
\AA010\100\ZA0\10110\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%65
\AA1\11\100\ZA\11\1110\ZZ  &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\%12
\AA\101\100\ZA10\1110\ZZ   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%11
\AA10\1001\ZA\10101\1\ZZ   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   \\%28
\AA\11\1001\ZA1\1101\1\ZZ  &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\%27
\AA10000\1\ZA\100011\ZZ    &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\%26
\AA0\10001\ZA01001\1\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\%48
\AA\11000\1\ZA1\10011\ZZ   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\%25
\AA0\1100\1\ZA01\1011\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\%47
\AA00\1100\ZA001\110\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%64
\AA000\110\ZA000100\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\%66
\hline
           &               &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{8}$ &
                               \pB     &   \pB     &   \pB     &   \pB     &   \pB     &
                            $\sqrt{8}$ &$\sqrt{8}$ &   \pB     &   \pB     &   \pB     &
                            $\sqrt{3}$ &$\sqrt{3}$ &$\sqrt{5}$ &   \pB     &   \pB     &
                               \pB     &$\sqrt{8}$ &   \pB     &   \pB     &   \pB     &
                            $\sqrt{3}$ &$\sqrt{3}$ &$\sqrt{10}$&$\sqrt{3}$ &$\sqrt{3}$ &
                            $\sqrt{10}$&$\sqrt{5}$ &$\sqrt{3}$ \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000010) dominant weight in (100000)$\otimes$(000100).}
The fundamental $(100000)$ irrep is marked as $\ol{F}$, and its highest weight state as $\mid \ol{F}\rangle$.
\AA n\ZZ\ is an abbreviation for \AA000010$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_7371bar}, \ref{t:paths_1728}, 
and eqs.(\ref{eq:100000in351P}--\ref{eq:100000in351}).
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351cgc000010part2}
\tiny
%                 12  34567890123  4567890123456789  01234  5
%\begin{tabular}{|cc||rrrrrrrrrrr||rrrrrrrrrrrrrrrr||rrrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{11}{|c|}{\mbox{ }} & 
      \multicolumn{16}{|c|}{\mbox{ }} & 
      \multicolumn{ 5}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{11}{|c|}{\normalsize \em (100100) } &
      \multicolumn{16}{|c|}{\normalsize \em (000011) } &
      \multicolumn{ 5}{|c|}{\normalsize \em (010000) } &
                           {\normalsize \em $\ol{F}$} \\
 & &  \multicolumn{11}{|c|}{continued from table \ref{t:351cgc000010}} & 
      \multicolumn{16}{|c|}{\mbox{ }} & 
      \multicolumn{ 5}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-35}
 & & 
          \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ& \AA 37\ZZ& \AA 38\ZZ& \AA 39\ZZ& \AA 40\ZZ& \AA 41\ZZ& \AA 42\ZZ& \AA 43\ZZ& 
                                                            \AA 44\ZZ& 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
                                                            \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& 
$\!\!$\BG 6\ZZ& \BG 3\ZZ& \BG 2\ZZ& \BG 1\ZZ& \BG 4\ZZ$\:$ & 
      $\,\,\mid\!\!\ol{F}\,\rangle$ \\ 
\hline
%
%*                                                                                                                                                            .
%*                                                                                                                                                            .
%                            4   5   6   7   8   9   0   1   2   3   4*  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6*  1   2   3   4   5*  1
%                          -----------------------------------------------------------------------------------------------------------------------------------------
\AA100000\ZA\100010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\pD&\pE&   &\p1\\%36
\AA100000\ZA\100010$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &\pD&   &\mF&   &\mB\\%34
\AA100000\ZA\100010$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &\pD&   &   &\pD&   &\pB\\%24
\AA100000\ZA\100010$_5$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &\mB&   &\mB\\%40
\AA100000\ZA\100010$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pC&\pD&\p1\\%20
%		      
\AA\100010\ZA100000$_2$\ZZ &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\qQ&\sQ&   &   &\pD&\pC&   &\p1\\%37
\AA\100010\ZA100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&\qQ&\sQ&   &   &   &   &   &\pD&   &\mB&   &\mB\\%29
\AA\100010\ZA100000$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &\pD&\pB\\%13
\AA\100010\ZA100000$_5$\ZZ &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &   &   &   &   &\pB&   &\mB\\% 2
\AA\100010\ZA100000$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &\sQ&   &   &   &   &   &   &   &   &\pD&   &   &\p1&   &\p1\\%21
%		      
\AA\110000\ZA1\10010$_2$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   &\p1&\pE&   &\m1\\%35
\AA\110000\ZA1\10010$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\sQ&\sQ&\sQ&   &   &\mD&\mF&\mF&   &\pB\\%33
\AA\110000\ZA1\10010$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&   &\sQ&   &   &   &   &   &   &   &\mD&   &\pD&\pD&   &\mB\\%23
\AA\110000\ZA1\10010$_5$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\sQ&   &   &\mB&\mB&   &\pB\\%39
\AA\110000\ZA1\10010$_6$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &\sQ&\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &\pC&\pC&\mD&\m1\\%19
%		      
\AA1\10010\ZA\110000$_2$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\sQ&   &   &   &\m1&\pC&   &\m1\\%38
\AA1\10010\ZA\110000$_3$\ZZ&\p1&   &\sQ&   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&\qQ&\qQ&\qQ&\qQ&\qQ&\qQ&   &\mD&\mB&\mB&   &\pB\\%30
\AA1\10010\ZA\110000$_4$\ZZ&   &   &   &\sQ&\p1&   &   &   &   &   &\sQ&   &\qQ&\qQ&\qQ&\qQ&   &   &   &   &   &\sQ&   &   &   &   &\sQ&   &   &   &   &\mD&\mB\\%14
\AA1\10010\ZA\110000$_5$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &\qQ&   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &\qQ&   &   &\pB&\pB&   &\pB\\% 4
\AA1\10010\ZA\110000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&\qQ&\qQ&\qQ&   &   &   &   &   &   &   &\mD&   &\p1&\p1&   &\m1\\%22
%		      
\AA0\11000\ZA01\1010$_2$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &\qQ&   &   &\p1&\p1&\mD&   &\p1\\%32
\AA0\11000\ZA01\1010$_3$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&\qQ&\sQ&   &   &\mB&\mF&   &   &\mB\\%62
\AA0\11000\ZA01\1010$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&\qQ&   &   &   &   &\sQ&   &   &\pD&\pD&\pD&   &   &\pB\\%46
\AA0\11000\ZA01\1010$_5$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\sQ&   &\mB&\mB&   &   &\mB\\%61
\AA0\11000\ZA01\1010$_6$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\sQ&\qQ&   &   &   &   &   &   &   &\sQ&   &   &   &   &\pC&\pC&   &\pD&\p1\\%60
%		      
\AA01\1010\ZA0\11000$_2$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&   &   &   &\sQ&   &   &\m1&\m1&\mD&   &\p1\\%31
\AA01\1010\ZA0\11000$_3$\ZZ&\p1&\p1&   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&\qQ&\qQ&   &\pB&\mB&   &   &\mB\\%63
\AA01\1010\ZA0\11000$_4$\ZZ&\p1&   &   &\sQ&\p1&   &   &   &   &\sQ&   &   &\sQ&   &\qQ&\sQ&   &   &   &   &   &   &\sQ&\qQ&   &   &\sQ&   &   &   &   &\pD&\pB\\%59
\AA01\1010\ZA0\11000$_5$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &\sQ&   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   &\qQ&   &\pB&\pB&   &   &\mB\\%10
\AA01\1010\ZA0\11000$_6$\ZZ&   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &\sQ&   &   &   &   &\qQ&   &   &\pD&\p1&\p1&   &   &\p1\\%45
%		      
\AA00\1101\ZA001\11\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &\qQ&   &   &   &\qQ&   &   &   &   &   &   &\p1&\p1&   &\pD&\p1&\m1\\%18
\AA00\1101\ZA001\11\1$_3$\ZZ&  &\p1&   &   &   &\p1&   &\p1&   &   &   &   &   &   &\qQ&\qQ&   &\qQ&   &\qQ&   &   &   &\qQ&\qQ&\qQ&   &\mB&\mB&\pD&   &\mB&\pB\\%44
\AA00\1101\ZA001\11\1$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\qQ&\sQ&   &   &   &   &\sQ&   &   &   &\pD&   &   &\pD&\mB\\%54
\AA00\1101\ZA001\11\1$_5$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &\sQ&   &   &   &\sQ&   &   &   &   &\mB&\mB&   &   &\mB&\pB\\%50
\AA00\1101\ZA001\11\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\sQ&   &   &   &\sQ&   &   &   &\pC&\pC&   &   &\m1&\m1\\%58
%		      
\AA001\11\1\ZA00\1101$_2$\ZZ&   &   &  &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\sQ&   &   &   &\sQ&\sQ&   &   &   &   &   &\m1&\m1&   &\pD&\m1&\m1\\%15
\AA001\11\1\ZA00\1101$_3$\ZZ&\p1&\p1&  &   &\p1&   &   &\p1&   &   &   &   &   &   &\sQ&\sQ&   &\sQ&   &\sQ&   &   &   &\sQ&\sQ&\sQ&\sQ&\pB&\pB&\pD&   &\pB&\pB\\%41
\AA001\11\1\ZA00\1101$_4$\ZZ&\p1&   &  &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &\sQ&\qQ&   &   &\sQ&\qQ&   &   &   &   &   &   &   &\mD&\mB\\%51
\AA001\11\1\ZA00\1101$_5$\ZZ&   &   &  &   &   &   &   &   &   &\sQ&   &\qQ&\qQ&   &   &   &   &   &\qQ&   &   &   &\qQ&   &   &   &   &\pB&\pB&   &   &\pB&\pB\\% 6
\AA001\11\1\ZA00\1101$_6$\ZZ&   &  &\sQ&   &   &   &   &   &   &   &\sQ&   &   &   &   &\qQ&   &   &   &\qQ&   &   &   &   &\qQ&   &   &\mC&\p1&   &   &\p1&\m1\\%55
%		      
\AA000\111\ZA00010\1$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &\sQ&   &   &   &   &   &   &   &   &   &   &\m1&   &   &\mD&\p1&\p1\\%17
\AA000\111\ZA00010\1$_3$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\qQ&\qQ&   &\sQ&   &\sQ&   &   &   &   &   &   &   &\pB&   &\mD&   &\mB&\mB\\%42
\AA000\111\ZA00010\1$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\sQ&\qQ&   &   &   &   &   &   &   &   &   &   &   &\pD&\pB\\%52
\AA000\111\ZA00010\1$_5$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&   &   &   &   &   &\qQ&   &   &   &   &   &   &   &   &\pB&   &   &   &\mB&\mB\\% 8
\AA000\111\ZA00010\1$_6$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\qQ&   &   &   &\qQ&\sQ&   &   &\mC&\mD&   &   &\m1&\p1\\%56
%		      
\AA00010\1\ZA000\111$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\qQ&   &   &   &   &\qQ&   &   &   &   &   &\p1&   &   &\mD&\m1&\p1\\%16
\AA00010\1\ZA000\111$_3$\ZZ&   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &\sQ&\sQ&   &\qQ&   &\qQ&   &   &   &   &   &   &\qQ&\mB&   &\mD&   &\pB&\mB\\%43
\AA00010\1\ZA000\111$_4$\ZZ&\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &\qQ&\sQ&   &   &\qQ&\sQ&\qQ&   &   &   &\mD&   &   &\mD&\pB\\%53
\AA00010\1\ZA000\111$_5$\ZZ&   &   &   &   &   &   &   &   &   &\sQ&   &   &\qQ&   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &\mB&   &   &   &\pB&\mB\\%49
\AA00010\1\ZA000\111$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\qQ&   &   &   &\sQ&   &   &   &   &   &   &   &\pC&   &   &   &\p1&\p1\\%57
%*		      
\AA\1001\10\ZA100\120\ZZ   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\qW&\qW&\sQ\\% 1
\AA1\101\10\ZA\110\120\ZZ  &   &   &   &\p1&   &   &   &   &   &   &   &\m1&\m1&\p1&\p1&   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &\qW&\qW&\sW&\qQ\\% 3
\AA01\11\10\ZA0\11\120\ZZ  &   &   &   &\p1&   &   &   &   &   &\p1&   &\p1&\p1&   &\p1&   &   &   &   &   &   &   &\p1&\p1&   &   &\p1&   &\qW&\qW&   &\qW&\sQ\\% 9
\AA0010\1\1\ZA00\1021\ZZ   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &\p1&\p1&   &   &\p1&\p1&   &   &   &\qW&\qW&   &   &   &\qQ\\% 5
\AA0000\11\ZA00002\1\ZZ    &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &\qW&   &   &   &   &\sQ\\% 7
\AA010\100\ZA0\10110\ZZ    &   &   &   &\p1&   &   &\p1&   &   &\p1&   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &\sW&   &\qW&\qQ\\%65
\AA1\11\100\ZA\11\1110\ZZ  &   &   &   &\p1&   &   &\p1&   &   &   &   &   &\m1&\p1&\p1&   &   &   &   &   &\p1&\m1&\m1&\p1&   &\p1&\m1&   &\sW&\sW&\sW&\sW&\sQ\\%12
\AA\101\100\ZA10\1110\ZZ   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &\p1&\p1&   &\p1&   &   &   &   &\p1&\m1&\p1&   &\p1&   &   &   &\qW&   &\sW&\qW&\qQ\\%11
\AA10\1001\ZA\10101\1\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\m1&\p1&\p1&\p1&\m1&\p1&   &   &   &\sW&\sW&   &\qW&   &\qQ\\%28
\AA\11\1001\ZA1\1101\1\ZZ  &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&   &\p1&\p1&\p1&   &\p1&\p1&\p1&\qW&\qW&\qW&\qW&   &\sQ\\%27
\AA10000\1\ZA\100011\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\m1&\p1&   &   &   &   &   &   &   &\sW&   &   &\sW&   &\sQ\\%26
\AA0\10001\ZA01001\1\ZZ    &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&\p1&\sW&   &\qW&   &   &\qQ\\%48
\AA\11000\1\ZA1\10011\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&   &   &   &   &   &   &   &   &\qW&   &\sW&\sW&   &\qQ\\%25
\AA0\1100\1\ZA01\1011\ZZ   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &\sW&\sW&\sW&   &   &\sQ\\%47
\AA00\1100\ZA001\110\ZZ    &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &\sW&   &   &\sW&\qQ\\%64
\AA000\110\ZA000100\ZZ     &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sW&\sQ\\%66
%
\hline
           &               &                                       $\sqrt{8}$  &$\sqrt{6}$  &
                            $\sqrt{10}$ &$\sqrt{8}$  &   \pB      &   \pB      &$\sqrt{10}$ &
                            $\sqrt{6}$  &$\sqrt{5}$  &$\sqrt{8}$  &$\sqrt{10}$ &
                                                                                $\sqrt{15}$ &
                            $\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &
                            $\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &
                            $\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &$\sqrt{20}$ &
%
                            \multicolumn{ 5}{@{$\,$}c@{$\,$}|}{$\sqrt{200}$ each state} &
%                            $\sqrt{200}$&$\sqrt{200}$&$\sqrt{200}$&$\sqrt{200}$&$\sqrt{200}$&
%
                            $\sqrt{52}$ \\
\hline
\end{tabular}
\end{table}

\clearpage



%******************************************************************************************************%
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%                                                                                                      %
%                                     CGG  for   27 x 351^prime                                        %
%                                                                                                      %
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%******************************************************************************************************%


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000011) dominant weight in (100000)$\otimes$(000020).}
%                                                         27  x  351^prime
%
Each entry should be divided by the respective number in the last row to keep
the states normalized to 1.
}
\label{t:351primecgc000011}
\footnotesize
%\tiny
%                 1  23456  7
%\begin{tabular}{|c||ccccc||c|}
\begin{tabular}{|@{\hspace{0.5mm}}c
                |c@{$\,$}c@{$\,$}c@{$\,$}c@{$\,$}c@{$\,$}
                |c@{$\,$}|}
\hline
\mbox{ } & \multicolumn{5}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\mbox{ } & \multicolumn{5}{ c|}{\normalsize \em (100020) } & {\normalsize \em (000011) } \\
\mbox{ } & \multicolumn{5}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\cline{2-7}
 & 
   $\,\mid$000011$_{5}\rangle$ & 
   $\,\mid$000011$_{4}\rangle$ & 
   $\,\mid$000011$_{3}\rangle$ & 
   $\,\mid$000011$_{2}\rangle$ & 
   $\,\mid$000011$_{1}\rangle$ & 
   $\,\mid$000011$        \rangle$ \\ 
\hline
%                         &   5     4      3      2      1     HWS          
%                         -----------------------------------------------------------------
\AA0000\11\MM000020\ZZ    &        1    &     &      &      &      & -$\sqrt{2}$   \\
\AA000\111\MM000100\ZZ    & $\sqrt{2}$  &  1  &      &      &      & $\;\,$1       \\
\AA00\1101\MM001\110\ZZ   &             &  1  &   1  &      &      &      -1       \\
\AA0\11000\MM01\1011\ZZ   &             &     &   1  &   1  &      & $\;\,$1       \\
\AA\110000\MM1\10011\ZZ   &             &     &      &   1  &   1  &      -1       \\
\AA100000\MM\100011\ZZ    &             &     &      &      &   1  & $\;\,$1       \\
\hline
                          & $\sqrt{3}$ & $\sqrt{2}$ & $\sqrt{2}$ & $\sqrt{2}$ & $\sqrt{2}$ &$\sqrt{7}$ \\
\hline
\end{tabular}
\end{table}







\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (010000) dominant weight in (100000)$\otimes$(000020).}
\AA n\ZZ\ is an abbreviation for \AA010000$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_7722bar}, and \ref{t:paths_1728}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351primecgc010000}
%\small
\footnotesize
%\tiny
%                 12  34567890123456  7890 
%\begin{tabular}{|cc||rrrrrrrrrrrrrr||rrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                                                        r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{$\,$}r@{$\,$}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{14}{|c|}{\mbox{ }} & 
      \multicolumn{ 4}{|c|}{\mbox{ }}  \\
 & &  \multicolumn{14}{|c|}{\normalsize \em (100020) } &
      \multicolumn{ 4}{|c|}{\normalsize \em (000011) }  \\
 & &  \multicolumn{14}{|c|}{\mbox{ }} & 
      \multicolumn{ 4}{|c|}{\mbox{ }}  \\
\cline{3-20}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
                                                            \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& 
$\!\!$\BG 6\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ$\:$ \\
\hline
%                               1   2   3   4   5   6   7   8   9   0   1   2   3   4*  6   3   4   5*
%                          ---------------------------------------------------------------------------------------
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\110000$_3$\ZZ &     &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   \\
\AA100000\ZA\110000$_4$\ZZ &     &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   \\
\AA100000\ZA\110000$_5$\ZZ &     &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ\\
\AA100000\ZA\110000$_6$\ZZ &     &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   \\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA100000$_3$\ZZ &     &   &\p1&   &   &   &   &   &\sQ&   &   &   &   &   &   &\qQ&   &   \\
\AA\110000\ZA100000$_4$\ZZ &     &   &   &\p1&   &   &   &   &   &   &   &   &\sQ&   &   &   &\qQ&   \\
\AA\110000\ZA100000$_5$\ZZ &     &   &   &   &\p1&   &   &   &   &   &   &   &   &\sQ&   &   &   &\qQ\\
\AA\110000\ZA100000$_6$\ZZ &  \sQ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   \\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA02\1000\ZZ    &  \p1&   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &\pB&\p1&   &   \\
\AA00\1101\ZA011\10\1\ZZ   &     &   &   &   &   &   &\p1&\p1&\p1&   &   &   &\p1&   &\m1&\p1&\p1&   \\
\AA000\111\ZA0101\1\1\ZZ   &     &   &   &   &   &\p1&\p1&   &   &   &   &   &\p1&\p1&\p1&   &\p1&\p1\\
\AA00010\1\ZA010\101\ZZ    &     &   &   &   &   &   &\p1&\p1&   &   &\p1&   &   &   &\m1&\mB&\m1&   \\
\AA0000\11\ZA01001\1\ZZ    &     &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&\m1&   &   &\p1\\
\AA001\11\1\ZA01\11\11\ZZ  &     &   &   &   &   &\p1&\p1&   &   &\p1&\p1&   &   &   &\p1&\p1&\m1&\m1\\
\AA0010\1\1\ZA01\1011\ZZ   &     &   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &\m1&\m1&   &\m1\\
\AA01\1010\ZA0010\10\ZZ    &     &   &   &   &   &   &   &   &   &\p1&\p1&\p1&   &   &   &\p1&\pB&\p1\\
\AA01\11\10\ZA001\110\ZZ   &     &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &\m1&\m1&\p1\\
\AA010\100\ZA000100\ZZ     &     &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\m1&\mB\\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\hline			    
       &       & $\sqrt{3}$&  \sQ       &  \sQ       &  \sQ       &  \sQ & 
                  \pB      &  \pB       & $\sqrt{3}$ &  \pB       &  \sQ &
                 $\sqrt{3}$& $\sqrt{3}$ &  \pB       &  \pB       &
                             $\sqrt{14}$& $\sqrt{14}$& $\sqrt{14}$& $\sqrt{14}$ \\
\hline
\end{tabular}
\end{table}


\clearpage



\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000010) dominant weight in (100000)$\otimes$(000020).}
\AA n\ZZ\ is an abbreviation for \AA000010$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_7722bar}, and \ref{t:paths_1728}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351primecgc000010}
%\small
%\footnotesize
\tiny
%                 12  34567890123456789012345678901  (29 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{29}{|c|}{\mbox{ }} \\
 & &  \multicolumn{29}{|c|}{\normalsize \em (100020) } \\
 & &  \multicolumn{29}{|c|}{\mbox{ }} \\
\cline{3-31}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ$\:$ \\
\hline
%
% 0 0 0 0 1 0                1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   
%                         ------------------------------------------------------------------------------------------------------------------------
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100010$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA100000\ZA\100010$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100010$_5$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100010$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA\100010\ZA100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &\p1&   &   &\sQ&   &   &   &   \\
\AA\100010\ZA100000$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA100000$_5$\ZZ &   &   &   &\sQ&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA100000$_6$\ZZ &   &   &   &   &   &   &   &   &   &\p1&   &   &\sW&   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA\110000\ZA1\10010$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   \\
\AA\110000\ZA1\10010$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10010$_5$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10010$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA1\10010\ZA\110000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &\p1&   &\sQ&   &   &   &\p1&   \\
\AA1\10010\ZA\110000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\110000$_5$\ZZ&   &   &\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\110000$_6$\ZZ&   &   &   &   &   &   &   &   &   &\p1&\p1&   &\sQ&   &\sQ&   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA0\11000\ZA01\1010$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1\\
\AA0\11000\ZA01\1010$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1010$_5$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA0\11000\ZA01\1010$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA01\1010\ZA0\11000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1\\
\AA01\1010\ZA0\11000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &\p1&   \\
\AA01\1010\ZA0\11000$_5$\ZZ&   &\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA01\1010\ZA0\11000$_6$\ZZ&   &   &   &   &   &   &   &   &   &\p1&\p1&   &\sQ&\p1&   &   &   &   &\p1&   &   &\pC&   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA00\1101\ZA001\11\1$_3$\ZZ&  &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1\\
\AA00\1101\ZA001\11\1$_4$\ZZ&  &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\11\1$_5$\ZZ&  &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA00\1101\ZA001\11\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA001\11\1\ZA00\1101$_3$\ZZ&  &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1\\
\AA001\11\1\ZA00\1101$_4$\ZZ&  &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\p1&\sQ&   &   &   &\p1&   \\
\AA001\11\1\ZA00\1101$_5$\ZZ&\sQ&\sQ&  &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA001\11\1\ZA00\1101$_6$\ZZ&  &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &\pC&   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA000\111\ZA00010\1$_3$\ZZ&   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA00010\1$_4$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA00010\1$_5$\ZZ&\sQ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA00010\1$_6$\ZZ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &\pB&   &   &\pB&   &   &   &   &   &   &\p1\\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                .
\AA00010\1\ZA000\111$_3$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\111$_4$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &\p1&   &\p1&\p1\\
\AA00010\1\ZA000\111$_5$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\111$_6$\ZZ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                                               .
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   
\AA\1001\10\ZA100\120\ZZ   &   &   &   &\p1&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\120\ZZ  &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\120\ZZ  &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
\AA0010\1\1\ZA00\1021\ZZ   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   \\
\AA0000\11\ZA00002\1\ZZ    &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10110\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1110\ZZ  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   \\
\AA\101\100\ZA10\1110\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   \\
\AA10\1001\ZA\10101\1\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &\p1&   &   &   &\p1&   &   &   &   &   \\
\AA\11\1001\ZA1\1101\1\ZZ  &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   \\
\AA10000\1\ZA\100011\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01001\1\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10011\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1011\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA00\1100\ZA001\110\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA000\110\ZA000100\ZZ     &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000020\ZZ     &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
          &                &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\hline
           &               &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{9}$ &
                               \pB     &   \pB     &   \pB     &   \pB     &$\sqrt{3}$ &
                               \pB     &   \pB     &   \pD     &$\sqrt{3}$ &$\sqrt{8}$ &
                               \pB     &$\sqrt{3}$ &$\sqrt{10}$&$\sqrt{8}$ &$\sqrt{8}$ &
                               \pB     &$\sqrt{24}$&$\sqrt{3}$ &$\sqrt{10}$&$\sqrt{6}$ &
                            $\sqrt{5}$ &$\sqrt{3}$ &$\sqrt{ 8}$&$\sqrt{6}$ \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000010) dominant weight in (100000)$\otimes$(000020).}
The fundamental $(100000)$ irrep is marked as $\ol{F}$, and its highest weight state as $\mid \ol{F}\rangle$.
\AA n\ZZ\ is an abbreviation for \AA000010$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_7722bar}, and \ref{t:paths_1728}.
%      
%
%   We may need to put a note how the paths of  1728bar  are  related to the  1728  paths.
%
%
%
%
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351primecgc000010part2}
\tiny
%                 12  34567890123  4567890123456789  0
%\begin{tabular}{|cc||rrrrrrrrrrr||rrrrrrrrrrrrrrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}
%                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
%                                r@{}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{11}{|c|}{\mbox{ }} & 
      \multicolumn{16}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{11}{|c|}{\normalsize \em (100020) } &
      \multicolumn{16}{|c|}{\normalsize \em (000011) } &
                           {\normalsize \em $\ol{F}$} \\
 & &  \multicolumn{11}{|c|}{continued from table \ref{t:351primecgc000010}} & 
      \multicolumn{16}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-30}
 & & 
          \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ& \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ& \AA 37\ZZ& \AA 38\ZZ& \AA 39\ZZ& \AA 40\ZZ& 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
                                                            \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& 
      $\,\,\mid\!\!\ol{F}\,\rangle$ \\ 
\hline
%
%*                                                                                                                                               .
%*                                                                                                                                               .
%                                0   1   2   3   4   5   6   7   8   9   0*  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6*  1   
%                          ----------------------------------------------------------------------------------------------------------------------
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
\AA100000\ZA\100010$_3$\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\sW&   &   &   &$ \sqrt{ 8}$\\
\AA100000\ZA\100010$_4$\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\sW&   &   &   &   &   &   &   &$-\sqrt{18}$\\
\AA100000\ZA\100010$_5$\ZZ     &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &$ \sqrt{32}$\\
\AA100000\ZA\100010$_6$\ZZ     &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\sQ&\sW&   &   &   &   &   &   &   &   &   &   &\sQ&   &$-\sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA\100010\ZA100000$_3$\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&\sQ&   &\sW&   &   &$ \sqrt{ 8}$\\
\AA\100010\ZA100000$_4$\ZZ     &   &   &   &   &   &   &\sQ&\p1&   &   &\sQ&   &\sQ&\sQ&   &\sW&   &   &   &   &   &\sQ&   &   &   &   &   &$-\sqrt{18}$\\
\AA\100010\ZA100000$_5$\ZZ     &   &   &   &   &   &   &   &   &\p1&   &   &\sQ&   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &   &$ \sqrt{32}$\\
\AA\100010\ZA100000$_6$\ZZ     &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &\sQ&\sW&\sQ&   &   &\sQ&   &   &   &\sQ&\sQ&$-\sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA\110000\ZA1\10010$_3$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\qQ&\sQ&\sQ&   &$-\sqrt{ 8}$\\
\AA\110000\ZA1\10010$_4$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&   &\qQ&   &   &   &   &   &   &   &$ \sqrt{18}$\\
\AA\110000\ZA1\10010$_5$\ZZ    &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\sQ&$-\sqrt{32}$\\
\AA\110000\ZA1\10010$_6$\ZZ    &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &\sQ&\qQ&\sQ&   &   &   &   &   &   &   &   &   &\qQ&   &$ \sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA1\10010\ZA\110000$_3$\ZZ    &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&\qQ&\sQ&\qQ&\sQ&\qQ&$-\sqrt{ 8}$\\
\AA1\10010\ZA\110000$_4$\ZZ    &\sQ&   &\p1&   &   &   &\sQ&\p1&   &   &   &   &\qQ&\sQ&\sQ&\qQ&   &   &   &   &   &\sQ&   &   &   &   &\sQ&$ \sqrt{18}$\\
\AA1\10010\ZA\110000$_5$\ZZ    &   &   &   &   &   &\p1&   &   &\p1&   &   &\qQ&   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &\qQ&$-\sqrt{32}$\\
\AA1\10010\ZA\110000$_6$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&\qQ&\sQ&   &\sQ&   &   &   &\qQ&   &$ \sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA0\11000\ZA01\1010$_3$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&\sQ&   &$ \sqrt{ 8}$\\
\AA0\11000\ZA01\1010$_4$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&\sQ&   &   &   &   &\sQ&   &   &$-\sqrt{18}$\\
\AA0\11000\ZA01\1010$_5$\ZZ    &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\sQ&$ \sqrt{32}$\\
\AA0\11000\ZA01\1010$_6$\ZZ    &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &\qW&\qQ&\qQ&   &   &   &   &\qQ&   &   &\qQ&   &\qQ&   &$-\sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA01\1010\ZA0\11000$_3$\ZZ    &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&\sQ&\sQ&\qQ&$ \sqrt{ 8}$\\
\AA01\1010\ZA0\11000$_4$\ZZ    &\sQ&   &\p1&   &   &   &   &   &   &\sQ&   &   &\sQ&   &\sQ&\sQ&   &   &   &   &   &   &\sQ&\sQ&   &   &\sQ&$-\sqrt{18}$\\
\AA01\1010\ZA0\11000$_5$\ZZ    &   &   &   &   &   &\p1&   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   &\qQ&$ \sqrt{32}$\\
\AA01\1010\ZA0\11000$_6$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qW&\qQ&   &\qQ&\qQ&\qQ&   &   &\qQ&\qQ&   &$-\sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA00\1101\ZA001\11\1$_3$\ZZ   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&   &\qQ&   &\sQ&   &   &   &\sQ&\qQ&\qQ&   &$-\sqrt{ 8}$\\
\AA00\1101\ZA001\11\1$_4$\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\qQ&\qQ&   &   &   &   &\sQ&   &   &$ \sqrt{18}$\\
\AA00\1101\ZA001\11\1$_5$\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &\sQ&   &   &   &\sQ&   &   &   &   &$-\sqrt{32}$\\
\AA00\1101\ZA001\11\1$_6$\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &\sQ&\qQ&   &   &\qQ&\qQ&   &   &\qQ&   &   &   &$ \sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA001\11\1\ZA00\1101$_3$\ZZ    &   &  &\p1&   &   &   &   &   &   &   &   &   &   &   &\qQ&\sQ&   &\sQ&   &\qQ&   &   &   &\qQ&\sQ&\qQ&\sQ&$-\sqrt{ 8}$\\
\AA001\11\1\ZA00\1101$_4$\ZZ    &   &  &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &\sQ&\sQ&   &   &\sQ&\sQ&   &   &   &$ \sqrt{18}$\\
\AA001\11\1\ZA00\1101$_5$\ZZ    &   &  &   &   &   &   &   &   &   &\sQ&   &\qQ&\qQ&   &   &   &   &   &\qQ&   &   &   &\qQ&   &   &   &   &$-\sqrt{32}$\\
\AA001\11\1\ZA00\1101$_6$\ZZ    &  &   &   &   &   &   &   &\p1&   &   &   &   &   &\qQ&   &\qQ&\sQ&   &   &\sQ&\qQ&\sQ&   &   &\qQ&   &   &$ \sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA000\111\ZA00010\1$_3$\ZZ    &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\sQ&\qQ&   &\sQ&   &\qQ&   &   &   &   &   &\sW&   &$ \sqrt{ 8}$\\
\AA000\111\ZA00010\1$_4$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &\sQ&\sQ&   &   &   &   &   &   &   &$-\sqrt{18}$\\
\AA000\111\ZA00010\1$_5$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ&   &   &   &   &   &\qQ&   &   &   &   &   &   &   &   &$ \sqrt{32}$\\
\AA000\111\ZA00010\1$_6$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &\sQ&\sQ&   &   &\sQ&\sW&   &   &\sQ&\sQ&   &   &$-\sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                .
\AA00010\1\ZA000\111$_3$\ZZ    &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &\qQ&\sQ&   &\qQ&   &\sQ&   &   &   &   &   &   &\qQ&$ \sqrt{ 8}$\\
\AA00010\1\ZA000\111$_4$\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &\qQ&\qQ&   &   &\qQ&\qQ&\qQ&   &   &$-\sqrt{18}$\\
\AA00010\1\ZA000\111$_5$\ZZ    &   &   &   &   &   &   &   &   &   &\sQ&   &   &\qQ&   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &$ \sqrt{32}$\\
\AA00010\1\ZA000\111$_6$\ZZ    &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &\qQ&   &\qQ&\qQ&   &   &\qQ&   &\qQ&   &   &   &   &   &$-\sqrt{ 2}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%*                                                                                                                                               .
%                                0   1   2   3   4   5   6   7   8   9   0*  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6*  1   
\AA\1001\10\ZA100\120\ZZ       &   &   &   &   &   &   &\p1&   &   &   &\p1&\p1&\p1&\m1&   &\m1&   &   &   &   &   &\p1&   &   &   &   &   &\mE\\
\AA1\101\10\ZA\110\120\ZZ      &\p1&   &   &   &   &   &\p1&   &   &   &   &\m1&\m1&\m1&\m1&   &   &   &   &   &   &\p1&   &   &   &   &\p1&\pE\\
\AA01\11\10\ZA0\11\120\ZZ      &\p1&   &   &   &   &   &   &   &   &\p1&   &\p1&\p1&   &\m1&   &   &   &   &   &   &   &\p1&\m1&   &   &\p1&\mE\\
\AA0010\1\1\ZA00\1021\ZZ       &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &\p1&\m1&   &   &\p1&\m1&   &   &   &\pE\\
\AA0000\11\ZA00002\1\ZZ        &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &   &\mE\\
\AA010\100\ZA0\10110\ZZ        &\p1&   &   &   &\p1&   &   &   &   &\p1&   &   &\p1&   &\m1&   &   &   &   &   &   &   &   &   &   &\m1&\m1&\pE\\
\AA1\11\100\ZA\11\1110\ZZ      &\p1&   &   &   &\p1&   &\p1&   &   &   &   &   &\m1&\m1&\m1&   &   &   &   &   &\m1&\m1&\m1&\m1&   &\m1&\m1&\mE\\
\AA\101\100\ZA10\1110\ZZ       &   &   &   &   &   &   &\p1&   &   &   &\p1&   &\p1&\m1&   &\m1&   &   &   &   &\m1&\m1&\p1&   &\m1&   &   &\pE\\
\AA10\1001\ZA\10101\1\ZZ       &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &\m1&\m1&\m1&\p1&\m1&\m1&   &   &   &\pE\\
\AA\11\1001\ZA1\1101\1\ZZ      &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&   &\m1&\p1&\p1&   &\m1&\m1&\p1&\mE\\
\AA10000\1\ZA\100011\ZZ        &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &\m1&\m1&   &\mB&   &   &   &   &   &\mE\\
\AA0\10001\ZA01001\1\ZZ        &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &\m1&\p1&\pE\\
\AA\11000\1\ZA1\10011\ZZ       &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&   &   &\mB&   &   &   &   &\mB&\pE\\
\AA0\1100\1\ZA01\1011\ZZ       &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &\mB&   &\m1&   &\mB&\mE\\
\AA00\1100\ZA001\110\ZZ        &   &   &   &   &   &   &   &   &   &   &\p1&   &\mB&   &   &\m1&   &   &   &   &   &   &\mB&   &\m1&   &   &\pE\\
\AA000\110\ZA000100\ZZ         &   &   &   &   &   &   &   &   &   &   &\p1&\mB&\mB&   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &\mE\\
\AA0000\10\ZA000020\ZZ         &   &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &$ \sqrt{50}$\\
          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                                                       .
\hline
           &               &                                                    $\sqrt{8}$  &
                            $\sqrt{ 3}$ &   \pB      &   \pB      &$\sqrt{10}$ &$\sqrt{ 5}$ &
                            $\sqrt{8}$  &   \pB      &$\sqrt{5}$  &$\sqrt{8}$  &$\sqrt{ 6}$ &
%
                            $\sqrt{21}$ &$\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &
                            $\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &
                            $\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &$\sqrt{28}$ &
                            $\sqrt{28}$ &
%
                            $\sqrt{650}$ \\
\hline
\end{tabular}
\end{table}

\clearpage



%******************************************************************************************************%
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%                                                                                                      %
%                                     CGG  for   27 x 351bar                                           %
%                                                                                                      %
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%******************************************************************************************************%


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (001000) dominant weight in (100000)$\otimes$(010000).}
%                                                         27  x       351bar
%
Each entry should be divided by the respective number in the last row to keep
the states normalized to 1.
}
\label{t:351barcgc001000}
\footnotesize
%\tiny
%                 1  23  4
%\begin{tabular}{|c||cc||c|}
\begin{tabular}{|@{\hspace{0.5mm}}c
                |c@{$\,$}c@{$\,$}
                |c@{$\,$}|}
\hline
\mbox{ } & \multicolumn{2}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\mbox{ } & \multicolumn{2}{ c|}{\normalsize \em (110000) } & {\normalsize \em (001000) } \\
\mbox{ } & \multicolumn{2}{|c|}{\mbox{ }}                  & {\mbox{ }}                  \\
\cline{2-4}
 & 
   $\,\mid$001000$_{1}\rangle$ & 
   $\,\mid$001000$_{2}\rangle$ & 
   $\,\mid$001000$        \rangle$ \\ 
\hline
%                         &        1         2        HWS          
%                         -----------------------------------------------------------------
\AA0\11000\MM010000\ZZ    &             &    1       &   $\:$1      \\
\AA\110000\MM1\11000\ZZ   &        1    &    1       &   -1         \\
\AA100000\MM\101000\ZZ    &        1    &            &   $\:$1      \\
\hline
                          & $\sqrt{2}$  & $\sqrt{2}$ &  $\sqrt{3}$     \\
\hline
\end{tabular}
\end{table}







\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (100010) dominant weight in (100000)$\otimes$(010000).}
\AA n\ZZ\ is an abbreviation for \AA100010$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}, and table II in \cite{abII}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc100010}
%\small
\footnotesize
%\tiny 
%                 12  34567890  1234  5
%\begin{tabular}{|cc||rrrrrrrr||rrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{ 8}{|c|}{\mbox{ }} & 
      \multicolumn{ 4}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{ 8}{|c|}{\normalsize \em (110000) } &
      \multicolumn{ 4}{|c|}{\normalsize \em (001000) } &
                           {\normalsize \em (100010) } \\
 & &  \multicolumn{ 8}{|c|}{\mbox{ }} & 
      \multicolumn{ 4}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-15}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& 
$\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ$\:$ & 
   $\,\mid$100010$\rangle\;\;$ \\ 
\hline
%                               1   2   3   4   5   6   7   8*  1   2   3   4*   1
%                          ---------------------------------------------------------------------------------------
          &                &     &   &   &   &   &   &   &   &   &   &   &   &                 \\
\AA100000\ZA000010$_1$\ZZ  &  \sQ&   &   &   &   &   &   &   &   &   &   &   &  -2$\;\;\;\;$   \\
\AA100000\ZA000010$_2$\ZZ  &     &\sQ&   &   &   &   &   &   &   &   &   &\sQ&   2$\;\;\;\;$   \\
\AA100000\ZA000010$_3$\ZZ  &     &   &   &   &   &   &\sQ&   &   &\sQ&   &   &  -2$\;\;\;\;$   \\
\AA100000\ZA000010$_4$\ZZ  &     &   &   &   &   &\sQ&   &   &   &   &\sQ&   &   1$\;\;\;\;$   \\
\AA100000\ZA000010$_6$\ZZ  &     &   &   &   &\sQ&   &   &   &\sQ&   &   &   &   1$\;\;\;\;$   \\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &                 \\
\AA\110000\ZA2\10010\ZZ    &  \p1&\p1&\p1&   &   &   &   &   &   &   &   &\m1&   $\sqrt{2}\;\;$\\
\AA0\11000\ZA11\1010\ZZ    &     &\p1&\p1&   &   &   &\p1&\p1&   &\m1&   &\m1&  -$\sqrt{2}\;\;$\\
\AA00\1101\ZA101\11\1\ZZ   &     &   &   &\p1&\p1&\p1&\p1&\p1&\m1&\m1&\m1&   &   $\sqrt{2}\;\;$\\
\AA000\111\ZA10010\1\ZZ    &     &   &   &\p1&   &\p1&   &   &\p1&   &\m1&   &  -$\sqrt{2}\;\;$\\
\AA00010\1\ZA100\111\ZZ    &     &   &   &\p1&\p1&   &   &   &\m1&   &\p1&   &  -$\sqrt{2}\;\;$\\
\AA001\11\1\ZA10\1101\ZZ   &     &   &   &\p1&   &   &   &\p1&\p1&\p1&\p1&   &   $\sqrt{2}\;\;$\\
\AA01\1010\ZA1\11000\ZZ    &     &   &\p1&   &   &   &   &\p1&   &\p1&   &\p1&  -$\sqrt{2}\;\;$\\
\AA1\10010\ZA010000\ZZ     &     &   &\p1&   &   &   &   &   &   &   &   &\p1&   $\sqrt{2}\;\;$\\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &                 \\
\hline			    
       &       & $\sqrt{3}$& \pB & \pB & \pB & \pB & 
                  \pB      & \pB & \pB & 
                              $\sqrt{6}$& $\sqrt{6}$& $\sqrt{6}$& $\sqrt{6}$&
                              $\sqrt{18}\;\;$  \\
\hline
\end{tabular}
\end{table}


\clearpage



\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000001) dominant weight states of the 5824-dimensional 
(110000) irrep in (100000)$\otimes$(010000).}
\AA n\ZZ\ is an abbreviation for \AA000001$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc000001}
%\small
%\footnotesize
\scriptsize
%\tiny
%                 12  345678901234567890123456  (24 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{24}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-26}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ$\:$ \\
\hline
%
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   
%                         ----------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.4cm}$\!\!$                                                                                                %.
\AA100000\ZA\100001$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA100000\ZA\100001$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   \\
\AA100000\ZA\100001$_3$\ZZ &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100001$_4$\ZZ &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100001$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ$\:$ \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA\110000\ZA1\10001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   \\
\AA\110000\ZA1\10001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   \\
\AA\110000\ZA1\10001$_3$\ZZ&   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10001$_4$\ZZ&   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10001$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\sQ$\:$ \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA0\11000\ZA01\1001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &\p1&\sQ&   &   \\
\AA0\11000\ZA01\1001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &   \\
\AA0\11000\ZA01\1001$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1001$_4$\ZZ&   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1001$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &\sQ&   \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA00\1101\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &   &   &\p1&\sQ&   &   &   &   &   &   \\
\AA00\1101\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &   &   &\p1&\sQ&   &\sQ&   &   \\
\AA00\1101\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\sQ&   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\100$_4$\ZZ&   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\100$_6$\ZZ&   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA000\111\ZA0001\10$_1$\ZZ&   &   &   &   &   &   &\sQ&\p1&   &\p1&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &\sQ$\:$ \\
\AA000\111\ZA0001\10$_2$\ZZ&   &   &   &   &\sQ&\p1&   &   &   &   &   &\p1&\sQ&   &   &   &   &   &   &   &   &   &\sQ&   \\
\AA000\111\ZA0001\10$_3$\ZZ&   &   &\sQ&\p1&   &   &   &   &   &   &   &   &   &\p1&\sQ&\sQ&   &   &   &\sQ&   &   &   &   \\
\AA000\111\ZA0001\10$_4$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\10$_6$\ZZ&\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA0000\11\ZA000010$_1$\ZZ &   &   &   &   &   &   &\sQ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA000010$_2$\ZZ &   &   &   &   &\sQ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA000010$_3$\ZZ &   &   &\sQ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA000010$_4$\ZZ &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA000010$_6$\ZZ &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                           .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   
\AA0010\1\1\ZA00\1012\ZZ   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA001\11\1\ZA00\11\12\ZZ  &\p1&\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\102\ZZ    &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\11\ZZ   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   \\
\AA1\10010\ZA\1100\11\ZZ   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1$\:$ \\
\AA01\11\10\ZA0\11\111\ZZ  &   &   &\p1&   &\p1&   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\p1&   &   &   &   \\
\AA010\100\ZA0\10101\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &\p1&   &   \\
\AA1\101\10\ZA\110\111\ZZ  &   &   &   &   &\p1&   &\p1&   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\11\ZZ    &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1$\:$ \\
\AA1\11\100\ZA\11\1101\ZZ  &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\p1&   &\p1&   &\p1&   &   \\
\AA\1001\10\ZA100\111\ZZ   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\101000\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   \\
\AA\101\100\ZA10\1101\ZZ   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA\11\1001\ZA1\11000\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   \\
\rule[-0.25cm]{0mm}{0.5cm}$\!\!$                                                                                               %.
\AA0\10001\ZA010000\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\hline
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &$\sqrt{6}$ &$\sqrt{6}$ &$\sqrt{8}$ &   \pB     &$\sqrt{8}$ &
                               \pB     &$\sqrt{8}$ &   \pB     &$\sqrt{3}$ &   \pB     &
                            $\sqrt{8}$ &   \pB     &$\sqrt{8}$ &$\sqrt{3}$ &$\sqrt{10}$&
                            $\sqrt{8}$ &   \pB     &$\sqrt{8}$ &$\sqrt{3}$ &$\sqrt{10}$&
                            $\sqrt{3}$ &$\sqrt{10}$&$\sqrt{8}$ &$\sqrt{8} \:$ \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000001) dominant weight in (100000)$\otimes$(010000).}
The adjoint $(000001)$ irrep is marked as $A$, and its highest weight state as $\mid \!A\rangle$.
\AA n\ZZ\ is an abbreviation for \AA000001$_n$\ZZ. Numbering of the degenerate
states is consistent with table II in \cite{abII}, and table I in \cite{abI}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc000001part2}
%\footnotesize
\scriptsize
%\tiny
%                 12  345678901234567  89012  3
%\begin{tabular}{|cc||rrrrrrrrrrrrrrr||rrrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
%                                r@{}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
% & &  \multicolumn{15}{|c|}{\mbox{ }} & 
%      \multicolumn{ 5}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{15}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (001000) } &
      \multicolumn{ 5}{|c|}{\normalsize \em (100010) } &
                           {\normalsize \em $A\;$} \\
% & &  \multicolumn{15}{|c|}{\mbox{ }} & 
%      \multicolumn{ 5}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-23}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
                                                            \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ&  
      $\,\,\mid\!\!A\,\rangle$ \\ 
\hline
%
%*                                                                                                                            .
%*                                                                                                                            .
%                                1   2   3   4   5   6   7   8   9   0   1   2   3   4   5*  1   2   3   4   5*  1   
%                          ------------------------------------------------------------------------------------------
%                                                                                                                            .
%          &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.4cm}$\!\!$                                                                                                %.
\AA100000\ZA\100001$_1$\ZZ     &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\mB&\mB&   &   &   &\mB\\
\AA100000\ZA\100001$_2$\ZZ     &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &\mB&   &   &\pD\\
\AA100000\ZA\100001$_3$\ZZ     &   &   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &\mB&   &   &\mB&   &\mF\\
\AA100000\ZA\100001$_4$\ZZ     &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &\p1&   &   &   &\mB&\pC\\
\AA100000\ZA\100001$_6$\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\p1&   &   &   &   &\pE\\
%
%         &                    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &            \\
%                                                                                                                 .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                 
\AA\110000\ZA1\10001$_1$\ZZ    &   &   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &\mB&   &   &   &   &\pB\\
\AA\110000\ZA1\10001$_2$\ZZ    &   &   &\sQ&   &\sQ&   &   &   &   &   &\sQ&   &   &   &   &\pB&\pB&\pB&   &   &\mD\\
\AA\110000\ZA1\10001$_3$\ZZ    &   &   &   &   &   &   &   &   &\sQ&   &   &\sQ&   &   &   &\mB&\mB&   &\pB&   &\pF\\
\AA\110000\ZA1\10001$_4$\ZZ    &   &   &   &   &   &\sQ&\sQ&   &   &   &   &   &   &   &   &\p1&\p1&   &   &\pB&\mC\\
\AA\110000\ZA1\10001$_6$\ZZ    &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\sQ&\p1&\p1&   &   &   &\mE\\
%                                                                                                                 .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                 
\AA0\11000\ZA01\1001$_1$\ZZ    &   &   &\qQ&   &   &   &   &   &   &   &\qQ&   &   &   &   &\pB&   &   &   &   &\mB\\
\AA0\11000\ZA01\1001$_2$\ZZ    &   &   &   &   &\qQ&   &   &   &   &   &\sQ&   &   &   &   &   &\pB&   &   &   &\pD\\
\AA0\11000\ZA01\1001$_3$\ZZ    &   &   &   &   &\sQ&   &   &   &\sQ&   &   &   &\sQ&\sQ&   &   &\mB&\mB&\mB&   &\mF\\
\AA0\11000\ZA01\1001$_4$\ZZ    &   &   &   &\sQ&   &\sQ&   &\sQ&   &   &   &   &   &   &   &   &\p1&\p1&   &\mB&\pC\\
\AA0\11000\ZA01\1001$_6$\ZZ    &   &\sQ&   &   &   &   &   &   &   &\sQ&   &   &   &   &   &   &\p1&\p1&   &   &\pE\\
%                                                                                                                 .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                 
\AA00\1101\ZA001\100$_1$\ZZ    &   &   &\qQ&   &   &   &   &   &   &   &   &\qQ&   &   &   &\mB&   &   &   &   &\pB\\
\AA00\1101\ZA001\100$_2$\ZZ    &   &   &   &   &\qQ&   &   &   &\qQ&   &\qQ&   &   &   &   &   &\mB&   &   &   &\mD\\
\AA00\1101\ZA001\100$_3$\ZZ    &   &   &   &   &\sQ&   &   &   &   &   &   &   &\qQ&\qQ&   &   &   &\mB&   &   &\pF\\
\AA00\1101\ZA001\100$_4$\ZZ    &   &   &   &\sQ&   &   &   &\qQ&   &   &   &   &   &\sQ&   &   &   &\p1&\p1&\pB&\mC\\
\AA00\1101\ZA001\100$_6$\ZZ    &\sQ&\sQ&   &   &   &   &   &   &   &   &   &   &\sQ&   &   &   &   &\p1&\p1&   &\mE\\
%                                                                                                                 .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                 
\AA000\111\ZA0001\10$_1$\ZZ    &   &   &   &   &   &   &\qQ&   &   &   &   &\qQ&   &   &\qQ&\pB&   &   &   &   &\mB\\
\AA000\111\ZA0001\10$_2$\ZZ    &   &   &   &   &   &\qQ&   &   &\qQ&\qQ&   &   &   &   &   &   &\pB&   &   &   &\pD\\
\AA000\111\ZA0001\10$_3$\ZZ    &   &\qQ&   &\qQ&\qQ&   &   &\qQ&   &   &   &   &\qQ&\qQ&   &   &   &\pB&   &   &\mF\\
\AA000\111\ZA0001\10$_4$\ZZ    &   &   &   &   &   &   &   &\sQ&   &   &   &   &   &\sQ&   &   &   &   &\p1&\m1&\pC\\
\AA000\111\ZA0001\10$_6$\ZZ    &\qQ&   &   &   &   &   &   &\sQ&   &   &   &   &\sQ&   &   &   &   &   &\p1&\p1&\pE\\
%                                                                                                                 .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                 
\AA0000\11\ZA000010$_1$\ZZ     &   &   &   &   &   &   &\qQ&   &   &   &   &   &   &   &\sQ&\mB&   &   &   &   &\pB\\
\AA0000\11\ZA000010$_2$\ZZ     &   &   &   &   &   &\qQ&   &   &   &\sQ&   &   &   &   &   &   &\mB&   &   &   &\mD\\
\AA0000\11\ZA000010$_3$\ZZ     &   &\sQ&   &\qQ&   &   &   &\qQ&   &   &   &   &   &   &   &   &   &\mB&   &   &\pF\\
\AA0000\11\ZA000010$_4$\ZZ     &   &   &   &   &   &   &   &\sQ&   &   &   &   &\sQ&\qQ&   &   &   &   &\mB&\m1&\mC\\
\AA0000\11\ZA000010$_6$\ZZ     &\sQ&   &   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   &   &   &\p1&\mE\\
%*                                                                                                                .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                               %.
%                                1   2   3   4   5   6   7   8   9   0   1   2   3   4   5*  1   2   3   4   5*  1   
\AA0010\1\1\ZA00\1012\ZZ       &\p1&\p1&   &\p1&\p1&   &   &\m1&   &   &   &   &   &   &   &   &   &\sQ&   &\qQ&\sW\\
\AA001\11\1\ZA00\11\12\ZZ      &\m1&\m1&   &\p1&   &   &   &\m1&   &   &   &   &\m1&\p1&   &   &   &\qQ&\qQ&\qQ&\qW\\
\AA00010\1\ZA000\102\ZZ        &\p1&   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &   &   &   &\qQ&   &\sW\\
\AA01\1010\ZA0\110\11\ZZ       &   &\m1&   &\p1&   &\p1&   &\p1&   &\m1&   &   &   &   &   &   &\qQ&\qQ&   &\sQ&\sW\\
\AA1\10010\ZA\1100\11\ZZ       &   &   &   &   &   &\p1&\p1&   &   &\m1&   &   &   &   &\m1&\qQ&\qQ&   &   &\qQ&\qW\\
\AA01\11\10\ZA0\11\111\ZZ      &   &\p1&   &\p1&\p1&\p1&   &\p1&\p1&\p1&   &   &\p1&\p1&   &   &\sQ&\sQ&\sQ&\sQ&\qW\\
\AA010\100\ZA0\10101\ZZ        &   &   &   &   &   &   &   &   &\p1&   &\p1&   &\p1&\p1&   &   &\qQ&   &\sQ&   &\sW\\
\AA1\101\10\ZA\110\111\ZZ      &   &   &   &   &   &\p1&\p1&   &\p1&\p1&   &\p1&   &   &\p1&\sQ&\sQ&   &\qQ&\qQ&\sW\\
\AA\100010\ZA1000\11\ZZ        &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\m1&\qQ&   &   &   &\sQ&\sW\\
\AA1\11\100\ZA\11\1101\ZZ      &   &   &\p1&   &\p1&   &   &   &\p1&   &\p1&\p1&   &   &   &\qQ&\qQ&\qQ&\qQ&   &\qW\\
\AA\1001\10\ZA100\111\ZZ       &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &\p1&\sQ&   &   &\sQ&\sQ&\qW\\
\AA10\1001\ZA\101000\ZZ        &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &\sQ&   &\qQ&   &   &\sW\\
\AA\101\100\ZA10\1101\ZZ       &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &\qQ&   &\sQ&\sQ&   &\sW\\
\AA\11\1001\ZA1\11000\ZZ       &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &\sQ&\sQ&\sQ&   &   &\qW\\
\rule[-0.25cm]{0mm}{0.5cm}$\!\!$                                                                                               %.
\AA0\10001\ZA010000\ZZ         &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\sQ&   &   &   &\sW\\
%                                                                                                                             .
\hline
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &    \pC     &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &
                            $\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &
                            $\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &
%
                                \pF     &    \pF     &    \pF     &    \pF     &    \pF     &
%
                            $\sqrt{180}$ \\
\hline
\end{tabular}
\end{table}

\clearpage





\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the first 32 (000000) dominant weight states of the 5824-dimensional
(110000) irrep in the product (100000)$\otimes$(010000).} (The remaining 32 states of this
irrep with the same weight are shown in table \ref{t:351barcgc000000part2}.)
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row of the table to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc000000part1}
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ$\:$ \\
\hline
%
%aaa                         1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2 
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                              %.
\AA100000\ZA\100000$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\110000\ZA1\10000$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA\110000\ZA1\10000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\11000\ZA01\1000$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA0\11000\ZA01\1000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1101\ZA001\10\1$_1$\ZZ&  &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_3$\ZZ&  &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\111\ZA0001\1\1$_1$\ZZ&  &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_2$\ZZ&  &   &   &   &\p1&   &\pE&\pB&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_3$\ZZ&\p1&  &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   & 11&\pE&   &   \\
\AA000\111\ZA0001\1\1$_4$\ZZ&  &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_6$\ZZ&  &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00010\1\ZA000\101$_1$\ZZ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_3$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\11\ZA00001\1$_1$\ZZ&   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_2$\ZZ&   &   &   &   &\p1&   &\pE&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_3$\ZZ&\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   & 11&\pE&   &   \\
\AA0000\11\ZA00001\1$_4$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_6$\ZZ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA001\11\1\ZA00\11\11$_1$\ZZ& &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_2$\ZZ& &   &   &   &\p1&   &\pE&\pB&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_3$\ZZ&\p1& &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\p1& 10&\pE&   &   \\
\AA001\11\1\ZA00\11\11$_4$\ZZ& &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_6$\ZZ& &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&  &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&  &   &   &   &\p1&   &\pE&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_3$\ZZ&\p1&  &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\p1& 10&\pE&   &   \\
\AA0010\1\1\ZA00\1011$_4$\ZZ&  &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&  &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\1010\ZA0\110\10$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   \\
\AA01\1010\ZA0\110\10$_2$\ZZ&\p1&\p1&  &   &\p1&   &\pE&\pC&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   & 10&\pE&   &   \\
\AA01\1010\ZA0\110\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&   &   &   &   \\
\AA01\1010\ZA0\110\10$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA01\1010\ZA0\110\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\11\10\ZA0\11\110$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   \\
\AA01\11\10\ZA0\11\110$_2$\ZZ&\p1&\p1& &   &\p1&   &\pE&\pC&\p1&\p1&   &\p1&   &   &   &   &\p1&\p1&   &   &   &   &\p1&   &   &   &   &   & 10&\pE&   &   \\
\AA01\11\10\ZA0\11\110$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&   &   &   &   \\
\AA01\11\10\ZA0\11\110$_4$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA01\11\10\ZA0\11\110$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\10010\ZA\1100\10$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   \\
\AA1\10010\ZA\1100\10$_2$\ZZ&\p1&\p1&\p1&\p1&  &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   & 10&\pE&   &   \\
\AA1\10010\ZA\1100\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&   &   &\p1&\p1\\
\AA1\10010\ZA\1100\10$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA1\10010\ZA\1100\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA010\100\ZA0\10100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_2$\ZZ&   &   &   &   &   &   &   &   &\p1&\p1&   &   &\p1&\p1&   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA010\100\ZA0\10100$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\hline
%\multicolumn{34}{|l|}{$\qquad$ \small \em continued on next page} \\
\multicolumn{18}{|c }{ } &
\multicolumn{16}{ r|}{\small \em continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
%              \caption{ 
%              {\bf CG coefficients for (000000) dominant weight in (100000)$\otimes$(010000).}
%              \AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
%              states is consistent with table \ref{t:paths_5824bar}.
%              Each CGC should be divided by the respective number in the last row to maintain
%              $\langle n\!\mid\! n\rangle = 1$.
%              }
%              \label{t:351barcgc000000part1}
\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{\hspace{0.5mm}}|}
%
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{\hspace{0.5mm}}|}
%
\hline
\multicolumn{34}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ$\:$ \\
\hline
%
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2 
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                               %.
\AA1\101\10\ZA\110\110$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   \\
\AA1\101\10\ZA\110\110$_2$\ZZ&\p1&\p1&\p1&\p1& &\p1&   &   &\p1&\p1&   &\p1&   &   &\p1&\p1&   &   &\p1&\p1&   &   &   &   &   &   &   &   & 10&\pE&   &   \\
\AA1\101\10\ZA\110\110$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&   &   &\p1&\p1\\
\AA1\101\10\ZA\110\110$_4$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA1\101\10\ZA\110\110$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\100010\ZA1000\10$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_2$\ZZ&   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1\\
\AA\100010\ZA1000\10$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\11\100\ZA\11\1100$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_2$\ZZ& &   &   &   &   &   &   &   &\p1&\p1&   &   &\p1&\p1&\p1&\p1&   &   &\p1&   &\p1&\p1&   &\p1&   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_4$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &\p1\\
\AA1\11\100\ZA\11\1100$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\1001\10\ZA100\110$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_2$\ZZ&  &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1\\
\AA\1001\10\ZA100\110$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10\1001\ZA\10100\1$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &\p1&\pB&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_4$\ZZ&  &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA10\1001\ZA\10100\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\101\100\ZA10\1100$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA\101\100\ZA10\1100$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10000\1\ZA\100001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&\pB&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_4$\ZZ&   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ& &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_3$\ZZ& &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_4$\ZZ& &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1\\
\AA\11\1001\ZA1\1100\1$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11000\1\ZA1\10001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_3$\ZZ&  &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_4$\ZZ&  &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA\11000\1\ZA1\10001$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\10001\ZA01000\1$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_3$\ZZ&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_4$\ZZ&   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
\AA0\10001\ZA01000\1$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_3$\ZZ&  &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_4$\ZZ&  &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA0\1100\1\ZA01\1001$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1100\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\110\ZA0001\10$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA000\110\ZA0001\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\10\ZA000010$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA0000\10\ZA000010$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA0000\10\ZA000010$_6$\ZZ &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                                                          .
\hline
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2 
%aaa                      ---------------------------------------------------------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &$\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{6}$  &
                            $\sqrt{8}$  &$\sqrt{176}$&$\sqrt{50}$ &$\sqrt{8}$  &$\sqrt{8}$  &
                            $\sqrt{6}$  &$\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{18}$ &$\sqrt{8}$  &
                            $\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{8}$  &$\sqrt{8}$  &
                            $\sqrt{6}$  &$\sqrt{10}$ &$\sqrt{6}$  &$\sqrt{6}$  &$\sqrt{6}$  &
                                \pD     &$\sqrt{10}$ &    \pD     &$\sqrt{888}$&$\sqrt{228}$&
                            $\sqrt{8}$  &$\sqrt{8} \:$ \\
\hline
\end{tabular}
\end{table}




\clearpage

%
%  end of the first table 
%

\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the remaining 32 (000000) dominant weight states of the 5824-dimensional
(110000) irrep in the product (100000)$\otimes$(010000).} (The first 32 states of this
irrep with the same weight are shown in table \ref{t:351barcgc000000part1}.)
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc000000part2}
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{\hspace{0.5mm}}|}
%
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{\hspace{0.5mm}}|}
%
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$                       \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ& \AA 37\ZZ& \AA 38\ZZ& \AA 39\ZZ& \AA 40\ZZ& 
          \AA 41\ZZ& \AA 42\ZZ& \AA 43\ZZ& \AA 44\ZZ& \AA 45\ZZ& \AA 46\ZZ& \AA 47\ZZ& \AA 48\ZZ& \AA 49\ZZ& \AA 50\ZZ& 
          \AA 51\ZZ& \AA 52\ZZ& \AA 53\ZZ& \AA 54\ZZ& \AA 55\ZZ& \AA 56\ZZ& \AA 57\ZZ& \AA 58\ZZ& \AA 59\ZZ& \AA 60\ZZ& 
          \AA 61\ZZ& \AA 62\ZZ& \AA 63\ZZ& \AA 64\ZZ$\:$ \\
\hline
%
%                            3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                              %.
\AA100000\ZA\100000$_1$\ZZ &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA100000\ZA\100000$_4$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\110000\ZA1\10000$_1$\ZZ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA\110000\ZA1\10000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\11000\ZA01\1000$_1$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA0\11000\ZA01\1000$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1101\ZA001\10\1$_1$\ZZ&  &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA00\1101\ZA001\10\1$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &\p1&   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\111\ZA0001\1\1$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_2$\ZZ&  &   &   & & &11$\;$ &\pE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1\\
\AA000\111\ZA0001\1\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_4$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_6$\ZZ&  &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00010\1\ZA000\101$_1$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1\\
\AA00010\1\ZA000\101$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\11\ZA00001\1$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_2$\ZZ&   &   &   & & &11$\;$ &\pE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_6$\ZZ&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA001\11\1\ZA00\11\11$_1$\ZZ& &   &   &   &   &\p1&\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_2$\ZZ& &   &   &\p1& &10$\;$ &\pE& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_3$\ZZ& &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA001\11\1\ZA00\11\11$_4$\ZZ& &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&  &   &   &   &   &\p1&\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\p1&   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&  &   &   &\p1& &10$\;$ &\pE& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1\\
\AA0010\1\1\ZA00\1011$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_4$\ZZ&  &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\1010\ZA0\110\10$_1$\ZZ&  &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_2$\ZZ&  &   &   &\p1& &10$\;$ &\pE& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA01\1010\ZA0\110\10$_4$\ZZ&  &   &   &   &\p1&   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\11\10\ZA0\11\110$_1$\ZZ& &   &   &   &   &\p1&\p1&   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_2$\ZZ& &   &   &\p1& &10$\;$ &\pE& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&\p1\\
\AA01\11\10\ZA0\11\110$_3$\ZZ& &   &   &   &   &   &   &   &   &\p1&\p1&   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_4$\ZZ& &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_6$\ZZ& &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\10010\ZA\1100\10$_1$\ZZ&  &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_3$\ZZ&  &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA1\10010\ZA\1100\10$_4$\ZZ&\p1&  &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &\p1&\p1&   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA010\100\ZA0\10100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\p1&\p1&   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &\p1&\pB\\
\AA010\100\ZA0\10100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_3$\ZZ&   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_4$\ZZ&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA010\100\ZA0\10100$_6$\ZZ&   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\hline
\multicolumn{18}{|c }{ } &
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\hline
\end{tabular}
\end{table}




\clearpage


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\begin{table}
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%              \label{t:351barcgc000000part2}
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\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
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                                r@{}r@{\hspace{0.5mm}}|}
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\hline
\multicolumn{34}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$                       \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ& \AA 37\ZZ& \AA 38\ZZ& \AA 39\ZZ& \AA 40\ZZ& 
          \AA 41\ZZ& \AA 42\ZZ& \AA 43\ZZ& \AA 44\ZZ& \AA 45\ZZ& \AA 46\ZZ& \AA 47\ZZ& \AA 48\ZZ& \AA 49\ZZ& \AA 50\ZZ& 
          \AA 51\ZZ& \AA 52\ZZ& \AA 53\ZZ& \AA 54\ZZ& \AA 55\ZZ& \AA 56\ZZ& \AA 57\ZZ& \AA 58\ZZ& \AA 59\ZZ& \AA 60\ZZ& 
          \AA 61\ZZ& \AA 62\ZZ& \AA 63\ZZ& \AA 64\ZZ$\:$ \\
\hline
%
%                            3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                               %.
\AA1\101\10\ZA\110\110$_1$\ZZ& &\p1&\p1&   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_3$\ZZ& &   &   &\p1&\p1&   &   &   &   &\p1&\p1&   &   &\p1&   &   &   &\p1&\p1&\p1&   &   &   &   &\p1&\p1&\p1&   &   &\pD&\p1&\p1\\
\AA1\101\10\ZA\110\110$_4$\ZZ&\p1& &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_6$\ZZ& &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\100010\ZA1000\10$_1$\ZZ&   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA\100010\ZA1000\10$_4$\ZZ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\11\100\ZA\11\1100$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\p1&\p1&   &   &   &   &\p1&\p1&   &   &   &   &   &\p1&\p1&   &   &\p1\\
\AA1\11\100\ZA\11\1100$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&\pB\\
\AA1\11\100\ZA\11\1100$_3$\ZZ& &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &\p1&\p1&   &   &   &\pB&   &   \\
\AA1\11\100\ZA\11\1100$_4$\ZZ&\p1& &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &\p1&   &   \\
\AA1\11\100\ZA\11\1100$_6$\ZZ& &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\1001\10\ZA100\110$_1$\ZZ&  &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &\p1&   &   &\pD&   &   \\
\AA\1001\10\ZA100\110$_4$\ZZ&\p1&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_6$\ZZ&  &   &   &   &   &   &   &   &\p1&\p1&\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10\1001\ZA\10100\1$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &\p1\\
\AA10\1001\ZA\10100\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&\pB\\
\AA10\1001\ZA\10100\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &\pB&   &   \\
\AA10\1001\ZA\10100\1$_4$\ZZ&  &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA10\1001\ZA\10100\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\101\100\ZA10\1100$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &\p1&\p1&   &   &   \\
\AA\101\100\ZA10\1100$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1\\
\AA\101\100\ZA10\1100$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &\pB&\p1&   \\
\AA\101\100\ZA10\1100$_4$\ZZ&\p1&  &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &\p1&   &   \\
\AA\101\100\ZA10\1100$_6$\ZZ&  &   &   &   &   &   &   &   &\p1&\p1&\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10000\1\ZA\100001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1\\
\AA10000\1\ZA\100001$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_4$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   \\
\AA\11\1001\ZA1\1100\1$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1\\
\AA\11\1001\ZA1\1100\1$_3$\ZZ& &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &\pB&\p1&   \\
\AA\11\1001\ZA1\1100\1$_4$\ZZ& &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA\11\1001\ZA1\1100\1$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11000\1\ZA1\10001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&\pB\\
\AA\11000\1\ZA1\10001$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_4$\ZZ&  &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\10001\ZA01000\1$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_3$\ZZ&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_4$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA0\1100\1\ZA01\1001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA0\1100\1\ZA01\1001$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\p1&   \\
\AA0\1100\1\ZA01\1001$_4$\ZZ&  &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1100\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   \\
\AA00\1100\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA00\1100\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   \\
\AA00\1100\ZA001\100$_4$\ZZ&   &   &   &\p1&   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\110\ZA0001\10$_1$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_3$\ZZ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_4$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\10\ZA000010$_1$\ZZ &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_3$\ZZ &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_4$\ZZ &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA0000\10\ZA000010$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                                                          .
\hline
%                            3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4
%bbb                      ---------------------------------------------------------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &                          $\sqrt{  6}$&$\sqrt{  8}$&$\sqrt{  8}$&
                            $\sqrt{ 10}$&$\sqrt{  8}$&$\sqrt{688}$&$\sqrt{178}$&$\sqrt{  6}$&
                            $\sqrt{  6}$&$\sqrt{ 18}$&$\sqrt{ 18}$&$\sqrt{  8}$&$\sqrt{  8}$&
                            $\sqrt{  6}$&$\sqrt{  6}$&$\sqrt{  6}$&$\sqrt{  6}$&$\sqrt{ 14}$&
                                \pD     &$\sqrt{ 10}$&$\sqrt{  8}$&$\sqrt{  8}$&$\sqrt{  6}$&
                            $\sqrt{  6}$&    \pD     &$\sqrt{ 14}$&$\sqrt{ 10}$&$\sqrt{  8}$&
                            $\sqrt{  8}$&$\sqrt{ 76}$&$\sqrt{ 12}$&    \pF $\:$ \\
\hline
\end{tabular}
\end{table}




\clearpage

%
%  end of the second table 
%

\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the first 36 (000000) dominant weight states of the 2925-dimensional
(001000) irrep in the product (100000)$\otimes$(010000).} (The remaining 9 states of this
irrep with the same weight are shown in table \ref{t:351barcgc000000part4}.)
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table II. in ref.\cite{abII}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc000000part3}
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  345678901234567890123456789012345678  (36 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{\hspace{0.5mm}}|}
%
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
%
\hline
 & &  \multicolumn{36}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (001000) } \\
\cline{3-38}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ& \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ$\:$ \\
\hline
%
%                             1   2  3  4  5  6  7  8  9  0  1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6
%                         ----------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                             %.
\AA100000\ZA\100000$_1$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA100000\ZA\100000$_2$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1$\;$\\
\AA100000\ZA\100000$_3$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA100000\ZA\100000$_4$\ZZ &    &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA100000\ZA\100000$_6$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA\110000\ZA1\10000$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\110000\ZA1\10000$_2$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  & 1$\;$\\
\AA\110000\ZA1\10000$_3$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  \\
\AA\110000\ZA1\10000$_4$\ZZ&    &  &  &  &  &  & 1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\110000\ZA1\10000$_6$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1& 1&  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA0\11000\ZA01\1000$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0\11000\ZA01\1000$_2$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  \\
\AA0\11000\ZA01\1000$_3$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  \\
\AA0\11000\ZA01\1000$_4$\ZZ&    &  &  &  &  &  & 1&  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0\11000\ZA01\1000$_6$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1& 1&  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA00\1101\ZA001\10\1$_1$\ZZ&   &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  \\
\AA00\1101\ZA001\10\1$_2$\ZZ&   &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA00\1101\ZA001\10\1$_3$\ZZ&   &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1& 1&  &  \\
\AA00\1101\ZA001\10\1$_4$\ZZ&   &  &  &  &  &  &  &  &  & 1&  &  &  & 1& 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA00\1101\ZA001\10\1$_6$\ZZ&   &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1$\;$\\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA000\111\ZA0001\1\1$_1$\ZZ&   &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  \\
\AA000\111\ZA0001\1\1$_2$\ZZ&   &  & 1&  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA000\111\ZA0001\1\1$_3$\ZZ& 1 &  &  &  & 1& 1&  &  &  &  &  &  &  &  & 1&  &  & 1&  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  \\
\AA000\111\ZA0001\1\1$_4$\ZZ&   &  &  & 1&  &  &  &  &  & 1&  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA000\111\ZA0001\1\1$_6$\ZZ&   & 1&  &  &  &  &  & 1&  &  & 1&  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1$\;$\\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA00010\1\ZA000\101$_1$\ZZ&    &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  \\
\AA00010\1\ZA000\101$_2$\ZZ&    &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA00010\1\ZA000\101$_3$\ZZ&    &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1& 1&  &  &  &  &  &  &  &  \\
\AA00010\1\ZA000\101$_4$\ZZ&    &  &  &  &  &  &  &  &  & 1&  &  &  & 1& 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA00010\1\ZA000\101$_6$\ZZ&    &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA0000\11\ZA00001\1$_1$\ZZ&    &  &-1&  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  \\
\AA0000\11\ZA00001\1$_2$\ZZ&    &  & 1&  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0000\11\ZA00001\1$_3$\ZZ&  1 &  &  &  &  &-1&  &  &  &  &  &  &  &  &-1&  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0000\11\ZA00001\1$_4$\ZZ&    &  &  & 1&  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0000\11\ZA00001\1$_6$\ZZ&    & 1&  &  &  &  &  &  &  &  &-1&  &  &  & 1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA001\11\1\ZA00\11\11$_1$\ZZ&  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &-1&  &  &  &  &  &  &  &  &  &  \\
\AA001\11\1\ZA00\11\11$_2$\ZZ&  &  & 1&  &  &  &  &  & 1&  &  & 1&  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA001\11\1\ZA00\11\11$_3$\ZZ&1 &  &  &  & 1& 1&  &  &  &  &  &  &  &  & 1& 1& 1&  &  &  &  & 1&  &  &  &  & 1& 1&  &  &  & 1&  & 1&  &  \\
\AA001\11\1\ZA00\11\11$_4$\ZZ&  &  &  & 1&  &  &  &  &  & 1&  &  &  &  &-1&  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA001\11\1\ZA00\11\11$_6$\ZZ&  & 1&  &  &  &  &  & 1&  &  & 1&  &  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&   &  &-1&  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &-1&  &  &  &  &  &  &  &  &  &  \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&   &  & 1&  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0010\1\1\ZA00\1011$_3$\ZZ& 1 &  &  &  &  &-1&  &  &  &  &  &  &  &  &-1& 1& 1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  &-1&  &  &  &  \\
\AA0010\1\1\ZA00\1011$_4$\ZZ&   &  &  & 1&  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&   & 1&  &  &  &  &  &  &  &  &-1&  &  &  & 1&  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA01\1010\ZA0\110\10$_1$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA01\1010\ZA0\110\10$_2$\ZZ& -1&  &-1&  &  & 1&  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA01\1010\ZA0\110\10$_3$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&-1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  & 1&  &  &  &  \\
\AA01\1010\ZA0\110\10$_4$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  & 1&  & 1&  & 1&-1&  &  &  &  &  &  &  &  \\
\AA01\1010\ZA0\110\10$_6$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  & 1&  &-1&  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA01\11\10\ZA0\11\110$_1$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &-1&  &  &  &  &-1&  &  &  &  &  &  &  \\
\AA01\11\10\ZA0\11\110$_2$\ZZ&-1&  &-1&  &-1&-1&  &  &-1&  &  &  &-1&  &-1&  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA01\11\10\ZA0\11\110$_3$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&-1&  &  &  &  & 1&  &  &  &  &-1&-1&  &  &  &-1&  &-1&  &  \\
\AA01\11\10\ZA0\11\110$_4$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  & 1&  & 1&  &  & 1&  &  &  &  &  &  &  &  \\
\AA01\11\10\ZA0\11\110$_6$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  & 1&  &-1&  &  &  &  &  &  & 1&  &  &-1&  &  &  & 1&  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA1\10010\ZA\1100\10$_1$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA1\10010\ZA\1100\10$_2$\ZZ& -1&-1&  &-1&  & 1&  &  &  &  & 1&  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA1\10010\ZA\1100\10$_3$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&-1&  &-1&  &-1&-1&-1&  &  &  &  &  &  &  &  & 1&  &  &  &  \\
\AA1\10010\ZA\1100\10$_4$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  & 1&  &  &  &-1&  & 1&-1&  &  &  &  &  &  &-1&  \\
\AA1\10010\ZA\1100\10$_6$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA010\100\ZA0\10100$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &-1&  &  &  \\
\AA010\100\ZA0\10100$_2$\ZZ&    &  &  &  &-1&  &-1&  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  \\
\AA010\100\ZA0\10100$_3$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&-1&  &  &  &  &  & 1&  &  \\
\AA010\100\ZA0\10100$_4$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1& 1&  &  &  &  &  &  &  & 1&  &  & 1&  &  &  &  &  &  &  &  \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                         %.
\AA010\100\ZA0\10100$_6$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  &  & 1&  \\
%                                                                                                                                                           .
\hline
\multicolumn{20}{|c }{ } &
\multicolumn{18}{ r|}{\small \em continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
%              \caption{ 
%              {\bf CG coefficients for (000000) dominant weight in (100000)$\otimes$(010000).}
%              \AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
%              states is consistent with table \ref{t:paths_5824bar}.
%              Each CGC should be divided by the respective number in the last row to maintain
%              $\langle n\!\mid\! n\rangle = 1$.
%              }
%              \label{t:351barcgc000000part3}
\tiny
%                 12  345678901234567890123456789012345678  (36 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{\hspace{0.5mm}}|}
%
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
%
\hline
\multicolumn{38}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{36}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (001000) } \\
\cline{3-38}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ& \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ$\,$ \\
\hline
%
%                              1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6
%                         ----------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                             %.
\AA1\101\10\ZA\110\110$_1$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &-1&  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &-1$\;$\\
\AA1\101\10\ZA\110\110$_2$\ZZ&-1&-1&  &-1&-1&-1&  &-1&  &-1&-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA1\101\10\ZA\110\110$_3$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&-1&  &-1&  &-1&-1&-1&  &  &  &-1&-1&  &  &  &-1&  &-1&-1&  \\
\AA1\101\10\ZA\110\110$_4$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  & 1&  &  &  &-1&  &  & 1&  &  &  &  &  &  & 1&  \\
\AA1\101\10\ZA\110\110$_6$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  & 1&  &  &-1&  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA\100010\ZA1000\10$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\100010\ZA1000\10$_2$\ZZ&    &-1&  & 1&  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\100010\ZA1000\10$_3$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &-1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\100010\ZA1000\10$_4$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &-1&  \\
\AA\100010\ZA1000\10$_6$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&-1&  &  &  & 1&  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA1\11\100\ZA\11\1100$_1$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &-1&  &  &-1$\;$\\
\AA1\11\100\ZA\11\1100$_2$\ZZ&  &  &  &  &-1&  &-1&-1&  &-1&  &-1&  &-1&-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  \\
\AA1\11\100\ZA\11\1100$_3$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&-1&  &  &  &  &  & 1&-1&  \\
\AA1\11\100\ZA\11\1100$_4$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1& 1&  & 1&  &-1&  &-1&  &-1&  &  & 1&  &  &  &  &  &  & 1&  \\
\AA1\11\100\ZA\11\1100$_6$\ZZ&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA\1001\10\ZA100\110$_1$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1$\;$\\
\AA\1001\10\ZA100\110$_2$\ZZ&   &-1&  & 1&  &  &  &-1&  & 1&-1&  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\1001\10\ZA100\110$_3$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &-1&  & 1&  &  &  &  &  &  &  &  &  &  &  &-1&  \\
\AA\1001\10\ZA100\110$_4$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  & 1&  \\
\AA\1001\10\ZA100\110$_6$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&-1&  &  &  & 1&  &  &  &  &  & 1&-1&  &  &-1&  &  &-1&  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA10\1001\ZA\10100\1$_1$\ZZ&   &  &  &  &  &  &-1&  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &-2&-1&  &  \\
\AA10\1001\ZA\10100\1$_2$\ZZ&   &  &  &  &  &  &  &  &  &  &  & 1&  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA10\1001\ZA\10100\1$_3$\ZZ&   &  &  &  &  &  &  & 1&  &-1&  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  \\
\AA10\1001\ZA\10100\1$_4$\ZZ&   & 1&  &-1&  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  & 1&  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA10\1001\ZA\10100\1$_6$\ZZ&   &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA\101\100\ZA10\1100$_1$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1$\;$\\
\AA\101\100\ZA10\1100$_2$\ZZ&   &  &  &  &  &  &  &-1&  & 1&  &-1&  & 1& 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\101\100\ZA10\1100$_3$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  \\
\AA\101\100\ZA10\1100$_4$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &-1& 1& 1&  & 1&  &  &  &  &  &  &  &  &  & 1&  \\
\AA\101\100\ZA10\1100$_6$\ZZ&   &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&-1&  &  & 1& 1&  & 1&  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA10000\1\ZA\100001$_1$\ZZ&    &  &  &  &  &  &-1&  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1& 1&  &  \\
\AA10000\1\ZA\100001$_2$\ZZ&    &  &  &  &  &  &  &  &  &  &  & 1&  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA10000\1\ZA\100001$_3$\ZZ&    &  &  &  &  &  &  & 1&  &-1&  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1$\;$\\
\AA10000\1\ZA\100001$_4$\ZZ&    & 1&  &-1&  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA10000\1\ZA\100001$_6$\ZZ&    &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ&  &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &-1&  &  &  \\
\AA\11\1001\ZA1\1100\1$_2$\ZZ&  &  &  &  &  &  & 1&  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  \\
\AA\11\1001\ZA1\1100\1$_3$\ZZ&  &  &  &  & 1&  &  & 1&  & 1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  & 1& 1&  &  &  &  &  & 1& 1&  \\
\AA\11\1001\ZA1\1100\1$_4$\ZZ& 1& 1&  & 1&  &  &  &  &  &  &  &  &  &  &  & 1& 1&  & 1&  & 1& 1& 1&  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\11\1001\ZA1\1100\1$_6$\ZZ&  &  &  &  &  & 1&  &  &  &  & 1&  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA\11000\1\ZA1\10001$_1$\ZZ&   &  &  &  &  &  &  &  &  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  &  \\
\AA\11000\1\ZA1\10001$_2$\ZZ&   &  &  &  &  &  & 1&  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1& 1&  &  \\
\AA\11000\1\ZA1\10001$_3$\ZZ&   &  &  &  & 1&  &  & 1&  & 1&  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  & 1$\;$\\
\AA\11000\1\ZA1\10001$_4$\ZZ&  1& 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA\11000\1\ZA1\10001$_6$\ZZ&   &  &  &  &  & 1&  &  &  &  & 1&  &  &  &-1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA0\10001\ZA01000\1$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  &-1&  &  &  \\
\AA0\10001\ZA01000\1$_2$\ZZ&    &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &-1&  &  \\
\AA0\10001\ZA01000\1$_3$\ZZ&    &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1& 1&  &  &  &  &  & 1&  &  \\
\AA0\10001\ZA01000\1$_4$\ZZ&   1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  & 1& 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0\10001\ZA01000\1$_6$\ZZ&    &  &  &  &  & 1&  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&   &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  & 1&  &  &  \\
\AA0\1100\1\ZA01\1001$_2$\ZZ&   &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1& 1&  &  \\
\AA0\1100\1\ZA01\1001$_3$\ZZ&   &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  \\
\AA0\1100\1\ZA01\1001$_4$\ZZ&  1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0\1100\1\ZA01\1001$_6$\ZZ&   &  &  &  &  & 1&  &  &  &  &  &  & 1&  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA00\1100\ZA001\100$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1$\;$\\
\AA00\1100\ZA001\100$_2$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  \\
\AA00\1100\ZA001\100$_3$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA00\1100\ZA001\100$_4$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  & 1&  & 1&  &  &  &  &  &  &  &  &  &  \\
\AA00\1100\ZA001\100$_6$\ZZ&    &  &  &  &  &  &  &  & 1&  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA000\110\ZA0001\10$_1$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1$\;$\\
\AA000\110\ZA0001\10$_2$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  \\
\AA000\110\ZA0001\10$_3$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA000\110\ZA0001\10$_4$\ZZ&    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  \\
\AA000\110\ZA0001\10$_6$\ZZ&    &  & 1&  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%                                                                                                                                         .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                             %.
\AA0000\10\ZA000010$_1$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0000\10\ZA000010$_2$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0000\10\ZA000010$_3$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  &  &  \\
\AA0000\10\ZA000010$_4$\ZZ &    &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  & 1&  &  &  &  &  &  &  &  &  &  \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                        %.
\AA0000\10\ZA000010$_6$\ZZ &    &  & 1&  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
%*                                                                                                                                                          .
%ccc                          1   2  3  4  5  6  7  8  9  0  1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6  7  8  9  0  1  2  3  4  5  6
%                         ----------------------------------------------------------------------------------------------------------------
\hline
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &
                             \multicolumn{12}{|c}{$\sqrt{ 12}$ each of the first twelve states} &
%
%                            $\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&
%                            $\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&
%                            $\sqrt{ 12}$&$\sqrt{ 12}$&
%
                                                          \pC     &    \pC     &$\sqrt{ 18}$&
                            $\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&
                            $\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&    \pC     &
                                \pC     &$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&    \pC     &
                                \pC     &    \pC     &$\sqrt{ 24}$&$\sqrt{ 18}$&$\sqrt{ 12}$&
                            $\sqrt{ 12} \:$ \\
\hline
\end{tabular}
\end{table}




\clearpage

%
%  end of the third table 
%


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the (000000) dominant weight states of the 2925-dimensional
(001000) irrep, 650-dimensional (100010) irrep, and 78-dimensional (000001) irrep
in the product (100000)$\otimes$(010000).} 
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table II. in ref.\cite{abII} and table I in ref.\cite{abI}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351barcgc000000part4}  
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  345678901  23456789012345678901  234567  (35 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrr||rrrrrrrrrrrrrrrrrrrr||rrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
 & &  \multicolumn{ 9}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (001000) } &
      \multicolumn{20}{|c|}{\normalsize \em (100010) } &
      \multicolumn{ 6}{|c|}{\normalsize \em (000001) } \\
\cline{3-37}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ$\:$ \\
\hline
%                             7   8  9  0  1  2  3  4  5*  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0*  1   2   3   4   5   6   
%                         ---------------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                                    %.
\AA100000\ZA\100000$_1$\ZZ &    &  &  &  &  &  &  &  &  &\mB&   &   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &\pD&   &\mB\\
\AA100000\ZA\100000$_2$\ZZ &    &  &  &  &  &  &  &  &  &\pB&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pD&   &   &\pD\\
\AA100000\ZA\100000$_3$\ZZ &    &  &  &  &  & 1&  &  &  &\mB&   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &\pD&   &   &   &\mF\\
\AA100000\ZA\100000$_4$\ZZ &    &  &  &  &  &  &  &  &  &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&   &\pD&   &   &   &   &\pC\\
\AA100000\ZA\100000$_6$\ZZ &    &  &  &  &  &  &  &  &  &\p1&   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pD&\pE\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA\110000\ZA1\10000$_1$\ZZ&    &  &  &  &  &  &  &  &  &\mB&   &\mB&   &   &   &   &   &   &   &   &\mB&\mB&   &\mB&   &   &   &   &   &   &   &   &\mD&\mB&\mB\\
\AA\110000\ZA1\10000$_2$\ZZ&    &  &  &  &  &  &  &  &  &\pB&\mB&\pB&   &\mB&   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &\mD&   &\pD&\pD\\
\AA\110000\ZA1\10000$_3$\ZZ&    &  &  &  &  & 1&  &  &  &\mB&   &\mB&\mB&   &   &\mB&\mB&\mB&   &   &   &   &   &   &   &   &   &   &   &   &\mD&   &   &\mF&\mF\\
\AA\110000\ZA1\10000$_4$\ZZ&    &  &  &  &  &  &  &  &  &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\mB&\mB&\mD&   &   &   &\pC&\pC\\
\AA\110000\ZA1\10000$_6$\ZZ&    &  &  &  &  &  &  &  &  &\p1&   &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pE\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA0\11000\ZA01\1000$_1$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\mB&   &   &\mB&   &   &   &   &   &   &\mB&\mB&\pB&   &   &   &   &   &   &\mB&   &\pD&\mB&   \\
\AA0\11000\ZA01\1000$_2$\ZZ&    & 1&  &  &  &  &  &  &  &   &   &\pB&   &\mB&\pB&   &\mB&   &\mB&\pB&   &   &   &   &   &   &   &   &   &   &\pD&\pD&   &\pD&   \\
\AA0\11000\ZA01\1000$_3$\ZZ&    &  &  &  &  &  &  & 1&  &   &   &\mB&\mB&   &\mB&\pB&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&   &   &\mF&   \\
\AA0\11000\ZA01\1000$_4$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\p1&   &   &\p1&\mB&   &   &   &   &   &   &   &   &   &   &\pB&   &\mB&\pD&\pC&   &   &\pC&   \\
\AA0\11000\ZA01\1000$_6$\ZZ&    &  &  &  & 1& 1&  &  &  &   &   &\p1&\pB&   &\p1&   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&\mD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA00\1101\ZA001\10\1$_1$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &   &\mB&   &   &   &\mB&   &   &   &\mB&\mB&\mB&\mB&   &   &   &\mB&\mB&\mB&\mD&   &   \\
\AA00\1101\ZA001\10\1$_2$\ZZ&   & 1&  & 1&  &  &  &  &  &   &   &   &   &   &\pB&   &\mB&   &\pB&\mB&   &   &   &   &\pB&   &\mB&   &   &\pD&\pD&   &   &   &   \\
\AA00\1101\ZA001\10\1$_3$\ZZ&   &  &  &  &  &  &  & 1&  &   &   &   &\pB&\pB&\mB&\pB&\pB&   &   &\pB&   &   &   &   &\mB&   &\pB&   &\pB&\mB&\mB&\mB&   &\pD&   \\
\AA00\1101\ZA001\10\1$_4$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &   &\p1&\mB&   &   &   &\mB&   &   &   &   &\p1&   &\mB&   &   &\m1&\pC&\pC&   &   &   \\
\AA00\1101\ZA001\10\1$_6$\ZZ&   &  &  &  & 1& 1&  &  &  &   &\pB&   &   &   &\p1&   &   &\pB&   &   &   &   &   &   &\p1&   &   &\pB&   &\p1&\p1&\p1&   &   &\pD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA000\111\ZA0001\1\1$_1$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &\pB&\mB&   &   &   &   &\pB&   &\mB&\pB&   &   \\
\AA000\111\ZA0001\1\1$_2$\ZZ&   & 1&  & 1&  &  &  & 1& 1&   &   &   &   &   &   &\mB&\pB&   &\pB&\mB&   &   &\pB&\mB&\pB&\pB&\pB&   &   &\mD&\mD&   &   &   &   \\
\AA000\111\ZA0001\1\1$_3$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &\pB&   &   &   &   &   &\pB&   &\pB&   &\pB&\mB&\mB&\mB&   &\mB&\pB&   &\mB&\mB&\mD&   \\
\AA000\111\ZA0001\1\1$_4$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &\mB&   &   &   &\mB&\p1&\p1&\pB&   &   &\p1&   &\pC&\pC&   &   \\
\AA000\111\ZA0001\1\1$_6$\ZZ&   &  &  &  &  &  &  &  &  &   &\pB&   &   &   &   &   &   &   &   &   &\pB&   &   &   &\p1&\p1&   &\mB&   &\m1&   &\p1&\p1&   &\mD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA00010\1\ZA000\101$_1$\ZZ&    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\mB&\mB&   &   &   &\mB&   &\pB&\pD&   &   \\
\AA00010\1\ZA000\101$_2$\ZZ&    &  &  &-1&  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &\pB&   &\mB&   &   &\pD&   &   &   &   &   \\
\AA00010\1\ZA000\101$_3$\ZZ&    &  &  & 1&  &  &  &  &  &   &   &   &   &\mB&   &   &   &   &\mB&\mB&   &   &   &   &\mB&   &\pB&   &\pB&\mB&   &\pB&   &\mD&   \\
\AA00010\1\ZA000\101$_4$\ZZ&    &  & 1&  &  &  &  & 1& 1&   &   &   &   &   &   &\pB&\mB&   &\p1&\pB&   &   &   &   &\p1&   &\mB&   &   &\m1&\mD&\mC&   &   &   \\
\AA00010\1\ZA000\101$_6$\ZZ&  1 &  &  &  &  &  &  &  &  &   &\mB&   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &\pB&   &\p1&   &\m1&   &   &\mD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA0000\11\ZA00001\1$_1$\ZZ&    &  &  & 1&  &  &  &  &  &   &   &   &   &   &   &   &   &   &\pB&\pB&   &   &   &\pB&\pB&   &   &   &   &\mB&   &\pD&\pB&   &   \\
\AA0000\11\ZA00001\1$_2$\ZZ&    &  &  &  &  &  &  &  &-1&   &   &   &   &   &\pB&\pB&   &   &   &   &   &   &\pB&\mB&   &\pB&   &   &   &\pD&   &   &   &   &   \\
\AA0000\11\ZA00001\1$_3$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\pB&   &   &   &   &   &   &   &   &   &\pB&   &\pB&   &\mB&   &   &\pB&\mB&   &   &\mB&\pD&   \\
\AA0000\11\ZA00001\1$_4$\ZZ&    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&   &\p1&\m1&   &   &\m1&\pD&   &\pC&   &   \\
\AA0000\11\ZA00001\1$_6$\ZZ&    &  &  &  &  &  &  &  &  &\pB&   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &\p1&\p1&\pB&   &\p1&   &   &\p1&   &\pD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA001\11\1\ZA00\11\11$_1$\ZZ&  &  &  &  &  &  &  &  &  &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &\mB&\mB&   &   &   &   &\pB&\pB&\pB&\mB&   &   \\
\AA001\11\1\ZA00\11\11$_2$\ZZ&  &  &  &-1&  &  &  &  & 1&   &   &   &   &   &\pB&\pB&   &   &   &\pB&   &   &   &\pB&\pB&\pB&\pB&   &   &\mD&   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_3$\ZZ&  &  &  & 1&  &  &  & 1& 1&   &   &   &\mB&\mB&\mB&\mB&\mB&   &\mB&\mB&   &   &\mB&\mB&\mB&\mB&\mB&   &\mB&\pB&\pB&\pB&\pB&\pD&   \\
\AA001\11\1\ZA00\11\11$_4$\ZZ&  &  & 1&  &  &  &  &  &-1&   &   &   &   &   &\p1&   &\pB&   &\p1&\pB&   &\mB&\p1&\pB&\p1&\p1&\pB&   &   &\p1&\p1&\mC&\mC&   &   \\
\AA001\11\1\ZA00\11\11$_6$\ZZ&1 &  &  &  & 1&  & 1&  &  &   &\mB&   &   &   &\p1&   &   &\mB&\p1&   &\mB&   &\p1&   &\p1&\p1&   &\mB&   &\m1&\m1&\m1&\m1&   &\pD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&   &-1&  &-1&  &  &  &  &  &   &   &   &   &   &\pB&   &   &   &   &\mB&   &   &   &\mB&\pB&   &   &   &   &\mB&\mB&\mD&\mB&   &   \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&   &  &  &  &  &  &  &-1&-1&   &   &   &   &   &\mB&\mB&   &   &   &   &   &   &   &\pB&   &\pB&   &   &   &\pD&   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_3$\ZZ&   &  &  &  &  &  &  &  &-1&   &   &\mB&\pB&   &   &\pB&   &   &   &   &   &\mB&\mB&\mB&   &\mB&   &   &\pB&\mB&\mB&   &\pB&\mD&   \\
\AA0010\1\1\ZA00\1011$_4$\ZZ&   &  &  &  &  &  &  &  & 1&   &   &   &   &   &   &   &\m1&   &   &   &   &   &\p1&\pB&   &\p1&\m1&   &   &\m1&\m1&   &\mC&   &   \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&   &  &  &  &  &  &-1&  & 1&\mB&   &   &   &   &   &   &\p1&\pB&   &   &\mB&   &\p1&   &   &\p1&\p1&\pB&   &\p1&\p1&   &\m1&   &\mD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA01\1010\ZA0\110\10$_1$\ZZ&   &  &  &  &  &  &  &  &  &   &   &\mB&   &   &\mB&   &   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &\pB&   &\pB&\pB&   \\
\AA01\1010\ZA0\110\10$_2$\ZZ&   &  &  &  &  &  &  &  & 1&   &   &\pB&   &   &\pB&\pB&   &   &   &   &   &   &   &\mB&   &\mB&\mB&   &\pB&\pD&   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_3$\ZZ&   &  &  &  &  &  &  & 1& 1&   &   &\mB&\pB&   &\mB&\mB&\mB&   &   &   &   &   &\mB&\pB&   &   &   &   &   &   &\pB&   &\mB&\mB&   \\
\AA01\1010\ZA0\110\10$_4$\ZZ&   &  & 2& 1&  &  &  &  &-1&   &   &\p1&   &\pB&\p1&   &\pB&   &   &\mB&   &\p1&\p1&\mB&   &   &   &   &   &   &\p1&\pD&\pC&\p1&   \\
\AA01\1010\ZA0\110\10$_6$\ZZ&   &  &  &  & 1&  & 1&  &  &   &   &\p1&\mB&   &\p1&   &   &\mB&   &   &\pB&\p1&\p1&   &   &   &   &   &   &   &\m1&   &\p1&\m1&\mD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA01\11\10\ZA0\11\110$_1$\ZZ&  &-1&  &  &  &  &  &  &  &   &   &\pB&   &   &\pB&   &   &   &   &\pB&   &   &   &\pB&   &   &   &   &   &   &\mB&\pB&\pB&\mB&   \\
\AA01\11\10\ZA0\11\110$_2$\ZZ&  &  &  &-1&  &  &  &-1&-1&   &   &\mB&   &   &\mB&\mB&   &   &   &\mB&   &   &   &\mB&\mB&\mB&   &   &\mB&\mD&   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_3$\ZZ&  &  &  & 1&  &  &  &  &-1&   &   &\pB&\mB&   &   &\pB&   &   &\mB&\pB&   &   &\mB&\pB&   &   &   &   &   &   &\mB&\mB&\mB&\pB&   \\
\AA01\11\10\ZA0\11\110$_4$\ZZ&  &  & 1&  &  &  &  &  & 1&   &   &   &   &\m1&   &   &\m1&   &\p1&   &   &\p1&\p1&\mB&   &   &   &   &   &   &\m1&\m1&\pC&\m1&   \\
\AA01\11\10\ZA0\11\110$_6$\ZZ&-1&  &  &  &  &  &-1&  & 1&\pB&\pB&   &\pB&\p1&   &   &\p1&\pB&\p1&   &\pB&\p1&\p1&   &   &   &   &   &   &   &\p1&\p1&\p1&\p1&\pD\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA1\10010\ZA\1100\10$_1$\ZZ&   &  &  &  &  &  &  &  &  &\mB&   &\mB&   &   &   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &\mB&\pB&\pB\\
\AA1\10010\ZA\1100\10$_2$\ZZ&   &  &  &  &  &  &  &  &  &\pB&   &\pB&   &   &   &   &   &   &   &   &   &   &   &\pB&   &\pB&\pB&\pB&\pB&\mD&   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_3$\ZZ&   &  &  &  &  &  & 1&  &-1&\mB&   &\mB&\pB&   &   &\pB&\pB&\pB&   &   &   &   &\pB&\mB&   &   &   &   &   &   &\mD&   &\pB&\mB&\mB\\
\AA1\10010\ZA\1100\10$_4$\ZZ&  1&  &-1&-1&  &  &  &  &  &\p1&\pB&\p1&   &\pB&   &   &   &   &\pB&\pB&\p1&\p1&   &\pB&   &   &   &   &   &   &   &\mD&\mC&\p1&\p1\\
\AA1\10010\ZA\1100\10$_6$\ZZ&   &  &  &  &  &  &  &  &  &\p1&   &\p1&\mB&   &   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &\m1&\m1&\pC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA010\100\ZA0\10100$_1$\ZZ&    &-1&  &  &  &  &  &-1&  &   &   &\mB&   &   &\mB&\pB&   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &\pD&\pB&   &\pB&   \\
\AA010\100\ZA0\10100$_2$\ZZ&    &  &  &-1&  &  &  &  &  &   &   &\pB&   &   &   &   &   &   &   &\mB&   &   &   &   &\mB&   &   &   &\pB&\pD&   &   &   &   &   \\
\AA010\100\ZA0\10100$_3$\ZZ&    &  &  & 1&  &  &  &  &  &   &   &\mB&\pB&   &   &   &   &   &\mB&\pB&   &   &   &   &   &   &   &   &   &   &   &\mB&   &\mB&   \\
\AA010\100\ZA0\10100$_4$\ZZ&    &  & 1&  &  &  &  &  &  &   &   &   &\p1&\m1&   &   &   &   &\p1&   &   &\mB&   &\pB&   &   &   &   &   &   &   &\m1&\mD&\p1&   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                                %.
\AA010\100\ZA0\10100$_6$\ZZ&  -1&  &  &  &  &  &  &  &  &\mB&\pB&   &\m1&\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\m1&\mD\\
%                                                                                                                                                           .
\hline
\multicolumn{19}{|c }{ } &
\multicolumn{18}{ r|}{\small \em continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
%              \caption{ 
%              {\bf CG coefficients for (000000) dominant weight in (100000)$\otimes$(010000).}
%              \AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
%              states is consistent with table \ref{t:paths_5824bar}.
%              Each CGC should be divided by the respective number in the last row to maintain
%              $\langle n\!\mid\! n\rangle = 1$.
%              }
%              \label{t:351barcgc000000part4}
\tiny
%                 12  345678901  23456789012345678901  234567  (35 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrr||rrrrrrrrrrrrrrrrrrrr||rrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
\multicolumn{37}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{ 9}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (001000) } &
      \multicolumn{20}{|c|}{\normalsize \em (100010) } &
      \multicolumn{ 6}{|c|}{\normalsize \em (000001) } \\
\cline{3-37}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
    $\,$  \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& 
    $\,$  \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
    $\,$  \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ$\:$ \\
\hline
%
%                             7   8  9  0  1  2  3  4  5*  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0*  1   2   3   4   5   6   
%                         ---------------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                                     %.
\AA1\101\10\ZA\110\110$_1$\ZZ&  &  &  &  &  &  &  &  &  &\pB&   &\pB&   &   &   &   &   &   &   &\mB&   &   &   &\mB&   &   &   &   &   &   &   &\mB&\mB&\mB&\mB\\
\AA1\101\10\ZA\110\110$_2$\ZZ&  &  &  &  &  &  &  &  &  &\mB&   &\mB&   &   &   &   &   &   &   &\pB&   &   &   &\pB&\pB&\pB&   &\mB&\mB&\pD&   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_3$\ZZ&-1&  &-1&-1&-1&  &-1&-1&  &\pB&   &\pB&\mB&   &\pB&\mB&   &\mB&\pB&\mB&   &   &\pB&\mB&   &   &   &   &   &   &\pD&\pB&\pB&\pB&\pB\\
\AA1\101\10\ZA\110\110$_4$\ZZ&  &  &-1&  &  &  &  &  &  &   &\m1&   &   &\m1&   &   &   &   &\mB&   &\p1&\p1&   &\pB&   &   &   &   &   &   &   &\p1&\mC&\m1&\m1\\
\AA1\101\10\ZA\110\110$_6$\ZZ& 1&  &  &  &  &  &  &  &  &\mB&\m1&   &\pB&\p1&   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&\mC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA\100010\ZA1000\10$_1$\ZZ&    &  &  &  &  &  &  &  &  &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &   &\pB&   &\pB\\
\AA\100010\ZA1000\10$_2$\ZZ&    &  &  &  &  &  &  &  &  &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&   &\mB&   &\pB&   &\pD&   &   &   &   &   \\
\AA\100010\ZA1000\10$_3$\ZZ&    &  &  &  &  &  & 1&  &  &\mB&   &   &   &   &   &   &   &\pB&   &   &   &   &\mB&\pB&   &   &   &   &   &   &\pD&   &\mB&   &\mB\\
\AA\100010\ZA1000\10$_4$\ZZ&   1&  & 1&  &  &  &  &  &  &\p1&\pB&   &   &   &   &   &   &   &\mB&   &\p1&   &   &\m1&   &   &   &   &   &   &   &\pD&\pC&   &\p1\\
\AA\100010\ZA1000\10$_6$\ZZ&    &  &  &  &  &  &  &  &  &\p1&   &   &\pB&   &   &   &   &   &   &   &\m1&\mB&   &\m1&   &   &   &   &   &   &   &   &\p1&\pD&\pC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA1\11\100\ZA\11\1100$_1$\ZZ&  &  &  &  &  &-1&  &  &  &\mB&   &\mB&   &   &   &\mB&   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &\mB&\mB&   &\pB&\pB\\
\AA1\11\100\ZA\11\1100$_2$\ZZ&  &  &  &  &-1&  &  &-1&  &\pB&   &\pB&   &   &\pB&\pB&   &   &   &\pB&   &   &   &   &\pB&   &   &\pB&\pB&\mD&   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_3$\ZZ&-1&  &-1&-1& 1&  &  &  &-1&\mB&   &\mB&\pB&   &\mB&   &   &\pB&\pB&\mB&   &   &   &   &   &   &   &   &   &   &\mB&\pB&   &\mB&\mB\\
\AA1\11\100\ZA\11\1100$_4$\ZZ&  &  &-1&  &  &  &  &  & 1&   &\m1&   &\m1&\m1&   &   &\m1&\m1&\mB&   &\mB&\mB&\mB&\mB&   &   &   &   &   &   &\p1&\p1&\pD&\p1&\p1\\
\AA1\11\100\ZA\11\1100$_6$\ZZ& 1&  &  &  &  &  & 1&  & 1&\pB&\m1&   &\m1&\p1&   &   &\p1&\m1&   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&   &\m1&\pC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA\1001\10\ZA100\110$_1$\ZZ&   &  &  &  &  &  &  &  &  &\pB&   &   &   &   &   &   &   &   &   &\pB&   &   &   &\pB&   &   &   &   &   &   &   &\pB&\pB&   &\mB\\
\AA\1001\10\ZA100\110$_2$\ZZ&   &  &  &  &  &  &  &  &  &\mB&   &   &   &   &   &   &   &   &   &\mB&   &   &   &\mB&\mB&\mB&\mB&\mB&   &\mD&   &   &   &   &   \\
\AA\1001\10\ZA100\110$_3$\ZZ& -1&  & 1&  &-1&  &-1&  &-1&\pB&   &   &   &   &\mB&   &\mB&\mB&\mB&\pB&   &   &\mB&\pB&   &   &   &   &   &   &\mD&\mB&\mB&   &\pB\\
\AA\1001\10\ZA100\110$_4$\ZZ&   &  & 1& 1&  &  &  &  &  &   &\m1&   &   &   &   &   &   &   &\pB&\m1&\p1&   &   &\m1&   &   &   &   &   &   &   &\m1&\pC&   &\m1\\
\AA\1001\10\ZA100\110$_6$\ZZ&  1&  &  &  &  &  &  &  &  &\mB&\m1&\mB&\mB&\mB&   &   &   &   &   &\m1&\m1&\mB&   &\m1&   &   &   &   &   &   &   &\p1&\p1&\pD&\mC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA10\1001\ZA\10100\1$_1$\ZZ&   &  &  &  &  &-1&  &  &  &\pB&   &\pB&   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &\mB&\mB&\mB&   &   &\mD&\mB\\
\AA10\1001\ZA\10100\1$_2$\ZZ&   &  &  &  &-1&  &  &-1&  &\mB&   &   &   &   &\pB&\pB&   &   &   &   &   &   &   &   &   &   &   &\mB&   &\pD&   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_3$\ZZ&  1&  &-1&  & 1&  &  &  &-1&\pB&\pB&   &   &   &\mB&   &   &\pB&\mB&\pB&   &   &   &   &\mB&   &   &\pB&   &\mB&\mB&\mD&   &   &\pB\\
\AA10\1001\ZA\10100\1$_4$\ZZ&   &  &  &  &  &  &  &  & 1&   &   &   &   &   &   &   &\m1&\m1&   &   &\pB&   &\mB&\pB&   &\mB&\m1&\m1&   &\p1&\p1&   &\mD&   &\m1\\
\AA10\1001\ZA\10100\1$_6$\ZZ&   &  &  &  &  &  & 1&  & 1&\mB&   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &   &   &\p1&\m1&   &\m1&\m1&   &   &   &\mC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA\101\100\ZA10\1100$_1$\ZZ&   &  &  &  &  &-1&  &  &  &\mB&   &   &   &   &   &\pB&   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &\pB&\pB&   &   &\pB\\
\AA\101\100\ZA10\1100$_2$\ZZ&   &  &  &  &-1&  &  &  &  &\pB&   &   &   &   &\mB&\mB&   &   &   &\mB&   &   &   &   &\mB&   &\pB&\pB&   &\pD&   &   &   &   &   \\
\AA\101\100\ZA10\1100$_3$\ZZ& -1&  & 1&  & 1&  &  & 1& 1&\mB&   &   &   &   &\pB&\pB&\pB&\pB&\mB&\pB&   &   &   &   &   &   &   &   &   &   &\pB&\mB&   &   &\mB\\
\AA\101\100\ZA10\1100$_4$\ZZ&   &  & 1& 1&  &  &  &  &  &   &\m1&   &   &   &   &\m1&   &\m1&\pB&\m1&\mB&   &\pB&   &   &   &   &   &   &   &\m1&\m1&\mD&   &\p1\\
\AA\101\100\ZA10\1100$_6$\ZZ&  1&  &  &  &  &  & 1&  &-1&\pB&\m1&\pB&   &\mB&   &\m1&\mB&\m1&   &\m1&   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &\pD&\pC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA10000\1\ZA\100001$_1$\ZZ&    &  &  &  &  &  &  &  &  &\mB&   &\mB&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\mB&   &   &   &\pD&\pB\\
\AA10000\1\ZA\100001$_2$\ZZ&    &  &  &  & 1& 1&  &  &  &\pB&   &   &   &   &\mB&\mB&   &\mB&   &   &   &   &   &   &   &   &   &\mB&   &\pD&\pD&   &   &   &   \\
\AA10000\1\ZA\100001$_3$\ZZ&    &  &  &  &  &  &  &  &  &\mB&\mB&   &   &   &   &   &   &   &   &\mB&   &   &   &   &\mB&   &   &\pB&   &\mB&   &\pD&   &   &\mB\\
\AA10000\1\ZA\100001$_4$\ZZ&    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &\mB&   &   &\mB&   &\mB&\m1&\m1&   &\p1&   &   &\pD&   &\p1\\
\AA10000\1\ZA\100001$_6$\ZZ&    &  &  &  &  &  &  &  &  &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &\m1&   &   &   &   &\pC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ&  &  &  &  &  &-1&  &  &  &\pB&   &   &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &\pB&\pB&\pB&   &   &\pB&\mB\\
\AA\11\1001\ZA1\1100\1$_2$\ZZ&  &  &  &  &-1&  &  &  &  &\mB&   &\mB&   &   &\mB&\mB&   &   &   &   &   &   &   &   &   &   &\mB&\mB&\mB&\mD&   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_3$\ZZ& 1&  & 1& 1& 1&  &  & 1& 1&\pB&\pB&\pB&\pB&\pB&\pB&\pB&\pB&\pB&\pB&   &   &   &   &   &\pB&   &\pB&\pB&\pB&\pB&\pB&\pD&   &\pB&\pB\\
\AA\11\1001\ZA1\1100\1$_4$\ZZ&  &  &  &  &  &  &  &  &  &   &   &   &\m1&   &   &\m1&   &\m1&   &   &\pB&\pB&\pB&   &   &\pB&   &\m1&\m1&\m1&\m1&   &\pD&\m1&\m1\\
\AA\11\1001\ZA1\1100\1$_6$\ZZ&  &  &  &  &  &  & 1&  &-1&\mB&   &\mB&\m1&   &   &\m1&\mB&\m1&   &   &   &   &   &   &   &   &\mB&\m1&\m1&\p1&\p1&   &   &\mC&\mC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA\11000\1\ZA1\10001$_1$\ZZ&   &  &  &  &  &  &  &  &  &\mB&   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\pB&   &   &   &\mB&\pB\\
\AA\11000\1\ZA1\10001$_2$\ZZ&   &  &  &  & 1& 1&  & 1&  &\pB&   &\pB&\mB&   &\pB&   &\mB&\mB&   &   &   &   &   &   &   &   &\mB&\mB&\mB&\mD&\mD&   &   &   &   \\
\AA\11000\1\ZA1\10001$_3$\ZZ&   &  &  &  &  &  &  &  &  &\mB&\mB&\mB&   &\mB&   &   &   &   &   &   &   &   &   &   &\pB&   &\pB&\pB&\pB&\pB&   &\mD&   &\mB&\mB\\
\AA\11000\1\ZA1\10001$_4$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &\mB&\mB&   &   &   &\pB&   &\m1&\m1&\m1&   &   &\mD&\p1&\p1\\
\AA\11000\1\ZA1\10001$_6$\ZZ&   &  &  &  &  &  &  &  &  &\pB&   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&\m1&\p1&   &   &   &\pC&\pC\\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA0\10001\ZA01000\1$_1$\ZZ&    &  &  &  &  &  &  &  &  &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\pB&\mB&   &   &   &\pB&\pD\\
\AA0\10001\ZA01000\1$_2$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &\mB&\pD&   &   &   &   &   \\
\AA0\10001\ZA01000\1$_3$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\pB&\pB&\pB&   &   &   &   &   &   &   &   &   &   &\mB&   &\mB&   &\pB&\mB&   &\mD&   &\pB&   \\
\AA0\10001\ZA01000\1$_4$\ZZ&    &  &  &  &  &  &  &  &  &   &   &   &\m1&   &   &   &   &   &   &   &   &\pB&   &   &   &\mB&   &   &\m1&\p1&   &   &\mD&\m1&   \\
\AA0\10001\ZA01000\1$_6$\ZZ&    &  &  &  &  &  &  & 1& 2&   &   &\mB&\m1&   &\mB&   &\pB&   &   &   &   &   &   &   &   &   &\pB&   &\m1&\m1&\mD&   &   &\mC&   \\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&   &  &  &  &  & 1&  &  &  &\pB&   &   &\pB&   &   &\pB&   &\pB&   &   &   &   &   &   &   &   &   &\pB&\pB&\mB&\mB&   &   &\mB&\mD\\
\AA0\1100\1\ZA01\1001$_2$\ZZ&   &  &  &  &  &  &  &  &  &   &   &\pB&\mB&   &   &\mB&\pB&   &   &   &   &   &   &   &   &   &\pB&   &\mB&\pD&\pD&   &   &   &   \\
\AA0\1100\1\ZA01\1001$_3$\ZZ&   & 1&  & 1&  &  &  & 1&  &   &   &\mB&   &\mB&\mB&\pB&\mB&   &\mB&   &   &   &   &   &\mB&   &\mB&   &\pB&\mB&\mB&\pD&   &\mB&   \\
\AA0\1100\1\ZA01\1001$_4$\ZZ&   &  &  &  &  &  &  &  &  &   &   &   &   &   &   &\m1&   &   &   &   &   &\mB&\mB&   &   &\mB&   &   &\m1&\p1&\p1&   &\pD&\p1&   \\
\AA0\1100\1\ZA01\1001$_6$\ZZ&   &  &  &  &  &  &  &  & 1&   &   &\pB&   &   &\pB&\m1&   &   &   &   &   &   &   &   &   &   &\pB&   &\m1&\m1&\pC&   &   &\pC&   \\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA00\1100\ZA001\100$_1$\ZZ&    &  &  &  &  & 1&  &  &  &\mB&\pB&   &   &   &   &\pB&   &\pB&   &\pB&   &   &   &   &   &   &   &   &   &   &\mB&\mB&   &   &\pD\\
\AA00\1100\ZA001\100$_2$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\mB&\pB&\pB&   &\mB&\pB&   &   &\mB&   &   &   &   &   &   &   &   &   &   &\pD&\pD&   &\pD&   \\
\AA00\1100\ZA001\100$_3$\ZZ&    & 1&  &  &  &  &  & 1&  &   &   &   &   &   &\mB&\pB&\mB&   &\pB&\pB&   &   &   &   &   &   &   &   &   &   &\mB&\mF&   &   &   \\
\AA00\1100\ZA001\100$_4$\ZZ&    &  &  & 1&  &  &  &  &  &   &   &   &   &   &   &\m1&   &   &\mB&\m1&   &   &\mB&   &   &   &   &   &   &   &\p1&\p1&\mD&   &   \\
\AA00\1100\ZA001\100$_6$\ZZ&    &  &  &  &  &  &  &  & 1&   &   &   &   &   &\pB&\m1&   &   &   &\m1&   &   &   &   &\pB&   &\mB&   &   &\pD&\pC&\pC&   &   &   \\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA000\110\ZA0001\10$_1$\ZZ&    &  &  &  &  &  &  &  &  &\pB&\pB&   &   &   &   &   &   &   &   &\pB&\pB&   &   &\pB&   &   &   &   &   &   &   &\mB&\mB&   &\mD\\
\AA000\110\ZA0001\10$_2$\ZZ&    &  &  &  &  &  &  &  &  &   &   &\pB&   &\pB&   &   &   &   &   &\mB&   &\pB&   &\mB&   &   &   &   &   &   &   &\pD&\pD&\mD&   \\
\AA000\110\ZA0001\10$_3$\ZZ&    & 1&  &  &  &  &  &  &  &   &   &   &   &   &\pB&   &\pB&   &\pB&\pB&   &   &\pB&\pB&   &   &   &   &   &   &\mD&\mF&\mF&   &   \\
\AA000\110\ZA0001\10$_4$\ZZ&    &  &  & 1&  &  &  &  &  &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &\m1&   &   &   &   &   &   &   &\p1&\pE&   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &\m1&\pB&\pB&\pB&   &   &\mD&   &\pC&\pC&   &   \\
%                                                                                                                                                                .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                                    %.
\AA0000\10\ZA000010$_1$\ZZ &    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   &\pB&   &   &   &   &   &   &   &   &\mB&   &\pD\\
\AA0000\10\ZA000010$_2$\ZZ &    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &\mB&   &   &   &   &   &   &   &   &\pD&\pD&   \\
\AA0000\10\ZA000010$_3$\ZZ &    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\pB&   &   &   &   &   &   &\pD&   &\mF&   &   \\
\AA0000\10\ZA000010$_4$\ZZ &    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &\m1&   &   &   &   &   &   &   &\pD&\pE&   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                                %.
\AA0000\10\ZA000010$_6$\ZZ &    &  &  &  &  &  &  &  &  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &\pB&   &   &   &\pD&   &   &\pC&   &   \\
%*                                                                                                                                                          .
\hline
%ddd                          7   8  9  0  1  2  3  4  5*  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0*  1   2   3   4   5   6   
%                         ---------------------------------------------------------------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &
                                         $\sqrt{ 12}$&    \pC     &$\sqrt{ 24}$&$\sqrt{ 18}$&
                            $\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 12}$&$\sqrt{ 18}$&$\sqrt{ 27}$&
%
      \multicolumn{20}{|c|}{ $\sqrt{ 72}$ each state} &
%                            $\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&
%                            $\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&
%                            $\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&
%                            $\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&$\sqrt{ 72}$&
%
      \multicolumn{6}{|c|}{ $\sqrt{360}$ each state} \\
%                            $\sqrt{360}$&$\sqrt{360}$&$\sqrt{360}$&$\sqrt{360}$&$\sqrt{360}$&
%                            $\sqrt{360} \:$ \\
%
%
%
%                            $\sqrt{   }$&$\sqrt{   }$&$\sqrt{   }$&$\sqrt{   }$&$\sqrt{   }$&
%
\hline
\end{tabular}
\end{table}




\clearpage

%
%  end of the fourth table 
%




%******************************************************************************************************%
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%                                                                                                      %
%                                     CGG  for   27 x 351prime_bar                                     %
%                                                                                                      %
%   * * *          * * *          * * *          * * *          * * *          * * *          * * *    %
%******************************************************************************************************%



\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (001000) dominant weight in (100000)$\otimes$(200000).}
%                                                         27  x       351prime_bar
%
Each entry should be divided by the respective number in the last row to keep
the states normalized to 1.
}
\label{t:351prime_barcgc001000}
\footnotesize
%\tiny
%                 1  2  34
%\begin{tabular}{|c||c||cc|}                 THERE ONLY IS 1 STATE IN 3003,  AND THERE IS NO 2925 IN THIS PRODUCT
\begin{tabular}{|@{\hspace{0.5mm}}c          
                |c@{$\,$}
                |c@{$\,$}c@{$\,$}|}
\hline
\mbox{ } & {\mbox{ }}                  & \multicolumn{2}{|c|}{\mbox{ }}                  \\
\mbox{ } & {\normalsize \em (300000) } & \multicolumn{2}{ c|}{\normalsize \em (110000) } \\
\mbox{ } & {\mbox{ }}                  & \multicolumn{2}{|c|}{\mbox{ }}                  \\
\cline{2-4}
 & 
   $\,\mid$001000$    \rangle$ & 
   $\,\mid$001000$_{1}\rangle$ & 
   $\,\mid$001000$_{2}\rangle$ \\ 
\hline
%                         & 3003 state       1         2        
%                         -----------------------------------------------------------------
\AA0\11000\MM010000\ZZ    &        1    &    $\:$2   &   $\:$1     \\
\AA\110000\MM1\11000\ZZ   &        1    &   -1       &   $\:$1     \\
\AA100000\MM\101000\ZZ    &        1    &   -1       &    -2       \\
\hline
                          & $\sqrt{3}$  & $\sqrt{6}$ &  $\sqrt{6}$     \\
\hline
\end{tabular}
\end{table}







\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (100010) dominant weight in (100000)$\otimes$(200000).}
\AA n\ZZ\ is an abbreviation for \AA100010$_n$\ZZ. Numbering of the degenerate
states is consistent with tables \ref{t:paths_3003} and \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc100010}
%\small
\footnotesize
%\tiny 
%                 12  3456  78901234  5
%\begin{tabular}{|cc||rrrr||rrrrrrrr||r|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{ 4}{|c|}{\mbox{ }} & 
      \multicolumn{ 8}{|c|}{\mbox{ }} & {\mbox{ }} \\
 & &  \multicolumn{ 4}{|c|}{\normalsize \em (300000) } &
      \multicolumn{ 8}{|c|}{\normalsize \em (110000) } &
                           {\normalsize \em (100010) } \\
 & &  \multicolumn{ 4}{|c|}{\mbox{ }} & 
      \multicolumn{ 8}{|c|}{\mbox{ }} & {\mbox{ }} \\
\cline{3-15}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ$\:$ & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& 
   $\,\mid$100010$\rangle\;\;$ \\ 
\hline
%                              1   2   3   4*  1   2   3   4   5   6   7   8*   1
%                          ---------------------------------------------------------------------------------------
          &                &     &   &   &   &   &   &   &   &   &   &   &   &                 \\
\AA100000\ZA000010$_1$\ZZ  &     &   &   &\pB&\qQ&   &   &   &   &   &   &   &  $\sqrt{32}\;\;$\\
\AA100000\ZA000010$_2$\ZZ  &     &   &\sQ&   &   &\qQ&\qW&   &   &   &   &   & -$\sqrt{18}\;\;$\\
\AA100000\ZA000010$_3$\ZZ  &     &\sQ&   &   &   &   &   &   &   &   &\qQ&\qW&  $\sqrt{ 8}\;\;$\\
\AA100000\ZA000010$_6$\ZZ  & \sQ &   &   &   &   &   &   &\qW&\qQ&\qQ&   &   & -$\sqrt{ 2}\;\;$\\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &                 \\
\AA\110000\ZA2\10010\ZZ    &     &   &\p1&\sQ&\m1&\m1&\p1&   &   &   &   &   &  -5$\;\;\;\;$   \\
\AA0\11000\ZA11\1010\ZZ    &     &\p1&\p1&   &   &\m1&\p1&   &   &   &\m1&\p1&   5$\;\;\;\;$   \\
\AA00\1101\ZA101\11\1\ZZ   & \p1 &\p1&   &   &   &   &   &\p1&\m1&\m1&\m1&\p1&  -5$\;\;\;\;$   \\
\AA000\111\ZA10010\1\ZZ    & \p1 &   &   &   &   &   &   &\p1&\pB&\m1&   &   &   5$\;\;\;\;$   \\
\AA00010\1\ZA100\111\ZZ    & \p1 &   &   &   &   &   &   &\p1&\m1&\pB&   &   &   5$\;\;\;\;$   \\
\AA001\11\1\ZA10\1101\ZZ   & \p1 &\p1&   &   &   &   &   &\p1&\pB&\pB&\pB&\p1&  -5$\;\;\;\;$   \\
\AA01\1010\ZA1\11000\ZZ    &     &\p1&\p1&   &   &\pB&\p1&   &   &   &\pB&\p1&   5$\;\;\;\;$   \\
\AA1\10010\ZA010000\ZZ     &     &   &\p1&\sQ&\pB&\pB&\p1&   &   &   &   &   &  -5$\;\;\;\;$   \\
\AA\100010\ZA200000\ZZ     &     &   &   &\p1&\sQ&   &   &   &   &   &   &   &  $\sqrt{50}\;\;$\\
          &                &     &   &   &   &   &   &   &   &   &   &   &   &                 \\
\hline			    
       &       & $\sqrt{ 6}$& $\sqrt{ 6}$& $\sqrt{ 6}$&       \pC  &       \pC  & 
                 $\sqrt{12}$& $\sqrt{12}$& $\sqrt{12}$& $\sqrt{12}$& $\sqrt{12}$&
                 $\sqrt{12}$& $\sqrt{12}$& 
%
                 $\sqrt{270}\;\;$  \\
\hline
\end{tabular}
\end{table}


\clearpage




\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000001) dominant weight states of the 3003-dimensional (300000) irrep
and 650-dimensional (100010) irrep in (100000)$\otimes$(200000).}
\AA n\ZZ\ is an abbreviation for \AA000001$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_3003}, and table I in \cite{abI}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc000001}
%\footnotesize
\scriptsize
%\tiny
%                 12  3456789012  34567  
%\begin{tabular}{|cc||rrrrrrrrrr||rrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
 & &  \multicolumn{10}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (300000) } &
      \multicolumn{ 5}{|c|}{\normalsize \em (100010) $\;$ } \\
\cline{3-17}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ \\
\hline
%*                                                                                                                             .
%                                1   2   3   4   5   6   7   8   9   0*      1          2          3         4          5 
%                             --------------------------------------------------------------------------------------------------
%                                                                                                                              .
%          &                   &   &   &   &   &   &   &   &   &   &   &          &          &          &          &          \\
%                                                                                                                              .
\rule[0cm]{0mm}{0.4cm}$\!\!$                                                               
\AA100000\ZA\100001$_1$\ZZ     &   &   &   &   &   &   &   &   &   &\sQ&$ \sqrt{32}$&$ \sqrt{50}$&            &            &            \\
\AA100000\ZA\100001$_2$\ZZ     &   &   &   &   &   &   &   &\sQ&   &   &$-\sqrt{18}$&            &$ \sqrt{50}$&            &            \\
\AA100000\ZA\100001$_3$\ZZ     &   &   &   &   &   &\sQ&   &   &   &   &     \sW    &            &            &$ \sqrt{50}$&            \\
\AA100000\ZA\100001$_6$\ZZ     &   &   &   &\sQ&   &   &   &   &   &   &     \qQ    &            &            &            &$ \sqrt{50}$\\
%                                                                                           .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                         
\AA\110000\ZA1\10001$_1$\ZZ    &   &   &   &   &   &   &   &   &   &\sQ&$ \sqrt{32}$&     \qQ    &            &            &            \\
\AA\110000\ZA1\10001$_2$\ZZ    &   &   &   &   &   &   &   &\sQ&\sQ&\sQ&$-\sqrt{18}$&$-\sqrt{18}$&$-\sqrt{50}$&            &            \\
\AA\110000\ZA1\10001$_3$\ZZ    &   &   &   &   &\sQ&\sQ&   &   &   &   &     \sW    &     \sW    &            &$-\sqrt{50}$&            \\
\AA\110000\ZA1\10001$_6$\ZZ    &   &   &\sQ&\sQ&   &   &   &   &   &   &     \qQ    &     \qQ    &            &            &$-\sqrt{50}$\\
%                                                                                          
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                         
\AA0\11000\ZA01\1001$_1$\ZZ    &   &   &   &   &   &   &   &\sQ&   &\sQ&$-\sqrt{50}$&     \qQ    &     \qQ    &            &            \\
\AA0\11000\ZA01\1001$_2$\ZZ    &   &   &   &   &   &   &   &   &\sQ&\sQ&            &$-\sqrt{18}$&     \sW    &            &            \\
\AA0\11000\ZA01\1001$_3$\ZZ    &   &   &   &   &\sQ&   &\sQ&   &\sQ&   &            &     \sW    &     \sW    &$ \sqrt{50}$&            \\
\AA0\11000\ZA01\1001$_6$\ZZ    &   &\sQ&\sQ&   &   &   &   &   &   &   &            &     \qQ    &     \qQ    &            &$ \sqrt{50}$\\
%                                                                                          
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                         
\AA00\1101\ZA001\100$_1$\ZZ    &   &   &   &   &   &\sQ&   &\sQ&   &   &$ \sqrt{50}$&            &    \qQ     &     \qQ    &            \\
\AA00\1101\ZA001\100$_2$\ZZ    &   &   &   &   &\sQ&   &   &   &\sQ&\sQ&            &$ \sqrt{50}$&    \sW     &     \sW    &            \\
\AA00\1101\ZA001\100$_3$\ZZ    &   &   &   &   &   &   &\sQ&   &\sQ&   &            &            &    \sW     &$-\sqrt{18}$&            \\
\AA00\1101\ZA001\100$_6$\ZZ    &\sQ&\sQ&   &   &   &   &\sQ&   &   &   &            &            &    \qQ     &     \qQ    &$-\sqrt{50}$\\
%                                                                                          
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                         
\AA000\111\ZA0001\10$_1$\ZZ    &   &   &   &\sQ&   &\sQ&   &   &   &   &$-\sqrt{50}$&            &            &     \qQ    &     \qQ    \\
\AA000\111\ZA0001\10$_2$\ZZ    &   &   &\sQ&   &\sQ&   &   &   &   &   &            &$-\sqrt{50}$&            &     \sW    &     \sW    \\
\AA000\111\ZA0001\10$_3$\ZZ    &   &\sQ&   &   &   &   &\sQ&   &\sQ&   &            &            &$-\sqrt{50}$&$-\sqrt{18}$&$-\sqrt{18}$\\
\AA000\111\ZA0001\10$_6$\ZZ    &\sQ&   &   &   &   &   &\sQ&   &   &   &            &            &            &     \qQ    &$ \sqrt{32}$\\
%                                                                                          
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                         
\AA0000\11\ZA000010$_1$\ZZ     &   &   &   &\sQ&   &   &   &   &   &   &$ \sqrt{50}$&            &            &            &     \qQ    \\
\AA0000\11\ZA000010$_2$\ZZ     &   &   &\sQ&   &   &   &   &   &   &   &            &$ \sqrt{50}$&            &            &     \sW    \\
\AA0000\11\ZA000010$_3$\ZZ     &   &\sQ&   &   &   &   &   &   &   &   &            &            &$ \sqrt{50}$&            &$-\sqrt{18}$\\
\AA0000\11\ZA000010$_6$\ZZ     &\sQ&   &   &   &   &   &\sQ&   &   &   &            &            &            &$ \sqrt{50}$&$ \sqrt{32}$\\
%*                                                                                         
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                               %.
%                                1   2   3   4   5   6   7   8   9   0*  1   2   3   4   5  
%                             ---------------------------------------------------------------
\AA0010\1\1\ZA00\1012\ZZ       &\p1&\p1&   &   &   &   &\pB&   &\p1&   &   &   &\mE&-10&\mE\\
\AA001\11\1\ZA00\11\12\ZZ      &\p1&\p1&   &   &   &   &\p1&   &   &   &   &   &\pE&\pE&\mE\\
\AA00010\1\ZA000\102\ZZ        &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &\pE& 10\\
\AA01\1010\ZA0\110\11\ZZ       &   &\p1&\p1&   &   &   &   &   &   &   &   &\pE&\pE&   &\pE\\
\AA1\10010\ZA\1100\11\ZZ       &   &   &\p1&\p1&   &   &   &   &   &   &\pE&\pE&   &   &\mE\\
\AA01\11\10\ZA0\11\111\ZZ      &   &\p1&\p1&   &\p1&   &\p1&   &\p1&   &   &\mE&\mE&\pE&\pE\\
\AA010\100\ZA0\10101\ZZ        &   &   &   &   &\p1&   &\p1&   &\pB&\p1&   &\pE& 10&\pE&   \\
\AA1\101\10\ZA\110\111\ZZ      &   &   &\p1&\p1&\p1&\p1&   &   &   &   &\mE&\mE&   &\mE&\mE\\
\AA\100010\ZA1000\11\ZZ        &   &   &   &\p1&   &   &   &   &   &   &\pE&   &   &   &\pE\\
\AA1\11\100\ZA\11\1101\ZZ      &   &   &   &   &\p1&\p1&   &\p1&\p1&\p1&\pE&\pE&\mE&\mE&   \\
\AA\1001\10\ZA100\111\ZZ       &   &   &   &\p1&   &\p1&   &   &   &   &\mE&   &   &\pE&\pE\\
\AA10\1001\ZA\101000\ZZ        &   &   &   &   &   &   &   &\p1&\p1&\pB&\mE&-10&\mE&   &   \\
\AA\101\100\ZA10\1101\ZZ       &   &   &   &   &   &\p1&   &\p1&   &   &\pE&   &\pE&\pE&   \\
\AA\11\1001\ZA1\11000\ZZ       &   &   &   &   &   &   &   &\p1&   &\p1&\mE&\pE&\pE&   &   \\
\rule[-0.25cm]{0mm}{0.5cm}$\!\!$                                                                       %.
\AA0\10001\ZA010000\ZZ         &   &   &   &   &   &   &   &   &   &\p1& 10&\pE&   &   &   \\
%                                                                                                                             .
\hline
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &    \pC     &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &
                            $\sqrt{12}$ &$\sqrt{24}$ &$\sqrt{12}$ &$\sqrt{24}$ &$\sqrt{24}$ &
%
                            $\sqrt{540}$&$\sqrt{540}$&$\sqrt{540}$&$\sqrt{540}$&$\sqrt{540}$\\
\hline
\end{tabular}
\end{table}

\clearpage



\protect
\begin{table}
\caption{ 
{\bf CG coefficients for (000001) dominant weight states of the 5824-dimensional 
(110000) irrep in (100000)$\otimes$(200000).}
\AA n\ZZ\ is an abbreviation for \AA000001$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc000001part2}
%\small
%\footnotesize
\scriptsize
%\tiny
%                 12  345678901234567890123456  (24 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{1.0mm}}|}
\hline
 & &  \multicolumn{24}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-26}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
    $\!\!$\BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ$\:$ \\
\hline
%
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   
%                         ----------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.4cm}$\!\!$                                                                                                %.
\AA100000\ZA\100001$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\qW&   &   \\
\AA100000\ZA\100001$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\qW&   &   &   &   &   &   \\
\AA100000\ZA\100001$_3$\ZZ &   &   &   &   &   &   &   &   &   &\m1&\qW&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100001$_6$\ZZ &   &   &   &   &   &   &\qW&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ$\:$ \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA\110000\ZA1\10001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\sQ&   &   \\
\AA\110000\ZA1\10001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\qW&\m1&\qW&   &\qW&   &   \\
\AA\110000\ZA1\10001$_3$\ZZ&   &   &   &   &   &   &   &   &   &\m1&\qW&\m1&\qW&   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10001$_6$\ZZ&   &   &   &   &\qW&\m1&\qW&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\qQ&\qQ$\:$ \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA0\11000\ZA01\1001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\sQ&   &   &\m1&\sQ&   &   \\
\AA0\11000\ZA01\1001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\sQ&   &\qW&   &   \\
\AA0\11000\ZA01\1001$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\m1&\qW&\m1&\qW&   &   &   &   &\qW&   &   &   &   \\
\AA0\11000\ZA01\1001$_6$\ZZ&   &   &\qW&\m1&\qW&\m1&   &   &   &   &   &   &   &   &   &\qQ&   &   &   &   &   &   &\qQ&   \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA00\1101\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &\m1&\sQ&   &   &   &   &   &\m1&\sQ&   &   &   &   &   &   \\
\AA00\1101\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\m1&\sQ&   &   &   &   &   &\m1&\sQ&   &\sQ&   &   \\
\AA00\1101\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\sQ&   &   &   &   &\qW&   &   &   &   \\
\AA00\1101\ZA001\100$_6$\ZZ&\qW&\qQ&\qW&\m1&   &   &   &   &\m1&   &   &   &   &   &\qW&\qQ&   &   &   &   &   &   &   &   \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA000\111\ZA0001\10$_1$\ZZ&   &   &   &   &   &   &\sQ&\m1&   &\m1&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &\qQ$\:$ \\
\AA000\111\ZA0001\10$_2$\ZZ&   &   &   &   &\sQ&\m1&   &   &   &   &   &\m1&\sQ&   &   &   &   &   &   &   &   &   &\qQ&   \\
\AA000\111\ZA0001\10$_3$\ZZ&   &   &\sQ&\m1&   &   &   &   &   &   &   &   &   &\m1&\sQ&\qQ&   &   &   &\sQ&   &   &   &   \\
\AA000\111\ZA0001\10$_6$\ZZ&\sQ&\qQ&   &   &   &   &   &   &\m1&   &   &   &   &   &\qW&   &   &   &   &   &   &   &   &   \\
%                                                                                                                            .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
\AA0000\11\ZA000010$_1$\ZZ &   &   &   &   &   &   &\sQ&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sW$\:$ \\
\AA0000\11\ZA000010$_2$\ZZ &   &   &   &   &\sQ&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sW&   \\
\AA0000\11\ZA000010$_3$\ZZ &   &   &\sQ&\m1&   &   &   &   &   &   &   &   &   &   &   &\sW&   &   &   &   &   &   &   &   \\
\AA0000\11\ZA000010$_6$\ZZ &\sQ&\sW&   &   &   &   &   &   &\m1&   &   &   &   &   &\sQ&   &   &   &   &   &   &   &   &   \\
%*                                                                                                                           .
\rule[0cm]{0mm}{0.5cm}$\!\!$                                                                                                %.
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   
\AA0010\1\1\ZA00\1012\ZZ   &\p1&\pB&\p1&\sQ&   &   &   &   &\sQ&   &   &   &   &\sQ&\pB&\pB&   &   &   &\p1&   &   &   &   \\
\AA001\11\1\ZA00\11\12\ZZ  &\p1&\m1&\p1&\sQ&   &   &   &   &\sQ&   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\102\ZZ    &\mB&\m1&   &   &   &   &   &   &\sQ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\11\ZZ   &   &   &\p1&\sQ&\p1&\sQ&   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &\m1&   \\
\AA1\10010\ZA\1100\11\ZZ   &   &   &   &   &\p1&\sQ&\p1&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1$\:$ \\
\AA01\11\10\ZA0\11\111\ZZ  &   &   &\p1&\sQ&\p1&\sQ&   &   &   &   &   &\sQ&\p1&\sQ&\p1&\pB&   &   &   &\p1&   &   &\pB&   \\
\AA010\100\ZA0\10101\ZZ    &   &   &   &   &   &   &   &   &   &   &   &\sQ&\p1&\sQ&\p1&   &   &   &\sQ&\pB&   &\p1&   &   \\
\AA1\101\10\ZA\110\111\ZZ  &   &   &   &   &\p1&\sQ&\p1&\sQ&   &\sQ&\p1&\sQ&\p1&   &   &   &   &   &   &   &   &   &\pB&\pB\\
\AA\100010\ZA1000\11\ZZ    &   &   &   &   &   &   &\p1&\sQ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1$\:$ \\
\AA1\11\100\ZA\11\1101\ZZ  &   &   &   &   &   &   &   &   &   &\sQ&\p1&\sQ&\p1&   &   &   &\sQ&\p1&\sQ&\p1&   &\p1&   &   \\
\AA\1001\10\ZA100\111\ZZ   &   &   &   &   &   &   &\p1&\sQ&   &\sQ&\p1&   &   &   &   &   &   &   &   &   &   &   &   &\pB\\
\AA10\1001\ZA\101000\ZZ    &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\p1&\sQ&\p1&\sQ&\pB&   &   \\
\AA\101\100\ZA10\1101\ZZ   &   &   &   &   &   &   &   &   &   &\sQ&\p1&   &   &   &   &   &\sQ&\p1&   &   &   &   &   &   \\
\AA\11\1001\ZA1\11000\ZZ   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\p1&   &   &\sQ&\p1&   &   \\
\rule[-0.25cm]{0mm}{0.5cm}$\!\!$                                                                                               %.
\AA0\10001\ZA010000\ZZ     &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\sQ&\p1&   &   \\
\hline
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &$\sqrt{18}$&$\sqrt{18}$&$\sqrt{24}$&$\sqrt{12}$&$\sqrt{24}$&
                            $\sqrt{12}$&$\sqrt{24}$&$\sqrt{12}$&   \pC     &$\sqrt{12}$&
                            $\sqrt{24}$&$\sqrt{12}$&$\sqrt{24}$&   \pC     &$\sqrt{30}$&
                            $\sqrt{24}$&$\sqrt{12}$&$\sqrt{24}$&   \pC     &$\sqrt{30}$&
                                \pC    &$\sqrt{30}$&$\sqrt{24}$&$\sqrt{24} \:$ \\
\hline
\end{tabular}
\end{table}




\clearpage



%mmmmm

\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the (000000) dominant weight states of the 3003-dimensional
(300000) irrep in the product (100000)$\otimes$(200000).} 
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_3003}.
Each CGC should be divided by the respective number in the last row of the table to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc000000part1}
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  345678901234567890123456  (26 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
 & &  \multicolumn{24}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (300000) } \\
\cline{3-26}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ$\:$ \\
\hline
%
%351prime_aaa                1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4 
%                         -------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                %.
\AA100000\ZA\100000$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA100000\ZA\100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   \\
\AA100000\ZA\100000$_6$\ZZ &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA\110000\ZA1\10000$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   \\
\AA\110000\ZA1\10000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   \\
\AA\110000\ZA1\10000$_6$\ZZ&   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA0\11000\ZA01\1000$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   \\
\AA0\11000\ZA01\1000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1\\
\AA0\11000\ZA01\1000$_6$\ZZ&   &   &   &   &   &\p1&\p1&   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA00\1101\ZA001\10\1$_1$\ZZ&  &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_2$\ZZ&  &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   \\
\AA00\1101\ZA001\10\1$_3$\ZZ&  &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_6$\ZZ&  &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&\p1\\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA000\111\ZA0001\1\1$_1$\ZZ&  &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_2$\ZZ&  &   &\p1&\pE&   &   &   &   &\p1&   &   &   &   &   &   & 11&   &   &   &   &   &\p1&   &\p1\\
\AA000\111\ZA0001\1\1$_3$\ZZ&\p1&  &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_6$\ZZ&  &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA00010\1\ZA000\101$_1$\ZZ&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_2$\ZZ&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_3$\ZZ&   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_6$\ZZ&   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1\\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA0000\11\ZA00001\1$_1$\ZZ&   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_2$\ZZ&   &   &\p1&\pE&   &   &   &   &   &   &   &   &   &   &   & 11&   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_3$\ZZ&\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_6$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA001\11\1\ZA00\11\11$_1$\ZZ& &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_2$\ZZ& &   &\p1&\pE&   &   &   &   &\p1&   &   &   &   &   &   & 10&   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_3$\ZZ&\p1& &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &\p1\\
\AA001\11\1\ZA00\11\11$_6$\ZZ& &\p1&   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&  &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &\p1\\
\AA0010\1\1\ZA00\1011$_2$\ZZ&  &   &\p1&\pE&   &   &   &   &   &   &   &   &   &   &   & 10&   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_3$\ZZ&\p1&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&  &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA01\1010\ZA0\110\10$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_2$\ZZ&\p1&  &\p1&\pE&   &   &   &   &   &   &   &   &   &   &   & 10&   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1\\
\AA01\1010\ZA0\110\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &\p1&\p1&   &   &   &   &\pC&\pB&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA01\11\10\ZA0\11\110$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&\p1&   &   &   &   &\p1&   &   \\
\AA01\11\10\ZA0\11\110$_2$\ZZ&\p1& &\p1&\pE&\p1&   &   &   &\p1&   &   &   &   &   &   & 10&   &   &   &   &   &   &   &\p1\\
\AA01\11\10\ZA0\11\110$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_6$\ZZ& &   &   &   &   &   &   &   &   &   &\p1&   &\p1&\p1&   &   &   &\pB&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA1\10010\ZA\1100\10$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_2$\ZZ&\p1&\p1&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\pC&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA010\100\ZA0\10100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &\p1&   &\p1\\
\AA010\100\ZA0\10100$_2$\ZZ&   &   &   &   &\p1&   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                           %.
\AA010\100\ZA0\10100$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\pB&   &\p1&   &   &   &   \\
%                                                                                                                                                           .
\hline
\multicolumn{10}{|c }{ } &
\multicolumn{16}{ r|}{\small \em continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
%              \caption{ 
%              {\bf CG coefficients for the (000000) dominant weight states of the 3003-dimensional
%              (300000) irrep in the product (100000)$\otimes$(200000).} 
%              \AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
%              states is consistent with table \ref{t:paths_3003}.
%              Each CGC should be divided by the respective number in the last row of the table to maintain
%              $\langle n\!\mid\! n\rangle = 1$.
%              }
%              \label{t:351prime_barcgc000000part1}
\tiny
%                 12  345678901234567890123456  (26 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
\multicolumn{26}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{24}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (300000) } \\
\cline{3-26}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ$\:$ \\
\hline
%
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4 
%                         -------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                               %.
\AA1\101\10\ZA\110\110$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &\p1&   &   &   &\p1&   &   &   \\
\AA1\101\10\ZA\110\110$_2$\ZZ&\p1&\p1& &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &\p1&   &\p1&   &   &   &\p1\\
\AA1\101\10\ZA\110\110$_6$\ZZ& &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\pB&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA\100010\ZA1000\10$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_2$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\pB&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA1\11\100\ZA\11\1100$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &\p1&   &\p1&   \\
\AA1\11\100\ZA\11\1100$_2$\ZZ& &   &   &   &\p1&   &\p1&\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA1\11\100\ZA\11\1100$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   \\
\AA1\11\100\ZA\11\1100$_6$\ZZ& &   &   &   &   &   &   &   &   &   &\p1&   &\p1&\p1&   &   &   &\pB&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA\1001\10\ZA100\110$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   \\
\AA\1001\10\ZA100\110$_2$\ZZ&  &\p1&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   &   \\
\AA\1001\10\ZA100\110$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &\pC&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA10\1001\ZA\10100\1$_1$\ZZ&  &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &\p1&   \\
\AA10\1001\ZA\10100\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA10\1001\ZA\10100\1$_3$\ZZ&  &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   \\
\AA10\1001\ZA\10100\1$_6$\ZZ&  &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA\101\100\ZA10\1100$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   \\
\AA\101\100\ZA10\1100$_2$\ZZ&  &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1\\
\AA\101\100\ZA10\1100$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &\pB&\p1&   &   &   &\pC&   &\p1&   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA10000\1\ZA\100001$_1$\ZZ&   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
\AA10000\1\ZA\100001$_2$\ZZ&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   \\
\AA10000\1\ZA\100001$_3$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   \\
\AA10000\1\ZA\100001$_6$\ZZ&   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ& &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   \\
\AA\11\1001\ZA1\1100\1$_2$\ZZ& &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_3$\ZZ& &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &\p1\\
\AA\11\1001\ZA1\1100\1$_6$\ZZ&\p1&\p1& &   &   &   &   &   &   &   &   &   &\p1&\p1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA\11000\1\ZA1\10001$_1$\ZZ&  &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_2$\ZZ&  &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&\p1\\
\AA\11000\1\ZA1\10001$_3$\ZZ&  &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   &   &   \\
\AA\11000\1\ZA1\10001$_6$\ZZ&\p1&\p1&  &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA0\10001\ZA01000\1$_1$\ZZ&   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
\AA0\10001\ZA01000\1$_2$\ZZ&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_3$\ZZ&   &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_6$\ZZ&\p1&   &\p1&\pE&   &   &   &   &   &   &   &   &\p1&   &   & 10&   &   &   &   &   &   &   &\p1\\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&  &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &\p1&   \\
\AA0\1100\1\ZA01\1001$_2$\ZZ&  &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_3$\ZZ&  &   &   &   &\p1&   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &\p1\\
\AA0\1100\1\ZA01\1001$_6$\ZZ&\p1&  &\p1&\pE&   &   &   &   &   &   &   &\p1&   &   &   & 11&   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA00\1100\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   \\
\AA00\1100\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   \\
\AA00\1100\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &\p1\\
\AA00\1100\ZA001\100$_6$\ZZ&   &   &\p1&\pF&   &   &   &   &\p1&   &   &   &   &   &   & 11&   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA000\110\ZA0001\10$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   &   \\
\AA000\110\ZA0001\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &\p1&   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&   &   &   &\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                               %.
\AA0000\10\ZA000010$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                           %.
\AA0000\10\ZA000010$_6$\ZZ &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                                                          .
\hline
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4 
%351prime_aaa              -------------------------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &$\sqrt{12}$ &$\sqrt{12}$ &    \pC     &$\sqrt{264}$&$\sqrt{12}$ &
                                \pC     &$\sqrt{12}$ &$\sqrt{12}$ &$\sqrt{12}$ &    \pC     &
                                \pC     &$\sqrt{12}$ &$\sqrt{24}$ &$\sqrt{12}$ &$\sqrt{12}$ &
                           $\sqrt{1032}$&$\sqrt{12}$ &$\sqrt{72}$ &$\sqrt{24}$ &$\sqrt{24}$ &
                            $\sqrt{12}$ &    \pC     &$\sqrt{12}$ &$\sqrt{18} \:$ \\
\hline
\end{tabular}
\end{table}




\clearpage

%
%  end of the first table 
%

\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the first 32 (000000) dominant weight states of the 5824-dimensional
(110000) irrep in the product (100000)$\otimes$(200000).} (The remaining 32 states of this
irrep with the same weight are shown in table \ref{t:351prime_barcgc000000part3}.)
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row of the table to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc000000part2}
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ$\:$ \\
\hline
%
%351prime_bbb                1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2 
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                              %.
\AA100000\ZA\100000$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &\mB&   &   &\mB&   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\110000\ZA1\10000$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   \\
\AA\110000\ZA1\10000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &\mB&   &\mB&\mC&   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\11000\ZA01\1000$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   \\
\AA0\11000\ZA01\1000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &\mB&   &\mB&\mC&   &   &   &   &   &   &\mB&\mB&   &\m1&   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1101\ZA001\10\1$_1$\ZZ&  &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_3$\ZZ&  &   &   &   &   &   &   &   &\m1&\mB&   &   &\mB&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &\m1&   &\mB&\m1&   &\m1&   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\111\ZA0001\1\1$_1$\ZZ&  &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_2$\ZZ&  &   &   &   &\m1&   &\mE&\mD&   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &\mB&   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_3$\ZZ&\m1&\mB&  &   &   &   &   &   &\m1&\mB&   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &-11&\mG&   &   \\
\AA000\111\ZA0001\1\1$_6$\ZZ&  &   &\m1&\mB&   &\m1&   &   &   &   &   &   &   &   &\m1&\mB&   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00010\1\ZA000\101$_1$\ZZ&   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_3$\ZZ&   &   &   &   &   &   &   &   &\m1&\mB&   &   &\mB&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &\m1&   &\mB&\m1&   &\m1&   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\11\ZA00001\1$_1$\ZZ&   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_2$\ZZ&   &   &   &   &\m1&   &\mE&\mD&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_3$\ZZ&\m1&\mB&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &-11&\mG&   &   \\
\AA0000\11\ZA00001\1$_6$\ZZ&   &   &\m1&\mB&   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA001\11\1\ZA00\11\11$_1$\ZZ& &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_2$\ZZ& &   &   &   &\m1&   &\mE&\mD&   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &\mB&   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_3$\ZZ&\m1&\mB& &   &   &   &   &   &\m1&\mB&   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\mB&\m1&-10&\mE&   &   \\
\AA001\11\1\ZA00\11\11$_6$\ZZ& &   &\m1&\mB&   &\m1&   &   &   &   &   &   &   &   &\m1&\mB&   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &\m1&\mB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&  &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&  &   &   &   &\m1&   &\mE&\mD&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_3$\ZZ&\m1&\mB&  &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\mB&\m1&-10&\mE&   &   \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&  &   &\m1&\mB&   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\m1&\mB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\1010\ZA0\110\10$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &   \\
\AA01\1010\ZA0\110\10$_2$\ZZ&\m1&\p1&  &   &\m1&   &\mE&\m1&   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &-10&\mE&   &   \\
\AA01\1010\ZA0\110\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&\m1&   &   &   &   \\
\AA01\1010\ZA0\110\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&\m1&   &   &\m1&\mB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\11\10\ZA0\11\110$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &   \\
\AA01\11\10\ZA0\11\110$_2$\ZZ&\m1&\p1& &   &\m1&   &\mE&\m1&\m1&\p1&   &\p1&   &   &   &   &\m1&\p1&   &   &   &   &\p1&   &   &   &   &   &-10&\mE&   &   \\
\AA01\11\10\ZA0\11\110$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&\m1&   &   &   &   \\
\AA01\11\10\ZA0\11\110$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&\m1&   &   &\m1&\mB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\10010\ZA\1100\10$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &   \\
\AA1\10010\ZA\1100\10$_2$\ZZ&\m1&\p1&\m1&\p1&  &\m1&   &   &   &   &   &\mB&   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &-10&\mE&   &   \\
\AA1\10010\ZA\1100\10$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&\m1&   &   &\m1&\p1\\
\AA1\10010\ZA\1100\10$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&\m1&   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA010\100\ZA0\10100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_2$\ZZ&   &   &   &   &   &   &   &   &\m1&\p1&   &   &\p1&\m1&   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA010\100\ZA0\10100$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&\m1&   &   &   &   \\
%                                                                                                                                                           .
\hline
\multicolumn{18}{|c }{ } &
\multicolumn{16}{ r|}{\small \em continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
%              \caption{ 
%              {\bf CG coefficients for (000000) dominant weight in (100000)$\otimes$(200000).}
%              \AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
%              states is consistent with table \ref{t:paths_5824bar}.
%              Each CGC should be divided by the respective number in the last row to maintain
%              $\langle n\!\mid\! n\rangle = 1$.
%              }
%              \label{t:351prime_barcgc000000part2}
\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{\hspace{0.5mm}}|}
%
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{\hspace{0.5mm}}|}
%
\hline
\multicolumn{34}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ& 
          \AA 21\ZZ& \AA 22\ZZ& \AA 23\ZZ& \AA 24\ZZ& \AA 25\ZZ& \AA 26\ZZ& \AA 27\ZZ& \AA 28\ZZ& \AA 29\ZZ& \AA 30\ZZ& 
          \AA 31\ZZ& \AA 32\ZZ$\:$ \\
\hline
%
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2 
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                               %.
\AA1\101\10\ZA\110\110$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &   \\
\AA1\101\10\ZA\110\110$_2$\ZZ&\m1&\p1&\m1&\p1& &\m1&   &   &\m1&\p1&   &\p1&   &   &\m1&\p1&   &   &\m1&\p1&   &   &   &   &   &   &   &   &-10&\mE&   &   \\
\AA1\101\10\ZA\110\110$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\p1&\m1&   &   &\m1&\p1\\
\AA1\101\10\ZA\110\110$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&\m1&   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\100010\ZA1000\10$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_2$\ZZ&   &   &\m1&\p1&   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1\\
\AA\100010\ZA1000\10$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&\pB&   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\11\100\ZA\11\1100$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_2$\ZZ& &   &   &   &   &   &   &   &\m1&\p1&   &   &\p1&\m1&\m1&\p1&   &   &\m1&   &\p1&\m1&   &\m1&   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_3$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_6$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&\m1&   &   &\pB&\p1\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\1001\10\ZA100\110$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_2$\ZZ&  &   &\m1&\p1&   &\pB&   &   &   &   &   &   &   &   &\p1&\m1&   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1\\
\AA\1001\10\ZA100\110$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&\pB&   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10\1001\ZA\10100\1$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &\m1&   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10\1001\ZA\10100\1$_6$\ZZ&  &   &\pB&\p1&   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\pB&\p1\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\101\100\ZA10\1100$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &   &\pB&   &\p1&\m1&   &\pB&   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\pD&\pB&\pD&   &   &\pB&\p1\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10000\1\ZA\100001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &\p1&   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &\m1&   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_6$\ZZ&   &   &\pB&\p1&   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ& &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &\p1&\pB&   &\pB&   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_3$\ZZ& &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &\pB&\p1&   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11\1001\ZA1\1100\1$_6$\ZZ&\pB&\p1&\pB&\p1& &\pB&   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &\pB&\pB&\p1&\pB& 20& 10&\pB&\p1\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11000\1\ZA1\10001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &\p1&\pB&   &\pB&   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_3$\ZZ&  &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &\pB&\p1&   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_6$\ZZ&\pB&\p1&\pB&\p1&  &\pB&   &   &   &   &   &\p1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   & 22& 11&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\10001\ZA01000\1$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_3$\ZZ&   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_6$\ZZ&\pB&\p1&   &   &\pB&   & 10&\pE&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &\pB&\pB&\p1&\pB& 20& 10&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &\p1&   &   &\p1&   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_3$\ZZ&  &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\1100\1\ZA01\1001$_6$\ZZ&\pB&\p1   &   &\pB&   & 10&\pE&   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   & 22  11&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1100\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_6$\ZZ&   &   &   &   &\pB&   & 12&\pF&   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &\p1&   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\110\ZA0001\10$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   \\
\AA000\110\ZA0001\10$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\10\ZA000010$_1$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   \\
\AA0000\10\ZA000010$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA0000\10\ZA000010$_6$\ZZ &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                                                          .
\hline
%                            1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2 
%351prime_bbb              ---------------------------------------------------------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &$\sqrt{24}$ &$\sqrt{24}$ &$\sqrt{24}$ &$\sqrt{24}$  &$\sqrt{18}$ &
                            $\sqrt{24}$ &$\sqrt{528}$&$\sqrt{150}$&$\sqrt{24}$  &$\sqrt{24}$ &
                            $\sqrt{18}$ &$\sqrt{24}$ &$\sqrt{24}$ &$\sqrt{54}$  &$\sqrt{24}$ &
                            $\sqrt{24}$ &$\sqrt{24}$ &$\sqrt{24}$ &$\sqrt{24}$  &$\sqrt{24}$ &
                            $\sqrt{18}$ &$\sqrt{30}$ &$\sqrt{18}$ &$\sqrt{18}$  &$\sqrt{18}$ &
                            $\sqrt{48}$ &$\sqrt{30}$ &$\sqrt{48}$ &$\sqrt{2664}$&$\sqrt{684}$&
                            $\sqrt{24}$ &$\sqrt{24} \:$ \\
\hline
\end{tabular}
\end{table}




\clearpage

%
%  end of the second table 
%

\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the remaining 32 (000000) dominant weight states of the 5824-dimensional
(110000) irrep in the product (100000)$\otimes$(200000).} (The first 32 states of this
irrep with the same weight are shown in table \ref{t:351prime_barcgc000000part2}.)
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table \ref{t:paths_5824bar}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc000000part3}
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
                                r@{}r@{\hspace{0.5mm}}|}
%
%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
%                                r@{$\,$}r@{\hspace{0.5mm}}|}
%
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$                       \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ& \AA 37\ZZ& \AA 38\ZZ& \AA 39\ZZ& \AA 40\ZZ& 
          \AA 41\ZZ& \AA 42\ZZ& \AA 43\ZZ& \AA 44\ZZ& \AA 45\ZZ& \AA 46\ZZ& \AA 47\ZZ& \AA 48\ZZ& \AA 49\ZZ& \AA 50\ZZ& 
          \AA 51\ZZ& \AA 52\ZZ& \AA 53\ZZ& \AA 54\ZZ& \AA 55\ZZ& \AA 56\ZZ& \AA 57\ZZ& \AA 58\ZZ& \AA 59\ZZ& \AA 60\ZZ& 
          \AA 61\ZZ& \AA 62\ZZ& \AA 63\ZZ& \AA 64\ZZ$\:$ \\
\hline
%
%351prime_ccc                3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                              %.
\AA100000\ZA\100000$_1$\ZZ &   &\mB&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &   &   &   &   &   &   \\
\AA100000\ZA\100000$_3$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   \\
\AA100000\ZA\100000$_6$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\110000\ZA1\10000$_1$\ZZ&   &\mB&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &   &   &   &   &   &   \\
\AA\110000\ZA1\10000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &\mB\\
\AA\110000\ZA1\10000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\11000\ZA01\1000$_1$\ZZ&   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   \\
\AA0\11000\ZA01\1000$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\mC\\
\AA0\11000\ZA01\1000$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&   &   &   &   &   &   &   &   &\m1&\mB&\m1&   &   &\mB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1101\ZA001\10\1$_1$\ZZ&  &   &   &   &   &\m1&\mB&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &\mB&   &   &   &   &   &   \\
\AA00\1101\ZA001\10\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &\m1&\mB&   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &\mB&\mC\\
\AA00\1101\ZA001\10\1$_6$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &\m1&\mB&\m1&   &   &\mB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\111\ZA0001\1\1$_1$\ZZ&  &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_2$\ZZ&  &   &   &\mB&   &-11&\mG&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &\mB&   &   &   &   &\mB&\m1\\
\AA000\111\ZA0001\1\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\m1&\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\111\ZA0001\1\1$_6$\ZZ&  &\mB&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00010\1\ZA000\101$_1$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_3$\ZZ&   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &\mB&   &   &   &   &   &   \\
\AA00010\1\ZA000\101$_6$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &\m1&   &\mB&   &   &\mD&\mB&\mC\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\11\ZA00001\1$_1$\ZZ&   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_2$\ZZ&   &   &   &\mB&   &-11&\mG&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\11\ZA00001\1$_6$\ZZ&   &\mB&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA001\11\1\ZA00\11\11$_1$\ZZ& &   &   &   &   &\m1&\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_2$\ZZ& &   &   &\p1&   &-10&\mE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA001\11\1\ZA00\11\11$_3$\ZZ& &   &   &\mB&   &   &   &   &   &\mB&\m1&   &   &\mB&   &   &   &\m1&   &   &   &   &   &   &   &\mB&   &   &   &   &\mB&\m1\\
\AA001\11\1\ZA00\11\11$_6$\ZZ& &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &\m1&   &\m1&   &   &\mH&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&  &   &   &   &   &\m1&\p1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &\p1&   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&  &   &   &\p1&   &-10&\mE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\m1\\
\AA0010\1\1\ZA00\1011$_3$\ZZ&  &   &   &\mB&   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0010\1\1\ZA00\1011$_6$\ZZ&  &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\1010\ZA0\110\10$_1$\ZZ&  &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_2$\ZZ&  &   &   &\p1&   &-10&\mE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA01\1010\ZA0\110\10$_3$\ZZ&  &   &   &\mB&   &   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\mB&\m1\\
\AA01\1010\ZA0\110\10$_6$\ZZ&\mB&  &   &   &\m1&   &   &   &\mB&\mC&\mC&   &   &   &   &   &   &\m1&\mD&\mB&   &   &   &   &\mB&\mB&\m1&   &   &\mH&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA01\11\10\ZA0\11\110$_1$\ZZ& &   &   &   &   &\m1&\p1&   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_2$\ZZ& &   &   &\p1&   &-10&\mE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &\p1&\m1\\
\AA01\11\10\ZA0\11\110$_3$\ZZ& &   &   &\mB&   &   &   &   &   &\p1&\m1&   &   &\p1&   &   &   &\m1&   &   &   &   &   &   &   &\mB&   &   &   &   &   &   \\
\AA01\11\10\ZA0\11\110$_6$\ZZ&\mB& &   &   &\m1&   &   &   &\p1&\m1&\p1&   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &\m1&   &   &   &   &\m1&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\10010\ZA\1100\10$_1$\ZZ&  &\p1&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\10010\ZA\1100\10$_3$\ZZ&  &   &   &\p1&\m1&   &   &   &   &   &   &   &   &\mB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   \\
\AA1\10010\ZA\1100\10$_6$\ZZ&\p1&  &   &   &   &   &   &   &\mB&\mC&\mC&   &   &   &   &   &   &\m1&\p1&\m1&   &   &   &   &\m1&\p1&   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA010\100\ZA0\10100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &\m1&\p1&   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &\p1&   \\
\AA010\100\ZA0\10100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
\AA010\100\ZA0\10100$_3$\ZZ&   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &\mB&   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA010\100\ZA0\10100$_6$\ZZ&\mB&   &   &   &   &   &   &   &\p1&\m1&\p1&   &   &   &   &   &   &   &\mB&\m1&   &   &   &   &\m1&   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\hline
\multicolumn{18}{|c }{ } &
\multicolumn{16}{ r|}{\small \em continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
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%              \label{t:351prime_barcgc000000part3}
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\tiny
%                 12  34567890123456789012345678901234  (32 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}
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%                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
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%
\hline
\multicolumn{34}{|l|}{$\qquad$ \small \em continued from previous page} \\
\hline
 & &  \multicolumn{32}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (110000) } \\
\cline{3-34}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$                       \AA 33\ZZ& \AA 34\ZZ& \AA 35\ZZ& \AA 36\ZZ& \AA 37\ZZ& \AA 38\ZZ& \AA 39\ZZ& \AA 40\ZZ& 
          \AA 41\ZZ& \AA 42\ZZ& \AA 43\ZZ& \AA 44\ZZ& \AA 45\ZZ& \AA 46\ZZ& \AA 47\ZZ& \AA 48\ZZ& \AA 49\ZZ& \AA 50\ZZ& 
          \AA 51\ZZ& \AA 52\ZZ& \AA 53\ZZ& \AA 54\ZZ& \AA 55\ZZ& \AA 56\ZZ& \AA 57\ZZ& \AA 58\ZZ& \AA 59\ZZ& \AA 60\ZZ& 
          \AA 61\ZZ& \AA 62\ZZ& \AA 63\ZZ& \AA 64\ZZ$\:$ \\
\hline
%
%                            3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4
%                         ---------------------------------------------------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                                                               %.
\AA1\101\10\ZA\110\110$_1$\ZZ& &\p1&\m1&   &   &   &   &   &   &   &   &\m1&\p1&   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_3$\ZZ& &   &   &\p1&\m1&   &   &   &   &\p1&\m1&   &   &\p1&   &   &   &\m1&\p1&\m1&   &   &   &   &\m1&\p1&\m1&   &   &\pD&\p1&\m1\\
\AA1\101\10\ZA\110\110$_6$\ZZ&\p1& &   &   &   &   &   &   &\p1&\m1&\p1&   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &\m1&   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\100010\ZA1000\10$_1$\ZZ&   &\p1&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\100010\ZA1000\10$_3$\ZZ&   &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   \\
\AA\100010\ZA1000\10$_6$\ZZ&\p1&   &   &   &   &   &   &   &\mB&\pB&\mB&   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &\pB&   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA1\11\100\ZA\11\1100$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &\m1&\p1&   &\m1&\p1&   &   &   &   &\p1&\m1&   &   &   &   &   &\p1&\m1&   &   &\p1\\
\AA1\11\100\ZA\11\1100$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &\m1&   &   &   &\p1&   \\
\AA1\11\100\ZA\11\1100$_3$\ZZ& &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &\pB&\p1&\m1&   &   &   &   &\m1&\p1&\pB&   &   &\pB&   &   \\
\AA1\11\100\ZA\11\1100$_6$\ZZ&\p1& &   &   &\m1&   &   &   &\p1&\m1&\p1&   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &\m1&   &   &   &   &\pB&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\1001\10\ZA100\110$_1$\ZZ&  &\p1&\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\1001\10\ZA100\110$_3$\ZZ&  &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &\pB&   &\m1&   &   &\pD&   &   \\
\AA\1001\10\ZA100\110$_6$\ZZ&\p1&  &   &   &   &   &   &   &\p1&\pC&\pC&   &   &   &   &   &   &\m1&\p1&\pB&   &   &   &   &\pB&\p1&   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10\1001\ZA\10100\1$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\p1&\m1&\m1&   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &\p1\\
\AA10\1001\ZA\10100\1$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &\p1&   \\
\AA10\1001\ZA\10100\1$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &\m1&   &\pB&   &   &\pB&   &   \\
\AA10\1001\ZA\10100\1$_6$\ZZ&  &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\101\100\ZA10\1100$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &   &\p1&\m1&   &   &   \\
\AA\101\100\ZA10\1100$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &\p1\\
\AA\101\100\ZA10\1100$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   &   &\pB&   &\pB&   &   &\pB&\p1&\pB\\
\AA\101\100\ZA10\1100$_6$\ZZ&\p1&  &   &\p1&\pB&   &   &   &\p1&\pC&\pC&   &   &\p1&   &   &   &\pB&\p1&\pB&   &   &   &   &\pB&\p1&   &   &   &\pB&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA10000\1\ZA\100001$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&\pB&   &   &\p1\\
\AA10000\1\ZA\100001$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &   &   &   &   \\
\AA10000\1\ZA\100001$_6$\ZZ&   &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &\p1&\m1&   &   &   \\
\AA\11\1001\ZA1\1100\1$_2$\ZZ& &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &\m1&   &   &   &   &\p1\\
\AA\11\1001\ZA1\1100\1$_3$\ZZ& &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &\pB&\p1&\pB&   &   &   &   &\pB&\p1&\pB&   &   &\pB&\p1&\pB\\
\AA\11\1001\ZA1\1100\1$_6$\ZZ& &   &   &\p1&\pB&   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA\11000\1\ZA1\10001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &\pB&   &   &   &   &   &   &   &   &\pB&\p1&\pB&   &\p1&\pD\\
\AA\11000\1\ZA1\10001$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &   &   &   &   \\
\AA\11000\1\ZA1\10001$_6$\ZZ&  &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\10001\ZA01000\1$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_3$\ZZ&   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &\pB&   &   &   &   &   &   \\
\AA0\10001\ZA01000\1$_6$\ZZ&   &   &   &\pB&   & 20& 10&   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB\\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0\1100\1\ZA01\1001$_1$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   \\
\AA0\1100\1\ZA01\1001$_2$\ZZ&  &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA0\1100\1\ZA01\1001$_3$\ZZ&  &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &\pB&\p1&   &\p1&   &   &   &   &\p1&\pB\\
\AA0\1100\1\ZA01\1001$_6$\ZZ&  &   &   &\p1&   & 22& 11&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA00\1100\ZA001\100$_1$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &\p1&\pB&   &   &   \\
\AA00\1100\ZA001\100$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1\\
\AA00\1100\ZA001\100$_3$\ZZ&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &\p1&\pB\\
\AA00\1100\ZA001\100$_6$\ZZ&   &   &   &\p1&   & 22& 11&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA000\110\ZA0001\10$_1$\ZZ&   &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&\pB&   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_2$\ZZ&   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_3$\ZZ&   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   \\
%                                                                                                                                                           .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                                                               %.
\AA0000\10\ZA000010$_1$\ZZ &   &\p1&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_2$\ZZ &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\AA0000\10\ZA000010$_3$\ZZ &   &   &   &   &   &\pB&\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                                                           %.
\AA0000\10\ZA000010$_6$\ZZ &   &   &   &   &   &   &   &\p1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\
%*                                                                                                                                                          .
\hline
%                            3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   1   2   3   4
%351prime_ccc              ---------------------------------------------------------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &                          $\sqrt{ 18}$ &$\sqrt{ 24}$&$\sqrt{ 24}$&
                            $\sqrt{ 30}$&$\sqrt{ 24}$&$\sqrt{2064}$&$\sqrt{534}$&$\sqrt{ 18}$&
                            $\sqrt{ 18}$&$\sqrt{ 54}$&$\sqrt{ 54}$ &$\sqrt{ 24}$&$\sqrt{ 24}$&
                            $\sqrt{ 18}$&$\sqrt{ 18}$&$\sqrt{ 18}$ &$\sqrt{ 18}$&$\sqrt{ 42}$&
                            $\sqrt{ 48}$&$\sqrt{ 30}$&$\sqrt{ 24}$ &$\sqrt{ 24}$&$\sqrt{ 18}$&
                            $\sqrt{ 18}$&$\sqrt{ 48}$&$\sqrt{ 42}$ &$\sqrt{ 30}$&$\sqrt{ 24}$&
                            $\sqrt{ 24}$&$\sqrt{228}$&    \pF      &$\sqrt{108} \:$ \\
\hline
\end{tabular}
\end{table}





\clearpage

%
%  end of the third table 
%


\protect
\begin{table}
\caption{ 
{\bf CG coefficients for the (000000) dominant weight states of the 650-dimensional (100010) irrep
in the product (100000)$\otimes$(200000).} 
\AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
states is consistent with table I in ref.\cite{abI}.
Each CGC should be divided by the respective number in the last row to maintain
$\langle n\!\mid\! n\rangle = 1$.
}
\label{t:351prime_barcgc000000part4}  
%\small
%\footnotesize
%\scriptsize
\tiny
%                 12  34567890123456789012  (20 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
 & &  
      \multicolumn{20}{|c|}{ \rule[-0.30cm]{0mm}{0.87cm}
                             \normalsize \em (100010) } \\
\cline{3-22}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ$\:$ \\
\hline
%351prime_ddd                  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   
%                         -------------------------------------------------------------------------------------
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                  %.
\AA100000\ZA\100000$_1$\ZZ &  \pD&   &   &   &   &   &   &   &   &   &   &\pE&   &   & 10&   &   &   &   &    \\
\AA100000\ZA\100000$_2$\ZZ &  \mC&\pE&   &   &   &   &   &   &   &   & 10&   &   &   &   &   &   &   &   &    \\
\AA100000\ZA\100000$_3$\ZZ &  \pB&   &   &   &   &   & 10&   &\pE&   &   &   &   &   &   &   &   &   &   &    \\
\AA100000\ZA\100000$_6$\ZZ &  \m1&   &   &\pE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pE& 10 \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA\110000\ZA1\10000$_1$\ZZ&  \pD&   &\pD&   &   &   &   &   &   &   &   &\pE&\pE&   &\mE&   &   &   &   &    \\
\AA\110000\ZA1\10000$_2$\ZZ&  \mC&\pE&\mC&   &\pE&   &   &   &   &   &\mE&   &   &   &   &   &   &   &   &    \\
\AA\110000\ZA1\10000$_3$\ZZ&  \pB&   &\pB&\pE&   &   &\mE&\pE&\pE&   &   &   &   &   &   &   &   &   &   &    \\
\AA\110000\ZA1\10000$_6$\ZZ&  \m1&   &\m1&\mE&   &   &   &   &   &   &   &   &   &   &   &   &   &\pE&\pE&\mE \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA0\11000\ZA01\1000$_1$\ZZ&     &   &\pD&   &   &\pD&   &   &   &   &   &   &\pE&\pE&\pE&   &   &   &   &    \\
\AA0\11000\ZA01\1000$_2$\ZZ&     &   &\mC&   &\pE&\mC&   &\pE&   &\pE&\pE&   &   &   &   &   &   &   &   &    \\
\AA0\11000\ZA01\1000$_3$\ZZ&     &   &\pB&\pE&   &\pB&\pE&\mE&   &   &   &   &   &   &   &   &   &   &   &    \\
\AA0\11000\ZA01\1000$_6$\ZZ&     &   &\m1&\mE&   &\m1&\mE&   &\mE&   &   &   &   &   &   &   &   &\mE&-10&\mE \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA00\1101\ZA001\10\1$_1$\ZZ&    &   &   &   &   &\pD&   &   &   &\pE&   &   &   &\pE&\mE&\pD&\pE&   &   &    \\
\AA00\1101\ZA001\10\1$_2$\ZZ&    &   &   &   &   &\mC&   &\pE&   &\mE&\mE&   &   &   &   &\mC&   &\pE&   &    \\
\AA00\1101\ZA001\10\1$_3$\ZZ&    &   &   &\mE&\mE&\pB&\pE&\mE&   &   &\pE&   &   &   &   &\pB&   &\mE&   &\pE \\
\AA00\1101\ZA001\10\1$_6$\ZZ&    &\mE&   &   &   &\m1&\mE&   &\mE&   &\mE&   &   &   &   &\m1&   &\pE&\pE&    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA000\111\ZA0001\1\1$_1$\ZZ&    &   &   &   &   &   &   &   &   &\pE&   &   &   &   &\pE&\pD&\m1&   &   &    \\
\AA000\111\ZA0001\1\1$_2$\ZZ&    &   &   &   &   &   &\mE&\mE&   &\mE&\mE&   &   &\mE&\mE&\mC&\mC&\mE&   &    \\
\AA000\111\ZA0001\1\1$_3$\ZZ&    &   &   &   &\mE&   &   &   &   &   &\pE&   &\mE&   &\pE&\pB&\pB&\pE&   &\mE \\
\AA000\111\ZA0001\1\1$_6$\ZZ&    &\mE&   &   &   &   &   &   &   &   &\mE&\mE&   &   &\mE&\m1&\m1&\mE&\mE&    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA00010\1\ZA000\101$_1$\ZZ&     &   &   &   &   &   &   &   &   &\m1&   &   &   &   &\pE&\pD&\pE&   &   &    \\
\AA00010\1\ZA000\101$_2$\ZZ&     &   &   &   &   &   &   &   &   &\pB&\pE&   &   &   &   &\mC&   &\pE&   &    \\
\AA00010\1\ZA000\101$_3$\ZZ&     &   &   & 10&\pE&   &   &   &   &\pB&\mE&   &   &   &   &\pB&   &\mE&   &\pE \\
\AA00010\1\ZA000\101$_6$\ZZ&     &\pE&   &   &   &   &\pE&\pE& 10&\m1&\pE&   &   &   &   &\m1&   &\pE&\pE&    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA0000\11\ZA00001\1$_1$\ZZ&     &   &   &   &   &   &   &   &   &\mE&\pE&   &   &   &\pE&\mE&\m1&\m1&   &    \\
\AA0000\11\ZA00001\1$_2$\ZZ&     &   &   &   &   &\mE&\pE&   &   &   &   &   &   &\mE&\mE&   &\mC&\pB&   &    \\
\AA0000\11\ZA00001\1$_3$\ZZ&     &   &\mE&   &   &   &   &   &   &   &   &   &\mE&   &\pE&   &\pB&\mC&   &\pE \\
\AA0000\11\ZA00001\1$_6$\ZZ&  \mE&   &   &   &   &   &   &   &   &   &   &\mE&   &   &\mE&   &\m1&\pD&\pE&    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA001\11\1\ZA00\11\11$_1$\ZZ&   &   &   &   &   &\pD&   &   &   &\m1&   &   &   &\m1&\mE&\p1&\m1&   &   &    \\
\AA001\11\1\ZA00\11\11$_2$\ZZ&   &   &   &   &   &\mC&\pE&   &   &\pB&\pE&   &   &\pB&\pE&\mC&\mC&\mE&   &    \\
\AA001\11\1\ZA00\11\11$_3$\ZZ&   &   &   &\mE&\pE&\pB&\mE&\pE&   &\pB&\mE&   &\pE&\pB&\mE&\mB&\pB&\pE&   &\mE \\
\AA001\11\1\ZA00\11\11$_6$\ZZ&   &\pE&   &   &   &\m1&   &\mE&\mE&\m1&\pE&\pE&   &\m1&\pE&\m1&\m1&\mE&\mE&    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA0010\1\1\ZA00\1011$_1$\ZZ&    &   &   &   &   &\mE&   &\m1&   &   &\mE&   &   &\m1&\mE&\mE&\m1&\m1&   &    \\
\AA0010\1\1\ZA00\1011$_2$\ZZ&    &   &   &   &   &\pE&\mE&\pB&   &   &   &   &   &\pB&\pE&   &\mC&\pB&   &    \\
\AA0010\1\1\ZA00\1011$_3$\ZZ&    &   &\pE&\pE&   &   &\pE&\mC&   &   &   &   &\pE&\pB&\mE&   &\pB&\mC&   &\pE \\
\AA0010\1\1\ZA00\1011$_6$\ZZ& \pE&   &   &   &   &   &   &\pD&\pE&   &   &\pE&   &\m1&\pE&   &\m1&\pD&\pE&    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA01\1010\ZA0\110\10$_1$\ZZ&    &   &\pD&   &   &\pD&   &   &   &   &   &   &\m1&\m1&\pE&   &   &   &   &    \\
\AA01\1010\ZA0\110\10$_2$\ZZ&    &   &\mC&   &   &\mC&\pE&   &   &   &   &   &\pB&\pB&\mE&   &\pE&\pE&   &\pE \\
\AA01\1010\ZA0\110\10$_3$\ZZ&    &   &\pB&\pE&   &\pB&\mE&\pE&   &   &   &   &\mC&\pB&\pE&   &   &   &   &    \\
\AA01\1010\ZA0\110\10$_6$\ZZ&    &-10&\m1&\mE&\mE&\m1&   &\mE&\mE&   &\mE&\mE&\m1&\m1&\mE&   &   &   &   &    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA01\11\10\ZA0\11\110$_1$\ZZ&   &   &\mE&   &\m1&\mE&   &\m1&   &\m1&\pE&   &\m1&\m1&\pE&   &   &   &   &    \\
\AA01\11\10\ZA0\11\110$_2$\ZZ&   &   &\pE&   &\pB&\pE&\mE&\pB&   &\pB&\mE&   &\pB&\pB&\mE&\pE&\pE&   &   &\mE \\
\AA01\11\10\ZA0\11\110$_3$\ZZ&   &   &\mE&\mE&\mC&   &\pE&\mC&   &\pB&\pE&   &\mC&\pB&\pE&   &   &   &   &    \\
\AA01\11\10\ZA0\11\110$_6$\ZZ&\mE&\pE&   &\pE&\pD&   &   &\pD&\pE&\m1&   &\mE&\m1&\m1&\mE&   &   &   &   &    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA1\10010\ZA\1100\10$_1$\ZZ& \pD&   &\pD&   &   &   &   &   &   &   &   &\m1&\m1&   &\mE&   &   &   &   &    \\
\AA1\10010\ZA\1100\10$_2$\ZZ& \mC&   &\mC&   &   &   &   &   &   &   &   &\pB&\pB&   &\pE&   &\mE&\mE&\pE&\pE \\
\AA1\10010\ZA\1100\10$_3$\ZZ& \pB&   &\pB&\pE&   &   &\pE&\mE&\pE&   &   &\mC&\mC&\mE&\mE&   &   &   &   &    \\
\AA1\10010\ZA\1100\10$_6$\ZZ& \m1&\pE&\m1&\mE&\mE&   &   &   &   &\mE&\pE&\pD&\m1&   &\pE&   &   &   &   &    \\
%                                                                                                              .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                  %.
\AA010\100\ZA0\10100$_1$\ZZ&     &   &\pE&\m1&\m1&\pE&\pE&   &   &\m1&\pE&   &   &   &   &   &   &   &   &    \\
\AA010\100\ZA0\10100$_2$\ZZ&     &   &\mE&\pB&\pB&   &   &   &   &\pB&\mE&   &   &   &   &\pE&   &   &   &\pE \\
\AA010\100\ZA0\10100$_3$\ZZ&     &   &\pE&\pB&\mC&   &   &   &   &\pB&\pE&   &   &   &   &   &   &   &   &    \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA010\100\ZA0\10100$_6$\ZZ&  \pE&\pE&   &\m1&\pD&   &   &   &   &\m1&   & 10&\pE&   &\pE&   &   &   &   &    \\
%                                                                                                                                                           .
\hline
\multicolumn{ 4}{|c }{ } &
\multicolumn{18}{ r|}{\rule[-0.20cm]{0mm}{0.67cm} \small \em continued on next page $\qquad$ } \\
%\multicolumn{22}{|l|}{\small \em $\qquad$ continued on next page $\qquad$ } \\
\hline
\end{tabular}
\end{table}




\clearpage


\protect
\begin{table}
%              \caption{ 
%              {\bf CG coefficients for the (000000) dominant weight states of the 650-dimensional (100010) irrep
%              in the product (100000)$\otimes$(200000).} 
%              \AA n\ZZ\ is an abbreviation for \AA000000$_n$\ZZ. Numbering of the degenerate
%              states is consistent with table I in ref.\cite{abI}.
%              Each CGC should be divided by the respective number in the last row to maintain
%              $\langle n\!\mid\! n\rangle = 1$.
%              }
%              \label{t:351prime_barcgc000000part4}  
%\scriptsize
\tiny
%                 12  34567890123456789012  (20 columns of CGCs)
%\begin{tabular}{|cc||rrrrrrrrrrrrrrrrrrrr|}
\begin{tabular}{|@{\hspace{0.5mm}}l@{\hspace{0.5mm}}l@{\hspace{1mm}}
                |@{\hspace{0mm}}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}
                                r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{$\,$}r@{\hspace{0.5mm}}|}
\hline
\multicolumn{22}{|l|}{\rule[-0.20cm]{0mm}{0.67cm} $\qquad$ \small \em continued from previous page} \\
\hline
 & &  
      \multicolumn{20}{|c|}{\rule[-0.30cm]{0mm}{0.87cm}
                            \normalsize \em (100010) } \\
\cline{3-22}
\rule[-0.18cm]{0mm}{0.5cm}
 & & 
     $\,$ \BG 1\ZZ& \BG 2\ZZ& \BG 3\ZZ& \BG 4\ZZ& \BG 5\ZZ& \BG 6\ZZ& \BG 7\ZZ& \BG 8\ZZ& \BG 9\ZZ& \AA 10\ZZ& 
          \AA 11\ZZ& \AA 12\ZZ& \AA 13\ZZ& \AA 14\ZZ& \AA 15\ZZ& \AA 16\ZZ& \AA 17\ZZ& \AA 18\ZZ& \AA 19\ZZ& \AA 20\ZZ$\:$ \\
\hline
%
%                               1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0   
%                         ------------------------------------------------------------------------------------
%\multicolumn{22}{|l|}{\rule[-0.20cm]{0mm}{0.67cm} $\qquad$ \small \em continued from previous page} \\
%\hline
\rule[0cm]{0mm}{0.3cm}$\!\!$                                                                                  %.
\AA1\101\10\ZA\110\110$_1$\ZZ&\mE&\m1&\mE&   &\m1&   &   &   &   &   &\mE&\m1&\m1&   &\mE&   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_2$\ZZ&\pE&\pB&\pE&   &\pB&   &   &   &   &   &\pE&\pB&\pB&   &\pE&\mE&\mE&   &\mE&\mE\\
\AA1\101\10\ZA\110\110$_3$\ZZ&\mE&\mC&\mE&\mE&\mC&\mE&\mE&   &\mE&\mE&\mE&\mC&\mC&\mE&\mE&   &   &   &   &   \\
\AA1\101\10\ZA\110\110$_6$\ZZ&\pE&\m1&   &\pE&\pD&   &   &   &   &\pE&   &\pD&\m1&   &\pE&   &   &   &   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA\100010\ZA1000\10$_1$\ZZ&  \pD&   &   &   &   &   &   &   &   &   &   &\m1&   &   &\pD&   &   &   &   &   \\
\AA\100010\ZA1000\10$_2$\ZZ&  \mC&   &   &   &   &   &   &   &   &   &   &\pB&   &   &\mC&   &\pE& 10&\pE&   \\
\AA\100010\ZA1000\10$_3$\ZZ&  \pB&   &   &   &   &   &   & 10&\pE&   &   &\mC&   &\pE&\pB&   &   &   &   &   \\
\AA\100010\ZA1000\10$_6$\ZZ&  \m1&\pE&   &\pE& 10&   &   &   &   &\pE&   &\pD&\pE&   &\m1&   &   &   &   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA1\11\100\ZA\11\1100$_1$\ZZ&\pE&\m1&\pE&\m1&\m1&   &\mE&\m1&\m1&   &\mE&   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_2$\ZZ&\mE&\pB&\mE&\pB&\pB&\mE&\pE&\pB&\pB&   &\pE&   &   &   &   &\mE&   &   &\pE&\pE\\
\AA1\11\100\ZA\11\1100$_3$\ZZ&\pE&\mC&\pE&\pB&\mC&\pE&   &\mC&\pB&\mE&\mE&   &   &   &   &   &   &   &   &   \\
\AA1\11\100\ZA\11\1100$_6$\ZZ&\mE&\m1&   &\m1&\pD&   &   &\pD&\m1&\pE&   &\mE&\pE&\pE&\mE&   &   &   &   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA\1001\10\ZA100\110$_1$\ZZ& \mE&\m1&   &   &   &   &   &   &   &   &\pD&\m1&   &   &\pD&   &   &   &   &   \\
\AA\1001\10\ZA100\110$_2$\ZZ& \pE&\pB&   &   &   &   &   &   &   &   &\mC&\pB&   &   &\mC&\pE&\pE&\mE&\mE&   \\
\AA\1001\10\ZA100\110$_3$\ZZ& \mE&\mC&   &   &   &\pE&   &\mE&\mE&\pE&\pB&\mC&   &\pE&\pB&   &   &   &   &   \\
\AA\1001\10\ZA100\110$_6$\ZZ& \pE&\m1&\pE&\mE&\mE&   &   &   &   &\mE&\m1&\pD&\pE&   &\m1&   &   &   &   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA10\1001\ZA\10100\1$_1$\ZZ& \mE&   &\mE&   &   &   &\mE&\m1&\m1&   &   &   &   &   &   &   &   &\m1&\m1&\mE\\
\AA10\1001\ZA\10100\1$_2$\ZZ& \pE&   &   &   &   &\mE&\pE&\pB&\pB&   &   &   &   &   &   &   &   &\pB&\mC&   \\
\AA10\1001\ZA\10100\1$_3$\ZZ& \mE&\pE&   &   &   &\pE&   &\mC&\pB&\pE&\pE&   &   &   &   &\pE&   &\mC&\pB&   \\
\AA10\1001\ZA\10100\1$_6$\ZZ& \pE&   &   &   &   &   &   &\pD&\m1&   &   &\pE&   &\pE&\pE&   &\pE&\pD&\m1&   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA\101\100\ZA10\1100$_1$\ZZ& \pE&\m1&   &   &   &   &\pD&   &\m1&   &\pD&   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_2$\ZZ& \mE&\pB&   &   &   &\pE&\mC&   &\pB&   &\mC&   &   &   &   &\pE&   &\pE&\pE&   \\
\AA\101\100\ZA10\1100$_3$\ZZ& \pE&\mC&   &   &   &\mE&\pB&\pE&\pB&\pE&\pB&   &   &   &   &   &   &   &   &   \\
\AA\101\100\ZA10\1100$_6$\ZZ& \mE&\m1&\mE&   &\mE&   &\m1&\mE&\m1&\mE&\m1&\mE&-10&\mE&   &   &   &   &   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA10000\1\ZA\100001$_1$\ZZ&  \pE&   &\pE&\mE&   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\m1&\mE\\
\AA10000\1\ZA\100001$_2$\ZZ&  \mE&   &   &   &   &\pE&\mE&   &\mE&   &   &   &   &   &   &   &   &\pB&\mC&   \\
\AA10000\1\ZA\100001$_3$\ZZ&  \pE&\mE&   &   &   &   &   &   &   &   &\mE&   &   &   &   &\pE&   &\mC&\pB&   \\
\AA10000\1\ZA\100001$_6$\ZZ&  \mE&   &   &   &   &   &   &   &   &   &   &\mE&   &   &\mE&   &\pE&\pD&\m1&   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA\11\1001\ZA1\1100\1$_1$\ZZ&\mE&   &   &\m1&   &   &\pD&   &\m1&   &   &   &   &   &   &   &   &   &\m1&\pD\\
\AA\11\1001\ZA1\1100\1$_2$\ZZ&\pE&   &\pE&\pB&   &\pE&\mC&   &\pB&   &   &   &   &   &   &   &   &\mE&\mC&\mC\\
\AA\11\1001\ZA1\1100\1$_3$\ZZ&\mE&\pE&\mE&\pB&\pE&\mE&\pB&\pE&\pB&\mE&   &   &   &   &   &\mE&   &\pE&\pB&\pB\\
\AA\11\1001\ZA1\1100\1$_6$\ZZ&\pE&   &\pE&\m1&   &   &\m1&\mE&\m1&   &   &\pE&\pE&\mE&   &   &\mE&\mE&\m1&\m1\\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA\11000\1\ZA1\10001$_1$\ZZ& \pE&   &   &\pE&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\m1&\pD\\
\AA\11000\1\ZA1\10001$_2$\ZZ& \mE&   &\mE&\mE&   &\mE&   &\mE&\mE&   &   &   &   &   &   &   &   &\mE&\mC&\mC\\
\AA\11000\1\ZA1\10001$_3$\ZZ& \pE&\mE&\pE&   &\mE&   &   &   &   &   &   &   &   &   &   &\mE&   &\pE&\pB&\pB\\
\AA\11000\1\ZA1\10001$_6$\ZZ& \mE&   &\mE&   &   &   &   &   &   &   &   &\mE&\mE&   &   &   &\mE&\mE&\m1&\m1\\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA0\10001\ZA01000\1$_1$\ZZ&  \pE&   &   &\m1&   &   &   &   &   &   &   &   &   &   &   &   &   &   &\pE&\pD\\
\AA0\10001\ZA01000\1$_2$\ZZ&     &   &\pE&\pB&   &   &   &   &   &   &   &   &   &   &   &   &   &\pE&   &\mC\\
\AA0\10001\ZA01000\1$_3$\ZZ&     &   &\mE&\pB&\pE&   &   &   &   & 10&   &   &   &   &   &\pE&   &\mE&   &\pB\\
\AA0\10001\ZA01000\1$_6$\ZZ&     &   &\pE&\m1&   &\pE&   &\pE&   &   &   &   &\pE& 10&   &   &\pE&\pE&   &\m1\\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA0\1100\1\ZA01\1001$_1$\ZZ& \mE&   &   &\pE&   &   &\pD&   &\pE&   &   &   &   &   &   &   &   &   &\pE&\pD\\
\AA0\1100\1\ZA01\1001$_2$\ZZ&    &   &\mE&\mE&   &   &\mC&\pE&   &   &   &   &   &   &   &   &   &\pE&   &\mC\\
\AA0\1100\1\ZA01\1001$_3$\ZZ&    &   &\pE&   &\mE&\pE&\pB&\mE&   &\mE&   &   &   &   &   &\pE&   &\mE&   &\pB\\
\AA0\1100\1\ZA01\1001$_6$\ZZ&    &   &\mE&   &   &\mE&\m1&   &   &   &   &   &\mE&\mE&   &   &\pE&\pE&   &\m1\\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA00\1100\ZA001\100$_1$\ZZ&  \pE&\pE&   &   &   &   &\pD&   &\pE&   &\pD&   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_2$\ZZ&     &   &\pE&\pE&\pE&   &\mC&\pE&   &   &\mC&   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_3$\ZZ&     &   &   &   &   &\pE&\pB&\mE&   &\pE&\pB&   &   &   &   &   &   &   &   &   \\
\AA00\1100\ZA001\100$_6$\ZZ&     &   &   &   &   &\mE&\m1&   &   &\mE&\m1&   &   &\mE&   &\mE&-10&\mE&   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA000\110\ZA0001\10$_1$\ZZ&  \mE&\pE&   &   &   &   &   &   &   &   &\pD&\pE&   &   &\pD&   &   &   &   &   \\
\AA000\110\ZA0001\10$_2$\ZZ&     &   &\mE&   &\pE&   &   &   &   &   &\mC&   &\pE&   &\mC&   &   &   &   &   \\
\AA000\110\ZA0001\10$_3$\ZZ&     &   &   &   &   &\mE&   &\pE&   &\pE&\pB&   &   &\pE&\pB&   &   &   &   &   \\
\AA000\110\ZA0001\10$_6$\ZZ&     &   &   &   &   &   &   &   &   &\mE&\m1&   &   &   &\m1&\mE&\pE&\pE&   &   \\
%                                                                                                             .
\rule[0cm]{0mm}{0.25cm}$\!\!$                                                                                 %.
\AA0000\10\ZA000010$_1$\ZZ &   10&   &   &   &   &   &   &   &   &   &   &\pE&   &   &\pD&   &   &   &   &   \\
\AA0000\10\ZA000010$_2$\ZZ &     &   & 10&   &   &   &   &   &   &   &   &   &\pE&   &\mC&   &   &   &   &   \\
\AA0000\10\ZA000010$_3$\ZZ &     &   &   &   &   & 10&   &   &   &   &   &   &   &\pE&\pB&   &   &   &   &   \\
\rule[-0.25cm]{0mm}{0.25cm}$\!\!$                                                                                %.
\AA0000\10\ZA000010$_6$\ZZ &     &   &   &   &   &   &   &   &   &\pE&   &   &   &   &\m1& 10&\pE&   &   &   \\
%*                                                                                                                .
\hline
%351prime_ddd                  1   2   3   4   5   6   7   8   9   0   1   2   3   4   5   6   7   8   9   0
%                         ----------------------------------------------------------------------------------
\rule[-0.2cm]{0mm}{0.7cm}                                                                                               %.
           &               &
      \multicolumn{20}{|c|}{ \scriptsize $\sqrt{540}$ each state} \\
%
%
%                            $\sqrt{   }$&$\sqrt{   }$&$\sqrt{   }$&$\sqrt{   }$&$\sqrt{   }$&
%
\hline
\end{tabular}
\end{table}




%\clearpage




\end{document}



hep-ph0101349.extracted_bib
\begin{thebibliography}{99}

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                             Phys. Lett. B{\bf 60} (1976) 177.

\bibitem{Froggatt_Nielsen}  C.D.~Froggatt and H.B.~Nielsen,
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\bibitem{pat_Verma} Sai-Ping Li, R.V.~Moody, M.~Nicolescu, and J.~Patera,
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\bibitem{pat_E6} In-Guy Koh, J.~Patera, and C.~Rousseau, 
                              J. Math. Phys. {\bf 25} (1984) 2863.

\bibitem{abI} G.~Anderson and T.~Bla\v{z}ek, 
                              J. Math. Phys. {\bf 41} (2000) 4808.

\bibitem{abII} G.~Anderson and T.~Bla\v{z}ek, 
                              J. Math. Phys. {\bf 41} (2000) 8170.

\bibitem{bky} M.~Bando, T.~Kugo, and K.~Yoshioka,
%                    hep-ph/0003220,
                     Prog. Theor. Phys. {\bf 104} (2000) 211. 

\bibitem{Slansky_PhysRep} R.~Slansky, Phys. Rep. {\bf 79} (1981) 1.


\bibitem{footnote_1} For roots and weights we use the notation where $\ol{x}\equiv -x$.

\bibitem{pat_CCO} R.~V.~Moody and J.~Patera,
                                J. Math. Phys. {\bf 25} (1984) 2838.

\bibitem{footnote_2} Some applications of the charge conjugation operators 
                     can be found in ref.\cite{abI}.


\bibitem{ab_models} G.~Anderson and T.~Bla\v{z}ek, in preparation.


\bibitem{footnote_3} It is not always possible to choose a single basis
                     for all of the states of degenerate weights in a representation 
                     such that lowering a particular state results
                     in a particluar state of lower weight (as opposed to a linear combination
                     of weights) for all possible lowerings.  The simplest example of 
                     this is the 27 dimensional representation of $SU(3)$.


\end{thebibliography}