Quantum marginal problem and representations of the symmetric group
Alexander Klyachko
We discuss existence of mixed state $\rho_{AB}$ of two (or multy-) component system ${ \mathcal{H}}_{AB}={ \mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}$ with reduced density matrices ${\rho_A}$, ${\rho_B}$ and given spectra $\lambda_{AB},\lambda_A,\lambda_B$. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group. %This question is crucial for %understanding theoretical restrictions on possible manipulations %with quantum states of a multi-component system like $N$-qubits. %Up to unitary transformations the local states are %characterized by their spectra $\lambda_A=\text{Spec}\,\rho_A$ and %$\lambda_B=\text{Spec}\,\rho_B$. %We'll discuss relations between spectra of %$\rho_{AB},\rho_{A},\rho_{B}$ based on reduction of this problem %to representation theory of the symmetric group.
Q_Margins.tex
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\begin{document}

\title[Quantum Margins]
{Quantum marginal problem and representations of the symmetric group}%
\author{Alexander Klyachko}%
\address{Bilkent University \newline\indent Bilkent, 06533 Ankara Turkey}%
\email{klyachko@fen.bilkent.edu.tr}%

%\thanks{}%
\subjclass[2000]{20C30}%
%\keywords{}%

%\date{}%
\dedicatory{To 60 anniversary of Alain~Lascoux}%
%\commby{}%
% ----------------------------------------------------------------

\begin{abstract}
 We discuss existence of mixed state
$\rho_{AB}$ of two (or multy-) component system ${
\mathcal{H}}_{AB}={
\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}$ with reduced density
matrices ${\rho_A}$, ${\rho_B}$ and given spectra
$\lambda_{AB},\lambda_A,\lambda_B$. We give a complete solution of
the problem in terms of linear inequalities on the spectra,
accompanied with extensive tables of marginal inequalities,
including arrays up to 4 qubits. In the second part of the paper
we pursue another approach based on reduction of the problem to
representation theory of the symmetric group.
%This question is crucial for
%understanding theoretical restrictions on possible manipulations
%with quantum states of a multi-component system like $N$-qubits.
%Up to unitary transformations the local states are
%characterized by their spectra $\lambda_A=\text{Spec}\,\rho_A$ and
%$\lambda_B=\text{Spec}\,\rho_B$.
%We'll discuss relations between spectra of
%$\rho_{AB},\rho_{A},\rho_{B}$ based on reduction of this problem
%to representation theory of the symmetric group.
\end{abstract}
\maketitle
% ----------------------------------------------------------------
\tableofcontents
%\addcontentsline{toc}{subsubsection}{New}
%\addtocontents{toc}{\it}

\section{Introduction}
%\marginpar{\footnotesize
%It is my pleasure to dedicate the paper  to Alain~Lascoux, who was
%one of the inventors and main contributor to theory of
%%first to study combinatorics of both
%Schubert polynomials \cite{L-Sch} and gave the first known
%combinatorial description of tensor product for a class of
%irreducible representations of the symmetric group \cite{Las}.
%Both topics turn out to be entangled in a surprising way with
%quantum marginal problem.}
%way decomposition of tensor product of
%representations of the symmetric group \cite{Las}, which
%%turn out to be entangled
%%interrelated
%in a surprising way are entangled with the quantum marginal
%problem.}

The quantum marginal problem is about relations between  spectrum
of mixed state $\rho_{AB}$ of two (or multi) component system
$\mathcal{H}_{AB}=\mathcal{H}_A\otimes \mathcal{H}_B$ and that of
reduced states $\rho_A$ and $\rho_B$. Relations of this type, for
example, impose certain restrictions on manipulations with qubits
in quantum information theory.
%Mathematically
The problem can be stated in plain language as follows. Let
$M=[m_{ijk}]$ be cubic complex matrix and $H_1,H_2,H_3$ be Gram
matrices formed by Hermitian dot products of parallel slices of
$M$. We seek for
%The problem we address here is about
relations between spectra of these matrices.

We pursue two different approach to this problem. The first is
based on reduction
%of the problem
to general  Berenstein-Sjamaar theorem \cite{Ber-Sja} applied to
subgroup $\SU(\mathcal{H}_A)\times
SU(\mathcal{H}_B)\subset\SU(\mathcal{H}_A\otimes \mathcal{H}_B)$.
The relevant geometry and combinatorics are explained in section
\ref{MargIneq2}. Theorems \ref{cubicleThm} and \ref{ExtrEdge} give
a pretty explicit  ansatz for producing quantum marginal
inequalities. For systems of rank $\le 4$ they are given in
Appendix. It covers all system with few hundreds, rather then
thousands, marginal inequalities. An important case of an array of
qubits considered separately in section \ref{n_qubit}. Modulo a
``standard" conjecture the marginal inequalities can be produced
in a very straightforward way, see Theorem \ref{Mod_Qbt}.

The number of marginal inequalities increases drastically with
rank of the system. This makes the above solution
%based  on the above approach
inefficient for systems of big rank. In section \ref{ReprTheor} we
develop another approach based on reduction of the marginal
problem to decomposition of tensor product of irreducible
representations of the symmetric group, see Theorem \ref{mainThm}.
This approach, for example, allows answer questions about maximal
eigenvalue of a mixed state with given margins, see Theorem
\ref{Max}, or its rank, see Theorem \ref{Rank}. However the main
point here is not in new results, but in new vision, based on
connections between apparently very remote subjects. In last
section \ref{Appl2Repr} we consider some applications of the above
correspondence to back to representation theory of the symmetric
group.

It is my pleasure to dedicate the paper  to Alain~Lascoux, who was
one of the inventors and main contributor to theory of
%first to study combinatorics of both
Schubert polynomials \cite{L-Sch} and gave the first known
combinatorial description of tensor product for a class of
irreducible representations of the symmetric group \cite{Las}.
Both topics turn out to be entangled in a surprising way with
quantum marginal problem.


%State the problem in plain language. Add some justification of its
%importance. The main point of the paper is not in new results, but
%in new vision, based on connections between apparently very remote
%subjects.

%Thanks to IQI, John Preskill,...
I am appreciated to Sergey Bravyi and  Patrick Hayden for useful
discussion during my visit to the Institute for Quantum
Information in Caltech,
%at the California Institute of Technology
and to John Pereskill for
%an opportunity to spent time in IQI
kind invitation. This work was supported in part by the National
Science Foundation under grant EIA-0086038 through the Institute
for Quantum Information at the California Institute of Technology.

The text was prepared as a talk at mini conference ``Turing Days
04: Classical \& Quantum Computing" in Istanbul Bilgi University,
29--30 May, 2004. By that reason it contains background
information which some readers may find redundant. I decided to
keep
%this information
it untouched to make the paper easily accessible both for
physicists and mathematicians.

 %\cite{{Aud},{Hoog},{KTW},{Ful1},{Ful2}}.
 \section{Classical marginal problem} \label{class}
The classical marginal problem is about existence of a ``body" or
probability density $p_I(x_I):=p(x_1,x_2,\ldots,x_n)$ in $
\mathbb{R}^I$, $I=\{1,2,\ldots,n\}$ with given projections onto
some coordinate subspaces $\mathbb{R}^J\subset
\mathbb{R}^I,J\subset I$
$$p_J(x_J)=\int_{\mathbb{R}^{\bar{J}}}p_I(x_I)dx_{\bar{J}},\quad \bar{J}=I\backslash J$$
called {\it margins\,} of $p_I$. The problem has a long history,
see \cite{Kl02} and references therein. Here we give only few
relevant examples.

%\begin{Ex}
\subsubsection{\!\!}{\it Univariant margins $p_i(x_i)$ are always
compatible\,.} Indeed consider $x_i$ as independent variables and
define joint distribution by equation
$$p_I(x_I)=\prod_{i\in I}p_i(x_i).$$
%\end{Ex}

%\begin{Ex}
\subsubsection{\!\!}{ The following inequality is necessary for compatibility of
bivariant margins}
$$\langle x_i|x_i\rangle+
\langle x_j| x_j\rangle+
\langle x_k| x_k\rangle+
2\langle x_i| x_j\rangle+ 2\langle x_j| x_k\rangle+ 2\langle x_k|
x_i\rangle\ge0.$$ Indeed LHS is equal to variance $\langle
x_i+x_j+x_k\mid x_i+x_j+x_k\rangle\ge0$ {\em provided} joint
distribution $p_{ijk}$ exists.
%\end{Ex}

%\begin{Ex}
\subsubsection{\!\!} So called {\em Bell's ineqalities}\, in quantum mechanics are
just compatibility conditions for marginal distributions
corresponding to {\em commuting\,} observables \cite{Kl02}.
%\end{Ex}

%\begin{Ex}
\subsubsection{\!\!} \label{Planar}{\em Discrete version of the marginal problem\,} is about
existence of, say, cubic matrix $p_{ijk}\ge0$ with given
projections onto its facets
$$a_{ij}=\sum_k p_{ijk},\quad b_{jk}=\sum_i p_{ijk},\quad c_{ki}=\sum_j
p_{ijk}.$$ Compatibility conditions for such projections are still
unknown (so called {\em Planar Transport Polytope Problem\,}, see
\cite{EKK84}).
%\end{Ex}

%\begin{Ex}
\subsubsection{\!\!} \label{restr}{\em Restricted marginal problem\,} is about existence of a
matrix with prescribed content, e.g. $0-1$, and given margins. In
this case even univariant marginal problem becomes nontrivial. As
an example recall the following classical result.
%\end{Ex}
\begin{thm*}[Gale \cite{Gale, G-R}, Ryser \cite{Ryser,G-R}]
Partitions $\lambda,\mu$ are margins of a rectangular $0-1$ matrix
iff $\lambda\prec \mu^t$.
%A rectangular $0-1$ matrix with margins
%$\lambda,\mu$ exists iff $\lambda\prec \mu^t$.
\end{thm*}

Here marginal values
$\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_m\ge0$ arranged in
decreasing order are treated as {\it Young diagram\,} $\lambda$
with $i$-th row of length $\lambda_i$, $\mu^t$ stands for
transpose diagram, and the {\em majorization\,} or {\it
dominance\,} order $\lambda\prec\mu^t$ is defined by inequalities
\begin{align*}\lambda_1&\le\mu^t_1\\
\lambda_1+\lambda_2&\le\mu^t_1+\mu^t_2\\
\lambda_1+\lambda_2+\lambda_3&\le\mu^t_1+\mu^t_2+\mu^t_3\\
\cdots&\cdots\cdots
\end{align*}
The number of $0-1$ matrices with margins $\lambda,\mu$ is equal
to the number of pairs of tableaux of conjugate shape and contents
$\lambda,\mu$, see \cite{Macdonald}.

 We'll deal with quantum version of this problem below.
%This is a classical version of the problem we'll address below.
%version is most close to a problem we'll deal in quantum case.
\section{Quantum marginal problem}
\subsection{Quantum margins} A background of a quantum system $A$ is
Hilbert space $\mathcal{H}_A$ called {\it state space}. We'll
consider only {\it finite systems\,}, for which
$\dim\mathcal{H}_A<\infty$. Actual state of the system is
described by unit vector $\psi\in\mathcal{H}_A$ for {\em pure
state\,} or by non negative Hermitian operator
$\rho:\mathcal{H}_A\rightarrow\mathcal{H}_A,
\operatorname{Tr}\rho=1$, called {\em density matrix\,}, for {\em mixed state\,}. Pure state $\psi$
corresponds to projection operator $|\psi\rangle\langle\psi|$ onto
$\psi$. Hence
\begin{equation}
\rho=\mbox{pure}\Longleftrightarrow\operatorname{rk}\rho=1
\Longleftrightarrow\operatorname{Spec}\rho=(1,0,0\cdots,0).
\end{equation}
{\it An observable\,} is Hermitian operator
$X_A:\mathcal{H}_A\rightarrow\mathcal{H}_A$. Taking  measurement
of $X_A$
%over the
while system is in state $\rho$ produces random quantity
$x_A\in\Spec X_A$ implicitly determined by expectations
$$\langle f(x_A)\rangle_\rho=\Tr(\rho
f(X_A))=\langle\psi|f(X_A)|\psi\rangle$$ of arbitrary function
$f(x)$ on $\Spec X_A$. The second equation holds for pure state
$\psi$.

Superposition principle of quantum mechanics implies that state
space of composite system $AB$ splits into tensor product
$$\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$$
of state spaces of the components, as opposed to direct product
$P_{AB}=P_A\times P_B$ of configuration spaces in classical
mechanics. Density matrix of composite system can be written as
linear combination
\begin{equation}
\rho_{AB}=\sum_\alpha a_\alpha L_A^\alpha\otimes L_B^\alpha
\end{equation}
where $L_A^\alpha, L_B^\alpha$ are linear operators in
$\mathcal{H}_A,\mathcal{H}_B$ respectively. Its {\it reduced
matrices\,} or {\em marginal states\,} are defined by equations
\begin{align}
\rho_A&=\sum_\alpha
a_\alpha\Tr(L_B^\alpha)L_A^\alpha:=\Tr_B(\rho_{AB}),\\
\rho_B&=\sum_\alpha
a_\alpha\Tr(L_A^\alpha)L_B^\alpha:=\Tr_A(\rho_{AB}).
\end{align}
%Marginal state $\rho_A$
They can be characterized by the following property
\begin{equation}
\langle X_A\rangle_{\rho_{AB}}=\Tr(\rho_{AB}X_A)=\Tr(\rho_A
X_A)=\langle X_A\rangle_{\rho_A},\quad
X_A:\mathcal{H}_A\rightarrow\mathcal{H}_A,
\end{equation}
which tells that {\it observation of subsystem $A$  gives the same
results as if $A$ would be in reduced state
$\rho_A=\Tr_B(\rho_{AB})$.} This justifies the terminology.

%if we observe only part $A$ of composite system $AB$ in state
%$\rho_{AB}$ we see it in state $\rho_A=\Tr_B(\rho_{AB})$.

Margins of state $\rho_I$ of multicomponent system
$$\mathcal{H}_I=\bigotimes_{i\in I}\mathcal{H}_i=
\mathcal{H}_J\otimes\mathcal{H}_{\bar{J}},\quad J\subset I,
\bar{J}=I\backslash J$$
are defined in a similar way: $\rho_J=\Tr_{\bar{J}}(\rho_I)$.

\begin{Ex} In tensor algebra the above reduction
%marginalization
$\rho_I\mapsto\rho_J,J\subset I$ is known as {\it contraction\,}.
Most mathematicians are familiar with this procedure from
differential geometry, where, say, Ricci curvature
$\operatorname{Ric}:\mathcal{T}\rightarrow\mathcal{T}$ is defined
as contraction of Riemann curvature
$R:\mathcal{T}\otimes\mathcal{T}\rightarrow\mathcal{T}\otimes\mathcal{T}$
(here  $\mathcal{T}$ stands for tangent bundle).
\end{Ex}
\begin{Ex} Let's identify pure state of two component system
$$\psi=\sum_{ij}\psi_{ij}\alpha_i\otimes\beta_j\in\mathcal{H}_A\otimes\mathcal{H}_B$$
with its matrix $[\psi_{ij}]$ in orthonormal bases
$\alpha_i,\beta_j$ of $\mathcal{H}_A,\mathcal{H}_B$. Then margins
of $\psi$ in respective bases are given by matrices
\begin{equation}\rho_A=\psi^\dag\psi,\quad\rho_B=\psi\psi^\dag.\label{2-comp}\end{equation}
In striking difference with classical case margins of a pure
quantum state are mixed ones (provided
$\psi\neq\psi_A\otimes\psi_B$).
\end{Ex}
\begin{Ex} \label{slice}
A similar description holds  for multicomponent systems. For
example, write orthonormal components of tensor
$\psi\in\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$
into a cubic matrix $[\psi_{ijk}]$. Then univariant margins of
$\psi$ are given by {\it Gram matrices\,} formed by Hermitian dot
products of parallel slices of $[\psi_{ijk}]$.
%This implies
It follows that $\rk\rho_C\le \rk\rho_A\cdot\rk\rho_B$, because
$\rho_C$ can be written as Gram matrix of the slices of dimension
$\rk\rho_A\cdot\rk\rho_B$. This inequality is a simplest
manifestation of general problem about relations between margins
of a pure state, which we address below.
%example of numerous
%restrictions on margins of a pure state which we address below.
%one of many restrictions
%For example, consider components of tensor
%$\psi\in\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$ in
%orthonormal bases as a {\it cubic matrix\,} $[\psi_{ijk}]$ . Then
%univariant margins of $\psi$ are given by {\it Gram matrices\,}
%formed by Hermitian dot products of parallel slices of
%$[\psi_{ijk}]$.
\end{Ex}


\subsection{Marginal problem} \label{margin} {\it General quantum marginal problem\,} is
about existence of mixed state $\rho_I$ of composite system
$$\mathcal{H}_I=\bigotimes_{i\in I}\mathcal{H}_i$$
with given margins $\rho_J$ for some $J\subset I$ (cf. with
classical settings $n^\circ$\ref{class}).
%See  Section 2 for its classical counterpart.

Additional restrictions on state $\rho_I$ may be relevant. Here we
consider only two variations:
%For example we get
 \begin{itemize}
\item {\it Pure marginal problem\,}

%if  $\rho_I$ supposed to be pure state,
\end{itemize}
 corresponding to pure state $\rho_I$, and more general
 \begin{itemize}
\item{\it Spectral marginal problem\,}
%if $\Spec\rho_I=\lambda_I$ is fixed.
\end{itemize} corresponding to a state with
given spectrum $\lambda_I=\Spec\rho_I$.

Both versions are nontrivial even for univariant margins (cf. with
Gale--Ryser theorm $n^\circ$\ref{restr}). In this case margins
$\rho_i$ can be diagonalized by local unitary transformations
%Therefore
and their compatibility depends only on spectra
$\lambda_i=\Spec\rho_i$.

Pure quantum marginal problem has no classical analogue, since
projection of a point (=``pure state") is a point. In simplest
univariant case it can be stated in plain language as follows.
\begin{prob}\label{spectral} %Let's consider say cubic complex matrix
Let $M=[m_{ijk}]$ be complex cubic matrix and $M_1,M_2,M_3$ be
Gram matrices
 formed by Hermitian dot products of parallel slices
of $M$ (see Example \ref{slice}). What are possible spectra of
matrices $M_1$, $M_2$, and $M_3$?
\end{prob}
%{\it Pure marginal problem\,} is about existence of a pure state
%with given margins. It has no classical analogue. Other
%restrictions on the state $\rho_I$ may be also relevant, e.g.
% margins when
%$\rho_I$ is pure, i.e. $\Spec\rho_I=(1,0,0,\ldots,0)$.

%{\it Spectral marginal problem\,} when $\Spec
%\rho_I=\lambda_I$ is fixed.

\subsection{Some known results}
In contrast with classical marginal problem its quantum
counterpart attracts attention only quite recently. Until  now
%Currently
there were very few known results in this direction, which are
listed below.

\subsubsection{Pure two component problem} \label{pure}
 %$\bullet$ {\it Pure two component problem.\,}
 Equation \ref{2-comp}
implies that margins of pure state $\psi\in
\mathcal{H}_A\otimes\mathcal{H}_B$ are {\it isospectral\,}
$\Spec\rho_A=\Spec\rho_B$ if we discard zero eigenvalues. Vice
versa, this condition is sufficient for compatibility. Indeed let
$\alpha_i,\beta_i$ be orthonormal eigenbases of $\rho_A,\rho_B$
with common eigenvalues $\lambda_i$. Then state
$\psi=\sum_i\lambda_i\alpha_i\otimes\beta_i$ has margins
$\rho_A,\rho_B$. This representation of $\psi$ is known as {\it
Schmidt decomposition}.
\subsubsection{Scalar margins}\label{scalar}
%$\bullet$ {\it Scalar margins.\,}
Completely entangled state
$\psi\in\mathcal{H}_I=\bigotimes_i\mathcal{H}_i$ can be
characterized by its univariant margins $\rho_i$ \cite{VDMV,Kl02}:
$$\psi \mbox{ is completely entangled}\Longleftrightarrow
\rho_i=\mbox{scalar}.$$
\begin{thm*}[Klyachko \cite{Kl02}] Pure state $\psi\in\mathcal{H}_I$ with scalar univariant margins
exists iff informational capacities of the components
$\delta_i=\log\dim\mathcal{H}_i$ satisfy polygonal inequalities
$$\delta_i\le\sum_{j(\neq i)}\delta_j.$$
\end{thm*}

\subsubsection{Pure $N$-qubit problem} \label{N-qubit}
%$\bullet$ {\it Pure $N$-qubit problem.\,}
In this case there is a simple criterion for compatibility
univariant margins.
%A simple criterion for compatibility of
%univariant margins was found recently.
\begin{thm*}[Higuchi et al. \cite{HSS}, Bravyi \cite{Brav}]  Pure
$N$-qubit state $\psi\in\mathcal{H}^N,
\dim
\mathcal{H}=2$ with univariant margins $\rho_i$ exists iff their minimal
eigenvalues $\lambda_i$
%of $\rho_i$
satisfy polygonal inequalities
\begin{equation}\label{poly}\lambda_i\le\sum_{j(\neq i)}\lambda_j.
\end{equation}
\end{thm*}

\subsubsection{Mixed $2$-qubit problem}\label{two_qubit}
%$\bullet$ {\it Mixed $2$-qubit problem.\,}
This problem was solved by Sergey Bravyi.
\begin{thm*}[Bravyi  \cite{Brav}]
Mixed two-qubit state $\rho_{AB}$ with spectrum
$\lambda_1\ge\lambda_2\ge\lambda_3\ge\lambda_4$ and margins
$\rho_A,\rho_B$ exists iff minimal eigenvalues
$\lambda_A,\lambda_B$ of the margins satisfy
%the following
inequalities
\begin{align}\lambda_A&\ge\lambda_3+\lambda_4,\quad\lambda_B\ge\lambda_3+\lambda_4,\nonumber\\
\lambda_A&+\lambda_B\ge\lambda_2+\lambda_3+2\lambda_4,\label{brav}\\
|\lambda_A&-\lambda_B|\le\min(\lambda_1-\lambda_3,\lambda_2-\lambda_4).\nonumber
\end{align}
\end{thm*}
\subsubsection{Pure 3-qutrit problem}\label{hig}
%$\bullet$ {\it Pure 3-qutrit problem.\,}
Astuchi Higuchi  found a criterion for compatibility of univarint
margins in 3-qutrit system
$\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$,
$\dim\mathcal{H}_*=3$ in terms of marginal spectra
$\lambda_1^*\le\lambda_2^*\le\lambda_3^*$, $*=A,B,C$.
\begin{thm*}[Higuchi \cite{Hig}] Pure state
$\psi\in\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$
with  margins $\rho_A,\rho_B,\rho_C$ exists iff the following
inequalities holds
\begin{align*}
\lambda_2^a+\lambda_1^a&\le\lambda_2^b+\lambda_1^b+
\lambda_2^c+\lambda_1^c,\\
\lambda_3^a+\lambda_1^a&\le\lambda_2^b+\lambda_1^b+
\lambda_3^c+\lambda_1^c,\\
\lambda_3^a+\lambda_2^a&\le\lambda_2^b+\lambda_1^b+
\lambda_3^c+\lambda_2^c,\\
2\lambda_2^a+\lambda_1^a&\le2\lambda_2^b+\lambda_1^b+
2\lambda_2^c+\lambda_1^c,\\
2\lambda_1^a+\lambda_2^a&\le2\lambda_2^b+\lambda_1^b+
2\lambda_1^c+\lambda_2^c,\\
2\lambda_2^a+\lambda_3^a&\le2\lambda_2^b+\lambda_1^b+
2\lambda_2^c+\lambda_3^c,\\
2\lambda_2^a+\lambda_3^a&\le2\lambda_1^b+\lambda_2^b+
2\lambda_3^c+\lambda_2^c,
\end{align*}
where $a,b,c$ is a permutation of $A,B,C$.
\end{thm*}
It takes 46 pages to prove this!
% the following result.
\subsubsection{Remark} All the above theorems deal with univariant margins. In
quantum field theory they are known as {\it mean fields\,}, and
higher rank margins as {\it $n$-point correlations\,}. Most
physical effects are governed by two-point correlations. However
%general
complete solution of bivariant marginal problem is hardly possible
(even in classical case, see $n^\circ$\ref{Planar}). Here is a
couple of sporadic facts beyond trivial
%transitivity
compatibility relations like
$\Tr_A\rho_{AB}=\rho_B=\Tr_C\rho_{BC}$.

%$\bullet$
\subsubsection{}
Strong subadditivity \cite{LR,HJPW} of quantum entropy
$S(\rho)=-\Tr(\rho\log\rho)$
$$S(\rho_{ABC})\le S(\rho_{AB})+S(\rho_{BC})-S(\rho_B)$$
imposes a restriction on bivariant margins.

%$\bullet$
\subsubsection{} There is no pure 4-qubit state with scalar bivariant
margins \cite{HS}.

\section{Marginal inequalities}\label{mineq}
%\marginpar{\footnotesize
%The main result of Berenstein and  Sjamaar \cite{Ber-Sja} is {\it
%stability criterion} for stability of complete flag with
%appropriate polarisation w.r. to a compact subgroup, see the
%manuscript p. 185. This should give a criterion for existence of
%completely entangled state in an irreducible quantum dynamic
%system $G:\mathcal{H}$. }
In this section we give a general recipe
for producing marginal inequalities for arbitrary multi-component
system based on Berenstin--Sjamaar theorem \cite{Ber-Sja}. For $n$
qubits our result, modulo a ``standard" conjecture, amounts to a
simple combinatorics, but the number of involved inequalities
increases drastically with $n$. The marginal inequalities up to 4
qubits, and for systems of formats $2\times3$, $3\times3$,
$2\times4$, and $2\times2\times3$ are given in Appendix.

\subsection{Main result}\label{MargIneq2}
To avoid technicalities we confine  ourselves to two component
system $\mathcal{H}_{AB}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$.
Generalization to multicomponent systems is straightforward. We
start with some auxiliary notions and results.
%To state the result we need some auxiliary notions.
\subsubsection{Filtrations} Let $\alpha$ be a {\it nonincreasing filtration\,} of space
$\mathcal{H}$, i.e.  one parametric system of subspaces
$\mathcal{H}^\alpha(t)\subset\mathcal{H}, t\in\mathbb{R}$ such
that $\mathcal{H}^\alpha(s)\subset\mathcal{H}^\alpha(t)$  for
$s\ge t$ and
\begin{equation}\mathcal{H}^\alpha(t)=
\begin{cases}\mathcal{H},& \text{for $t\ll0$},\\
                      0,&\text{for $t\gg0$}.
\end{cases}
\end{equation}
The filtration supposed to be left continuous:
$\lim_{\varepsilon\rightarrow+0}\mathcal{H}^\alpha(t-\varepsilon)=\mathcal{H}^\alpha(t)$.

Denote by
\begin{equation}\mathcal{H}^{[\alpha]}(t)=\lim_{\varepsilon\rightarrow+0}
\frac{\mathcal{H}^\alpha(t)}{\mathcal{H}^\alpha(t+\varepsilon)}\end{equation}
{\it composition factors\,} of the filtration. The dimension
$m_\alpha(t)=\dim\mathcal{H}^{[\alpha]}(t)$ called {\it
multiplicity\,} of $t$ in spectrum of $\alpha$. Finite set of real
numbers $t\in\mathbb{R}$ with positive multiplicity
$m_\alpha(t)>0$ called  {\it spectrum\,} of filtration $\alpha$.
%The spectrum consists of
This are just discontinuity points of {\it dimension function}
\begin{equation}
d_\alpha(t)=\dim\mathcal{H}^\alpha(t).
\end{equation}
We always arrange the spectral values (counted according the
multiplicities) in nonincreasing order
\begin{equation}\label{spec}\Spec(\alpha):a_1\ge a_2\ge\cdots\ge a_n,\quad
n=\dim\mathcal{H}_A.\end{equation}
\begin{Ex} Hermitian operator
$\alpha:\mathcal{H}\rightarrow\mathcal{H}$ defines {\it spectral
filtration\,}
$$\mathcal{H}^\alpha(t)=\left\{\parbox{5cm}{subspace
spanned by eigenspaces of $\alpha$ with eigenvalues $\ge
t$}\right\}.$$ Spectrum of this filtration is equal to spectrum of
the operator $\alpha$. Every filtration  of Hilbert space
$\mathcal{H}$ is a spectral one. Hence filtration is a metric
independent substitution for Hermitian operator.
\end{Ex}
We often refer to spectrum of a filtration  as its {\it type}.
%and in this context use {\it simplified notation\,}
%Since the
%type often appears as superscript we our notation for spectrum
%$$\{\alpha\}:=\Spec(\alpha).$$
Filtrations $\alpha$ of fixed type $a=\Spec(\alpha)$ form {\it
flag variety\,} $\Flag_{a}(\mathcal{H})$
%It depends only on multiplicity of the spectral values, rather then the spectrum
%itself, and
consisting of all chains of subspaces
\begin{equation}\label{flag}\mathcal{H}=F^1\supset
F^2\supset\cdots\supset F^\ell\supset F^{\ell+1}=0,\end{equation}
with composition factors $F^i/F^{i+1}$ of dimension equal to
multiplicity of $i$-th spectral value in $a$.

\subsubsection{Composition of filtrations}
Let now $\alpha$, $\beta$ be filtrations in spaces
$\mathcal{H}_A$, $\mathcal{H}_B$ respectively. Define their {\it
composition\,} $\alpha\beta$ as filtration of
$\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ given by
equation
%Filtrations  $\alpha$, $\beta$ of spaces $\mathcal{H}_A$,
%$\mathcal{H}_B$ define filtration of
%$\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$
\begin{equation}\mathcal{H}_{AB}^{\alpha\beta}(t)=\sum_{r+s=t}
\mathcal{H}_A^\alpha(r)\otimes\mathcal{H}_B^\beta(s).\label{comp}\end{equation}
%called {\it composition\,} of filtrations $\alpha,\beta$.
%It has composition factors
%\begin{equation}\mathcal{H}_{AB}^{[\alpha\beta]}(z)\cong\bigoplus_{x+y=z}
%\mathcal{H}_A^{[\alpha]}(x)\otimes\mathcal{H}_B^{[\beta]}(y),\end{equation}
%and therefore type of the composition is given by equation
%filtration $\mathcal{H}_{AB}^{\alpha\beta}$ is determined by the
%types of filtrations $\alpha,\beta$. Specifically, let
The type of the composition $\alpha\beta$ depends only on spectra
$a=\Spec(\alpha)$ and $b=\Spec(\beta)$
%\begin{align*}\{\alpha\}&=\{\alpha_1\ge\alpha_2\ge\cdots\ge\alpha_n\},\\
%\{\beta\}&=\{\beta_1\ge\beta_2\ge\cdots\ge\beta_m\}
%\end{align*}
%of the components:
%Then
\begin{equation}\label{SpecComp}\Spec(\alpha\beta)=\{a_i+b_j
\text{ arranged in nonincreasing order}\}.
\end{equation}
%for filtrations $\alpha,\beta$ of types
%$\{\alpha\}=\{\alpha_1\ge\alpha_2\ge\cdots\ge\alpha_n\}$ and
%$\{\beta\}=\{\beta_1\ge\beta_2\ge\cdots\ge\beta_m\}$.
%\begin{equation}m_{\alpha\beta}(z)=\sum_{x+y=z}m_\alpha(x)m_\beta(y),\end{equation}
%and $\Spec(\alpha\beta)$ consists of numbers $\alpha_i+\beta_j$,
%$\alpha_i\in\Spec\alpha$, $\beta_j\in\Spec\beta$ arranged in
%nonincreasing order.
%Every filtration $\alpha$ of space $\mathcal{H}$ defines {\it
%flag\,}
%\begin{equation}\Flag(\alpha):\mathcal{H}=F^1\supset
%F^2\supset\cdots\supset F^\ell\supset F^{\ell+1}=0\end{equation}
%consisting of all different subspaces $F^i\subseteq \mathcal{H}$
%of the form $F^i=\mathcal{H}^\alpha(x)$ for some $x\in
%\mathbb{R}$. Dimensions of composition factors $F^i/F^{i+1}$ of
%the flag coincide with the multiplicities of spectral values in
%$\Spec(\alpha)$.
%We denote ordered set of these dimensions by
%$[\alpha]$ and call it {\it type\,} of the flag $\Flag(\alpha)$.
%Flags of a given type $[\alpha]$ form {\it flag variety\,}
%$\Flag^{[\alpha]}(\mathcal{H})$.
Therefore the composition of filtrations of {\it fixed\,} types
$a$  and $b$ gives rise to morphism of flag varieties
%\marginpar{\footnotesize May be more natural notation for the
%composition would be $a+b$.}
\begin{equation}\label{morph}
\varphi_{a,b}:
\Flag_{a}(\mathcal{H}_A)\times\Flag_{b}(\mathcal{H}_B)\rightarrow
\Flag_{ab}(\mathcal{H}_{AB}),\quad
\alpha\times\beta\mapsto\alpha\beta,
\end{equation}
where composition of spectra $ab$ is defined by RHS of equation
(\ref{SpecComp}).

%In a similar way one can define morphism
%\begin{equation}\label{morph3}
%\varphi_{a,b,c}:
%\Flag_{a}(\mathcal{H}_A)\times\Flag_{b}(\mathcal{H}_B)\times\Flag_{c}(\mathcal{H}_C)\rightarrow
%\Flag_{abc}(\mathcal{H}_{ABC}),
%\end{equation}
%for three or more spectra $a,b,c$.
\subsubsection{Cubicles and extremal edges} Morphism (\ref{morph}) depends only on
the {\it order\,} of quantities $a_i+b_j$ in spectrum
(\ref{SpecComp}). We call pairs of spectra $(a,b)$ and
$(\tilde{a},\tilde{b})$ to be equivalent iff they define the same
morphism $\varphi_{a,b}=\varphi_{\tilde{a},\tilde{b}}$. This means
that the quantities $a_i+b_j$ and $\tilde{a}_i+\tilde{b}_j$ come
in the same order
\begin{equation}a_i+b_j\le a_k+b_l\Leftrightarrow
\tilde{a}_i+\tilde{b}_j\le\tilde{a}_k+\tilde{b}_l.
\end{equation}
Note that affine transformations of the spectra
\begin{equation}
a_i\mapsto p\cdot a_i+q,\qquad b_j\mapsto p\cdot b_j+r,\quad
p>0;q,r\in\Real,
\end{equation}
preserve the equivalence classes. This allows reduce the spectra
$a,b$ to Weyl chambers
\begin{equation}\label{Weyl}
\begin{aligned}
\Delta_m&:a_1\ge a_2\ge\cdots\ge a_m,&{\sum}_i a_i&=0,& m&=\dim \mathcal{H}_A;\\
\Delta_n&:b_1\ge b_2\ge\cdots\ge b_n,&{\sum}_j b_j&=0,&n&=\dim \mathcal{H}_B.
\end{aligned}
\end{equation}
The equivalence produces a decomposition of the product
$\Delta_m\times\Delta_n$ into relatively open polyhedral cones. We
are mostly interested in cones of maximal and minimal dimension,
called {\it cubicles\,} and {\it extremal edges\,} respectively.
%The cones of maximal dimension $(m-1)(n-1)$ are called {\it
%cubicles\,}.
The cubicles  are just pieces into which hyperplanes
\begin{equation}\label{facet}
a_i+b_j=a_k+b_l.
\end{equation}
cut $\Delta_m\times\Delta_n$. Extremal edges are given by a system
of equations of this form with one dimensional space of solutions.
%unique  nonzero solution.


A cubicle consists of spectra $(a,b)$ with pairwise distinct
quantities $a_i+b_j$ coming in {\it fixed  order\,}. It can be
described by $m\times n$ matrix $T$ with entries $1,2,\ldots,mn$
written in the {\it opposite\,} order to that of matrix
$[a_i+b_j]$:
$$T_{ij}\ge T_{kl}\Leftrightarrow a_i+b_j\le
a_k+b_l.$$ In other words, $T$
%is just
shows a way of counting entries of matrix $[a_i+b_j]$ in {\it
decreasing\,} order. The entries of $T$ strictly increase in rows
and columns, i.e. $T$ is a {\it standard tableau\,} of rectangular
shape $m\times n$.
%The number of such tableaux is given by
There is a famous {\it hook formula\,} \cite{Macdonald} for the
number of such tableaux, which in
%our case
current settings takes form
\begin{equation}\label{hook}
\#\{\text{ standard $m\times n$
tableaux }\}=\frac{(mn)!}{\prod\limits_{\substack{1\le i\le
m\\1\le j\le n}}(i+j-1)}
\end{equation}
and gives an {\it upper bound\,} for the number of cubicles.
\begin{Ex} \label{2xn} For system of format $2\times n$ every standard tableau
corresponds to a cubicle. Hence by (\ref{hook}) the number of
cubicles is equal to {\it Catalan number\,}
\begin{equation*}\label{Catalan} \#\{\text{ cubicles
}\}=\frac{1}{n+1}{2n\choose n}.
\end{equation*}
A typical extremal edge in this case  comes from spectra
$$a=(1/2,-1/2),\qquad b=(b_1,b_2,\ldots,b_n),$$
with $b_i-b_{i+1}=0,1; \quad\sum b_i=0$.
%We have to add to the list edges of $\Delta_n$
The remaining extremal edges are that of $\Delta_n$
$$ a=(0,0),\qquad b=(\underbrace{k,k,\ldots,k}_{n-k},\underbrace{k-n,k-n,\ldots,k-n}_k).$$
This amounts altogether to $2^{n-1}+n-1$ extremal edges, which is
of order square root of the number of cubicles. In simplest case
of two qubits they are
\begin{equation}\label{2x2}
\begin{aligned}
a&=(1,-1);&a&=(0,\;\;\,0);&a&=(1,-1);\\
b&=(1,-1);&b&=(1,-1);&b&=(0,\;\;\, 0);
\end{aligned}
\end{equation}
\end{Ex}
\begin{Ex}
For two qutrits (=system of format $3\times3$) there are $36$
cubicles represented by the following tableaux and their transpose
\begin{center}
\vspace{3mm}
\begin{tabular}{|c|c|c|}
\hline
1&4&7\\
\hline
2&5&8\\
\hline
3&6&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&4&6\\
\hline
2&5&8\\
\hline
3&7&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&4&6\\
\hline
2&5&7\\
\hline
3&8&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&4&5\\
\hline
2&6&8\\
\hline
3&7&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&4&5\\
\hline
2&6&7\\
\hline
3&8&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&5&6\\
\hline
2&3&8\\
\hline
4&7&9\\
\hline
\end{tabular}

\vspace{3mm}
\begin{tabular}{|c|c|c|}
\hline
1& 3&6\\
\hline
2&5&7\\
\hline
4&8&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&3&7\\
\hline
2&5&8\\
\hline
4&6&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&3&7\\
\hline
2&4&8\\
\hline
5&6&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&3&6\\
\hline
2&4&8\\
\hline
5&7&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&3&6\\
\hline
2&4&7\\
\hline
5&8&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&2&6\\
\hline
3&4&8\\
\hline
5&7&9\\
\hline
\end{tabular}

\vspace{3mm}
\begin{tabular}{|c|c|c|}
\hline
1&2&7\\
\hline
3&4&8\\
\hline
5&6&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&2&5\\
\hline
3&6&8\\
\hline
4&7&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&2&6\\
\hline
3&5&8\\
\hline
4&7&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&2&7\\
\hline
3&5&8\\
\hline
4&6&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&3&5\\
\hline
2&6&7\\
\hline
4&8&9\\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|c|}
\hline
1&3&5\\
\hline
2&6&8\\
\hline
4&7&9\\
\hline
\end{tabular}
\end{center}
\vspace{3mm}
By hook formula (\ref{hook}) there are 42 standard $3\times3$
tableaux, 6 of them correspond to no cubicle.
%There are $42$ standard $3\times3$ tableaux, and 6 of them do not
%represent a cubicle.
%The remaining 6 tableaux correspond to no cubicle.
There are 17
extremal edges spanned by spectra $(a,b)$ or $(b,a)$ below
\begin{align*}
a&=(1,\;\;\,0,-1);&a&=(3,\;\;\,0,-3);&a&=(0,\;\;\,0,\;\;\,0);&a&=(3,\;\;\,0,-3);\\
b&=(1,\;\;\,0,-1);&b&=(5,-1,-4);&b&=(2,-1,-1);&b&=(1,\;\;\,1,-2);\\[3mm]
a&=(2,-1,-1);&a&=(3,\;\;\,0,-3);&a&=(0,\;\;\,0,\;\;\,0);&a&=(3,\;\;\,0,-3);\\
b&=(2,-1,-1);&b&=(4,\;\;\,1,-5);&b&=(1,\;\;\,1,-2);&b&=(2,-1,-1);\\[3mm]
&&a&=(1,\;\;\,1,-2);&a&=(2,-1,-1);&&\\
&&b&=(1,\;\;\,1,-2);&b&=(1,\;\;\,1,-2).&&
\end{align*}
\end{Ex}

%We'll need also in cones of dimension one called {\it extremal
%edges}. They are given by a system of equations of the form
%(\ref{facet}) with unique  nonzero solution.



\subsubsection{Cohomology of flag varieties}\label{cohom}
Let $T$ be a cubicle. By definition, morphism $\varphi_{ab}$
defined by equation (\ref{morph}) is independent of $(a,b)\in T$.
%(\ref{morph})
%depends only on $T$
Hence
%therefore will be denoted by
the notation $\varphi_{ab}=\varphi_T$.
%Moreover in this case
Besides, all three spectra $a,b,ab$ are simple and the
corresponding flag varieties $\Flag_a(\mathcal{H}_A)$,
$\Flag_b(\mathcal{H}_B)$, $\Flag_{ab}(\mathcal{H}_{AB})$ are {\it
complete\,}, i.e. consist of filtrations with composition factors
of dimension one. Thus every cubicle gives rise to a well defined
morphism of complete flag varieties
\begin{equation}
\varphi_T:
\Flag(\mathcal{H}_A)\times\Flag(\mathcal{H}_B)\rightarrow
\Flag(\mathcal{H}_{AB})
\end{equation}
 and that of their cohomology
\begin{equation}\label{CohomMorph}
\varphi_T^*:
H^*(\Flag(\mathcal{H}_{AB}))\rightarrow
H^*(\Flag(\mathcal{H}_A))\otimes H^*(\Flag(\mathcal{H}_B)).
\end{equation}
Cohomology ring of complete flag variety $H^*(\Flag(\mathcal{H}))$
is generated by characteristic classes $x_i=c_1(\mathcal{L}_i)$ of
line bundles $\mathcal{L}_i$ with fibers equal to $i$-th
composition factor of the flag (\ref{flag}). We call $x_i$ {\it
canonical generators\,}. Elementary symmetric functions
$\sigma_i(x)$ of the canonical generators are characteristic
classes of trivial bundle $\mathcal{H}$ and thus vanish. This
identifies the cohomology ring with factor
\begin{equation}\label{canonical}
H^*(\Flag(\mathcal{H}))=\mathbb{Z}[x_1,x_2,\ldots,x_n]/(\sigma_1,\sigma_2,\ldots,\sigma_n).
\end{equation}
In term of the canonical generators $x_i,y_j,z_k$  of cohomology
of flag varieties $\Flag(\mathcal{H}_A)$, $\Flag(\mathcal{H}_B)$,
and $\Flag(\mathcal{H}_{AB})$ morphism $\varphi_T^*$ can be
described as follows
\begin{equation}\label{special}
\varphi_T^*: z_k\mapsto x_i+ y_j \text{ for } k=T_{ij},
\end{equation}
where we identify cubicle with the corresponding tableau $T$ and
for simplicity write $x_i,y_j$ instead of $x_i\otimes 1, 1\otimes
y_j$. In other words, $z_k\mapsto x_i+y_j$ iff $k$-th term of the
composition $ab$ is $a_i+b_j$ for any $(a,b)$ in cubicle $T$.
%Here $x_i,y_j$ stand for $x_i\otimes 1, 1\otimes y_j$.

The cohomology ring $H^*(\Flag(\mathcal{H}))$ has a natural
geometric basis consisting of so called {\it Schubert cocycles\,}
$\sigma_w$, where $w\in S_n$ is a permutation of degree $n=\dim
\mathcal{H}$. They can be expressed via
characteristic classes $x_i$ in terms of {\it difference
operators\,}
\begin{equation}
\partial_i:f(x_1,x_2,\ldots,x_n)\mapsto
\frac{f(\ldots,x_i,x_{i+1},\ldots)-f(\ldots,x_{i+1},x_i,\ldots)}
{x_i-x_{i+1}}
\end{equation}
as follows. Write permutation $w\in S_n$ as product of minimal
number of transpositions $s_i=(i,i+1)$
\begin{equation}\label{decomp}
w=s_{i_1}s_{i_2}\cdots s_{i_\ell}.
\end{equation}
The number of factors $\ell(w)=\#\{i<j\mid w(i)>w(j)\}$ called
{\it length\,} of permutation $w$.
%It is equal to the number of inversions in $w$
%$$\ell(w)=\#\{i<j\mid w(i)>w(j)\}.$$
The product
\begin{equation}
\partial_w:=\partial_{i_1}\partial_{i_2}\cdots\partial_{i_\ell}
\end{equation}
is independent of reduced decomposition (\ref{decomp}) and  in
terms of these operators Schubert cocycle $\sigma_w$ is given by
equation
\begin{equation}\label{Schub}
\sigma_w=\partial_{w^{-1}w_0}(x_1^{n-1}x_2^{n-2}\cdots x_{n-1}),
\end{equation}
where  $w_0=(n,n-1,\ldots,2,1)$ is unique permutation of maximal
length.
%$\ell(w_0)=\frac{n(n-1)}{2}$.

Right hand side of equation (\ref{Schub}) called {\it Schubert
polynomial\,} $S_w(x_1,x_2,\ldots,x_n)$, $\deg S_w=\ell(w)$. These
polynomials where first introduced by Lascoux and
Sch\"utzen\-berger \cite{L-Sch} who studied them in a long series
of papers. See \cite{Mac91} for further references and a concise
exposition of the theory. We borrowed  from \cite{L-Sch}  the
following table, in which $x,y,z$ stand for $x_1,x_2,x_3$. % of Schubert polynomials.
%The following table of
%Schubert polynomials is borrowed from \cite{L-Sch}.
\begin{center}{\small\label{SchPol}
\begin{tabular}[b]{|c|c||c|c||c|c||c|c|}
\hline%\cline{1-8}
%\vspace{1pt}
$w$ & $S_w$&$w$&$S_w$&$w$&$S_w$&$w$&$S_w$\\
\hline%
3201&{\footnotesize $x^3y^2z$}&2301&{\footnotesize $x^2y^2$}&
2031&{\footnotesize $x^2y+x^2z$}&1203&{\footnotesize $xy$}\\
\hline
2310&{\footnotesize $x^2y^2z$}&3021&{\footnotesize $x^3y+x^3z$}&
2103&{\footnotesize $x^2y$}&2013&{\footnotesize $x^2$}\\
\hline
3120&{\footnotesize $x^3yz$}&3102&{\footnotesize $x^3y$}&
3012&{\footnotesize $x^3$}&0132&{\footnotesize $x+y+z$}\\
\hline
3201&{\footnotesize $x^3y^2$}&1230&{\footnotesize $xyz$}&
0231&{\footnotesize $xy+yz+zx$}&0213&{\footnotesize $x+y$}\\
\hline
1320&{\footnotesize $x^2yz+xy^2z$}&0321&{\footnotesize $x^2y+x^2z+xy^2$}&
0312&{\footnotesize $x^2+xy+y^2$}&1023&{\footnotesize $x$}\\
\hline
2130&{\footnotesize $x^2yz$}&1302&{\footnotesize $x^2y+xy^2$}&
1032&{\footnotesize $x^2+xy+xz$}&0123&{\footnotesize $1$}\\
\hline
\end{tabular}}
\end{center}
%\vspace{2mm}

%Insertion
Extra variables $x_{n+1},x_{n+2},\ldots$ being added to
(\ref{Schub}) leave Schubert polynomials unaltered.
%Schubert polynomials remain
%unchanged if we add
%%to definition (\ref{Schub}) dummy
%extra variables $x_{n+1},x_{n+2},\ldots$ in (\ref{Schub}).
%The definition is invariant
%with respect to extension of the set of variables $x_i$.
By that reason they are usually treated as
%it is convenient to treat them as
polynomials in infinite alphabet  $X=(x_1,x_2,\ldots)$. With this
understanding every homogeneous polynomial can be decomposed into
Schubert components
%over Schubert ones
as follows
\begin{equation}
f(X)=\sum_{\ell(w)=\deg(f)} \partial_w f\cdot S_w(X).
\end{equation}
Applying this to specialization (\ref{special}) of Schubert
polynomial $S_w(Z)$ we get decomposition
\begin{equation}\label{SchMap}
\varphi_T(S_w(Z))=\sum_{\ell(u)+\ell(v)=\ell(w)} c_{uv}^w(T)\cdot
S_u(X)S_v(Y),
\end{equation}
where
%\marginpar{\footnotesize Can this be written more explicitly?}
\begin{equation}\label{SchCoeff}
c_{uv}^w(T)=\partial_u\partial_v
S_w(Z)\left|_{z_k=x_i+y_j}\right.,\qquad k=T_{ij}.
\end{equation}
Here operators $\partial_u,\partial_v$ act on variables $x,y$
respectively. Reduction of (\ref{SchMap}) modulo elementary
symmetric functions in $x$ and  in $y$ gives cohomology morphism
(\ref{CohomMorph}) in terms of Schubert cocycles
%The cohomology morphism (\ref{CohomMorph}) in terms of Schubert
%cocycles  is just a reduction of (\ref{SchMap}) modulo elementary
%symmetric functions in $x$ and  in $y$
%Equation (\ref{SchMap}) taken
%modulo elementary symmetric functions in $x$ and in $y$ evaluates
%cohomology morphism
\begin{equation}\label{CocycleMap}
\varphi_T(\sigma_w)=\sum_{\ell(u)+\ell(v)=\ell(w)} c_{uv}^w(T)\cdot
\sigma_u\otimes\sigma_v.
\end{equation}
\subsubsection{Marginal inequalities} Now we are in position to
state a solution of the quantum marginal problem.
%To avoid
%technicalities we start with two component systems.
\begin{thm}\label{cubicleThm}
Let $\mathcal{H}_{AB}=\mathcal{H}_A\otimes \mathcal{H}_B$ be two
component system.
%of format $m\times n$.
Then   the following
conditions on spectra $\lambda^A,\lambda^B,\lambda^{AB}$
%\begin{align*}
%\lambda&:\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_m,\\
%\mu&:\mu_1\ge\mu_2\ge\cdots\ge\mu_n,\\
%\nu&:\nu_1\ge\nu_2\ge\cdots\ge\nu_{mn},
%\end{align*}
are equivalent.
%\begin{enumerate}
%\item
\newline
${}$\qquad {\rm (1)} There exists mixed state $\rho_{AB}$ with
margins $\rho_A$, $\rho_B$ and spectra
$\lambda^{AB},\lambda^A,\lambda^B$.
%$$\Spec\rho_A=\lambda,\quad\Spec\rho_B=\mu,
%\quad\Spec\rho_{AB}=\nu.$$
%\item
\newline
${}$\qquad {\rm (2)}
%For every cubicle $T$ and permutations $u\in
%S_n,v\in S_m,w\in S_{mn}$ such that coefficient $c_{uv}^w(T)$
%given by equation $(\ref{SchCoeff})$ is nonzero the following
%inequality
Each time the coefficient $c_{uv}^w(T)$ of decomposition
$(\ref{CocycleMap})$
%given by equation $(\ref{SchCoeff})$
is nonzero, the following inequality holds for all $(a,b)$ in the
cubicle $T$
\begin{equation}\label{MargIneq}
\sum_i a_i\lambda^A_{u(i)}+\sum_j b_j\lambda^B_{v(j)}\ge \sum_{k}
(a_i+b_j)_k\lambda^{AB}_{w(k)},
\end{equation}
where $(a_i+b_j)_k$ is $k$-th element of the sequence $a_i+b_j$
arranged in decreasing order.
%where $T_{ij}$ are entries of the tableau corresponding to cubicle
%$T$.
%\end{enumerate}
\end{thm}
\begin{proof}
The quantum marginal problem amounts to decomposition of
projection of a coadjoint orbit of group
$\SU(\mathcal{H}_A\otimes\mathcal{H}_B)$ into coadjoint orbits of
subgroup $\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)$.

The above discussion just puts the result into
%The result follows from the above discussion which put it into
framework of Berenstein-Sjamaar theorem \cite[Thm 3.2.1]{Ber-Sja}
applied  these groups.
%to subgroup $\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)\subset
%\SU(\mathcal{H}_{A}\otimes\mathcal{H}_B)$.
%The theorem is just a restatement with
%minor modification of Berenstein-Sjamaar Theorem 3.2.1
%\cite{Ber-Sja} applied to subgroup
%$\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)\subset
%\SU(\mathcal{H}_{AB})$.
\end{proof}
\begin{rem}
%Since Every point in a
Every cubicle is interior of convex hull  of its extremal edges.
Therefore inequality (\ref{MargIneq}) enough to check for extremal
edges only.
%This allows
%completely eliminate cubicles from the statement of the theorem.
%Thus there are only finitely
%many marginal inequalities.
%Note
%that it is enough to check linear inequality (\ref{MargIneq}) for
%extremal edges of the cubicle. Linear inequality (\ref{MargIneq})
%holds for all $(a,b)$ in the cubicle $T$ iff it holds for all its
%extremal edges.
\end{rem}
\begin{Ex} Note that $c_{uv}^w(T)=1$ for identical permutations $u,v,w$.
Hence the following inequality holds for every extremal edge
$(a,b)$
\begin{equation}\label{basic}
\sum_i a_i\lambda^A_{i}+\sum_j b_j\lambda^B_{j}\ge \sum_{k}
(a_i+b_j)_k\lambda^{AB}_{k}.
\end{equation}
%where in the right hand side spectral values $\lambda^{AB}_k$
%eigenvalues $\nu_k,k=1,2,\ldots,mn$
%follows in the same order as quantities $a_i+b_j$, so that the
%right hand side is maximal possible.
%indices $k(i,j)=1,2,\ldots,mn$ follows in the same order as
%$$\lambda^{AB}_{ij}<\lambda^{AB}_{pq}\Rightarrow a_i+b_j\le a_p+b_q.$$
%An ambiguity of this definition is irrelevant, it just makes RHS
%of (\ref{basic}) maximal.
We call these marginal inequalities {\it basic\,} ones.
\end{Ex}
\begin{Ex}\label{basic2x2} {\it ``Easy" two qubit inequalities.} Extremal edges for two qubits from Example \ref{2xn}
give rise to basic inequalities
%The basic
%inequalities for two qubits follows from the description of
%extremal edges
%system of format $2\times n$ follows from the
%description of extremal edges
%in Example \ref{2xn} equation (\ref{2x2})
\begin{align*}
&\lambda^A_1-\lambda^A_2+\lambda^B_1-\lambda^B_2\le 2(\lambda^{AB}_1-\lambda^{AB}_4),\\
&\lambda^A_1-\lambda^A_2\le \lambda^{AB}_1+\lambda^{AB}_2-\lambda^{AB}_3-\lambda^{AB}_4,\\
&\lambda^B_1-\lambda^B_2\le
\lambda^{AB}_1+\lambda^{AB}_2-\lambda^{AB}_3-\lambda^{AB}_4,
\end{align*}
which are equivalent to the first three inequalities in
(\ref{brav}). S.~Bravyi \cite{Brav} calls  them {\it easy\,} ones.
\end{Ex}
%The remaining marginal inequalities can be obtained from basic
%ones (\ref{basic}) as follows.
Extremal edge $(a,b)$ gives rise to
cohomology map
%Let $(a,b)$ be extremal edge and map
\begin{equation}\label{IncplMorph}
\varphi_{ab}^*:H^*(\Flag_{ab}(\mathcal{H}_{AB}))\rightarrow
H^*(\Flag_{a}(\mathcal{H}_{A}))\otimes
H^*(\Flag_{b}(\mathcal{H}_{B}))
\end{equation}
%be morphism of cohomologies
induced by morphism $\varphi_{ab}$
%defined  by
from equation (\ref{morph}). In contrast with cubicle case,
spectrum $ab$ is never simple and we have to deal with varieties
of {\it incomplete\,} flags.
%for extremal edge $(a,b)$
%spectra $a,b,ab$ are almost never simple.
%usually multiple.
%As a rule spectra $a,b,ab$
%are multiple.

Let $\alpha$ be a filtration of space $\mathcal{H}$ of dimension
$m$ and spectrum $a=\Spec\alpha$. Denote by
$\mu(a)=(m_1,m_2,\ldots,m_\ell)$, $\sum_i m_i=m$ the
multiplicities of the spectrum.
%
%Let $\mu(a)=(m_1,m_2,\ldots,m_\ell)$, $\sum_i m_i=m$ be
%multiplicities of $a_1>a_2>\cdots>a_\ell$ in spectrum
%$a=\Spec\alpha$ of filtration $\mathcal{H}^\alpha(x)$,
%$\dim\mathcal{H}=m$.
%Let values
%$a_1>a_2>\cdots>a_\ell$ enter in $\Spec(a)$ with multiplicities
%$\mu=(m_1,m_2,\ldots,m_\ell)$, $\sum m_i=m=\dim
%\mathcal{H}_A$. The space
Recall that flag variety $\Flag_{a}(\mathcal{H})$ consists of
%flags (\ref{flag})
the filtrations with composition factors of dimension $m_i$. Its
cohomology
%ring can be identified with invariants
known to be subring of invarinats
\begin{equation}\label{IncFlag}
H^*(\Flag_{a}(\mathcal{H}))=H^*(\Flag(\mathcal{H}))^{S_\mu}\subset
H^*(\Flag(\mathcal{H}))
\end{equation}
with respect to Schur subgroup $S_\mu=S_{m_1}\times
S_{m_2}\times\cdots\times S_{m_\ell}\subset S_m$ acting on
cohomolohy of {\it complete\,} flags by permutations of the
%characteristic classes
canonical generators $x_i$, see \cite{BGG}. An important example
of such invariant comes from Schubert cocycle $\sigma_w$
corresponding to a {\it shuffle of type $\mu(a)$}, i.e.
permutation $w\in S_m$ which preserves order of the first $m_1$
elements, the next $m_2$ elements, and so on. Such Schubert
cocycles form a basis of the cohomology ring (\ref{IncFlag}) of
incomplete flags.

%Keeping in mind this interpretation,  morphism (\ref{IncplMorph})
%can be described just as restriction of
In view of this interpretation morphism (\ref{IncplMorph}) becomes
just a restriction of
\begin{equation}
\varphi_T^*:
H^*(\Flag(\mathcal{H}_{AB}))\rightarrow
H^*(\Flag(\mathcal{H}_A))\otimes H^*(\Flag(\mathcal{H}_B))
\end{equation}
onto the algebras of invariants for any cubicle $T$ containing
extremal edge $(a,b)$ in its closure. Hence by
(\ref{CocycleMap})-(\ref{SchCoeff})
%In  particular,
coefficients of the decomposition
\begin{equation}\label{EdgeMorph}
\varphi_{ab}(\sigma_w)=\sum_{\ell(u)+\ell(v)=\ell(w)} c_{uv}^w(a,b)\cdot
\sigma_u\otimes \sigma_v,
\end{equation}
are given by equation
\begin{equation}
c_{uv}^w(a,b)=\partial_u\partial_v
S_w(Z)\left|_{z_k=x_i+y_j}\right.,\qquad k=T_{ij},
\end{equation}
where $u,v,w$ are shuffles of types $\mu(a),\mu(b),\mu(ab)$
respectively. This leads to the following description of marginal
inequalities in terms of extremal edges only.

%Now we can describe marginal inequalities in terms of extremal
%edges only.

\begin{thm}\label{ExtrEdge} All marginal inequalities for system
$\mathcal{H}_{AB}=\mathcal{H}_A\otimes \mathcal{H}_B$ can be
obtained from basic ones $(\ref{basic})$ by shuffles $u,v,w$ of
types $\mu(a),\mu(b),\mu(ab)$
\begin{equation}\label{modifiedINQ}
\sum_i a_i\lambda^A_{u(i)}+\sum_j b_j\lambda^B_{v(j)}\le\sum_{k}
(a_i+b_j)_k\lambda^{AB}_{w(k)}
\end{equation}
such that coefficient $c_{uv}^w(a,b)$ of decomposition
$(\ref{EdgeMorph})$ is nonzero.
\end{thm}
Extensive tables of marginal inequalities deduced from this
theorem are given in Addendum. Details of the calculation will be
published elsewhere \cite{Kl-Oz}.
%\marginpar{\footnotesize Reference to joint paper with I.~\"Ozen.}  Here we consider only a simple illustrative
%example.

\begin{Ex} {\it ``Difficult" two qubit inequality\,.}
Let's find modifications (\ref{modifiedINQ}) of the first basic
inequality form Example \ref{basic2x2}
\begin{equation}\label{first}
\lambda^A_1-\lambda^A_2+\lambda^B_1-\lambda^B_2\le
2(\lambda^{AB}_1-\lambda^{AB}_4),
\end{equation}
coming from extremal edge $a=(1,-1),b=(1,-1)$. There are two ways
of counting entries of matrix
%Let's return to two qubit system with three extremal edges $(a,b)$
%given  by equation (\ref{2x2}). The corresponding matrices
%$[a_i+b_j]$ look as follows
\begin{equation*}\label{ab}
%\begin{center}
%\vspace{3mm}
[a_i+b_j]=\begin{tabular}{|c|c|}
\hline
2&0\\
\hline
0&-2\\
\hline
\end{tabular}
%\qquad
%\begin{tabular}{|c|c|}
%\hline
%1&-1\\
%\hline
%1&-1\\
%\hline
%\end{tabular}
%\qquad
%\begin{tabular}{|c|c|}
%\hline
%1&1\\
%\hline
%-1&-1\\
%\hline
%\end{tabular}
\end{equation*}
%\end{center}
%There are two choices of standard tableau (=cubicle) corresponding
%to the first matrix depending on the order in which one counts off
%diagonal zeros:
%There are two ways to count entries of the first matrix
in decreasing order
%given by  tableaux
\begin{equation}\label{2x2cubicle}
\begin{tabular}{|c|c|}
\hline
1&3\\
\hline
2&4\\
\hline
\end{tabular}
\qquad \text{ or }\qquad
\begin{tabular}{|c|c|}
\hline
1&2\\
\hline
3&4\\
\hline
\end{tabular}
\end{equation}
%These tableaux
%corresponding  to
which exhibit  two cubicles containing $(a,b)$ in theirs closure.
%The same tableaux correspond to unique cubicles
%containing the second and the third edges respectively.
%Let's start with basic inequality coming from the first edge
%$a=(1,-1), b=(1,-1)$
%\begin{equation}\label{first}
%\lambda^A_1-\lambda^A_2+\lambda^B_1-\lambda^B_2\le
%2(\lambda^{AB}_1-\lambda^{AB}_4).
%\end{equation}
%To find all its modifications
%(\ref{modifiedINQ})

By Theorem \ref{ExtrEdge}
%we have to evaluate
the modifications come from components of
$\varphi_{ab}^*(\sigma_w)$ for shuffles $w$ of type $(1,2,1)$ and
length $\ell(w)\le 2$. Using first cubicle in (\ref{2x2cubicle})
the calculation amounts to specialization (\ref{special})
$$z_1=x_1+y_1,\quad z_2=x_2+y_1, \quad z_3=x_1+y_2, \quad z_4=x_2+y_2  $$
of Schubert polynomial $S_w(Z)$ taken modulo relations
$$x_1+x_2=y_1+y_2=0,\qquad x_1x_2=y_1y_2=0.$$
%A simple calculation based on table page
From the table
%of Schubert polynomials
on page \pageref{SchPol} it follows
%one finds out
\begin{gather*}
\varphi_{ab}^*(\sigma_{2134})=\varphi_{ab}^*(\sigma_{1243})=x_1+y_1=
\sigma_{21}\otimes\sigma_{12}+\sigma_{12}\otimes\sigma_{21},\\
\varphi_{ab}^*(\sigma_{2143})=\varphi_{ab}^*(\sigma_{3124})
=\varphi_{ab}^*(\sigma_{1342})=2x_1y_1=2\sigma_{21}\otimes\sigma_{21}.
\end{gather*}
The first line gives rise to 4 modified inequalities
%modifications of the basic inequality (\ref{first})
\begin{align*}
\lambda^A_2-\lambda^A_1+\lambda^B_1-\lambda^B_2&\le
2(\lambda^{AB}_2-\lambda^{AB}_4),\\
\lambda^A_1-\lambda^A_2+\lambda^B_2-\lambda^B_1&\le
2(\lambda^{AB}_2-\lambda^{AB}_4),\\
\lambda^A_2-\lambda^A_1+\lambda^B_1-\lambda^B_2&\le
2(\lambda^{AB}_1-\lambda^{AB}_3),\\
\lambda^A_1-\lambda^A_2+\lambda^B_2-\lambda^B_1&\le
2(\lambda^{AB}_1-\lambda^{AB}_3),
\end{align*}
which can be squeezed into the following one
%amount to nonlinear inequality
$${\mid(\lambda^A_1-\lambda^A_2)-(\lambda^B_1-\lambda^B_2)\mid}\le
2\min(\lambda^{AB}_1-\lambda^{AB}_3,\lambda^{AB}_2-\lambda^{AB}_4).$$
%which is equivalent to the last one in the list (\ref{brav}).
This is the last inequality in (\ref{brav})
%is what
designated by S.~Bravyi \cite{Brav} as a {\it difficult\,} one.
%marginal inequality.
All the other modified inequalities are redundant.
We'll discuss the underlying reason for this in the next section.
\end{Ex}
%\marginpar{\footnotesize Add system of format $2\times3$ with 41 marginal inequalities.}
\subsubsection{Redundancy}
%From practical point of view
Redundancy of marginal inequalities
%({modifiedINQ}) of theorem \ref{ExtrEdge}
is an important issue. For example, for two qutrits there are 17
extremal edges $(a,b)$, 2298 permutations $w\in S_9$ of length
$\le 12$, in addition $\varphi_{ab}^*(\sigma_w)$ may have many
components, and each gives a marginal inequality. Among those tens
of thousands inequalities only 397 are independent.
%The following
%``standard" conjecture is supported by
\begin{conj}\label{conj} All marginal constraints are given by
inequalities $(\ref{modifiedINQ})$ of theorem \ref{ExtrEdge} with
$c_{uv}^w=1$.
\end{conj}
This ``standard" conjecture may be true in framework of general
Berenstein-Sjamaar theorem \cite{Ber-Sja}. It is valid for
multicomponent quantum systems of rank $\le 4$, see Appendix.
Apparently the conjecture should follow from rigidity argument of
Belkale \cite{Bel}, who proved it for Hermitian spectral problem,
see section \ref{AddSpec}. In this case the inequalities listed in
the conjecture are also independent \cite{KTW}. However for
quantum marginal problem this is not the case. For example, for
system of format $2\times2\times3$ even some basic inequalities
are redundant. Redundancy also occurs in a counterpart of the
Hermitian spectral problem for groups other then $\SU(n)$. In this
case Belkale and Kumar \cite{Bel-Kum} have a refined procedure
%version of the conjecture
which gives independent inequalities, at least in some examples.


\subsection{Arrays of qubits}\label{n_qubit} In this section we study in details
the system of $n$ qubit, which is of fundamental importance for
quantum information. In this case, modulo Conjecture \ref{conj},
the marginal inequalities can be written down  in a
straightforward manner from the list of extremal edges.
\subsubsection{Cubicles and extremal edges} Let's consider
array of qubits
$$\mathcal{H}^{\otimes n} = \mathcal{H}_1\otimes \mathcal{H}_2\otimes\cdots
\otimes \mathcal{H}_n,\quad\dim \mathcal{H}_i=2$$
and denote by  $\alpha_i$ a filtration of $\mathcal{H}_i$,
normalized to Weyl chamber (\ref{Weyl}), so that
$$\Spec\alpha_i=\pm a_i,\quad a_i\ge0.$$
Henceforth we'll identify the spectrum with nonnegative number
$a_i\ge0$. As in (\ref{morph}) composition of filtrations gives
rise to morphism
\begin{equation}\label{comp_morph}
\varphi_a:\Flag_{a_1}(\mathcal{H}_1)\times\Flag_{a_2}(\mathcal{H}_2)\times
\cdots\times\Flag_{a_n}(\mathcal{H}_n)\rightarrow
\Flag_{A}(\mathcal{H}_1\otimes\mathcal{H}_2\otimes\cdots\otimes\mathcal{H}_n),
\end{equation}
where on the left hand side $a$
%means the sequence
stands for vector of spectra $(a_1,a_2,\ldots,a_n)$, while on the
right hand side $A$ refers to their composition
$$A=a_1a_2\cdots a_n=\left\{\pm a_1\pm a_2\pm\cdots\pm a_n
\text{ arranged in nonincreasing order}\right\}.$$
 Note that flag variety $\Flag_{a_i} (\mathcal{H}_i)$
amounts to projective line $\mathbb{P}^1$ (=Riemann sphere) for
$a_i>0$, or to a point for $a_i=0$. Similar to two component
system, $\varphi_a$ depends only on the order of $2^n$ quantities
\begin{equation}\label{proj}
\pm a_1\pm a_2\pm \cdots\pm a_n
\end{equation}
%and a cubicle consists of nonnegative vectors $a$ with pairwise
%distinct quantities (\ref{proj}) in fixed order.
%This means that
%projections of vertices of the $n$-cube onto direction $a$
%Speaking geometrically a cubicle consist called cubicle which are
which are just dot products
%of nonnegative vector
of vector $a$ and a vertex $(\pm1,\pm1,\ldots,\pm1)$ of $n$-cube.
Hence $n$-qubit  cubicle is determined by the order of pairwise
distinct projections of vertices of the $n$-cube onto positive
direction $a$. Equivalently, cubicles are pieces into which
hyperplanes
\begin{equation}\label{wall}
%\begin{aligned}
\varepsilon_1a_1+\varepsilon_1a_1+\cdots+\varepsilon_n a_n=
\delta_1a_1+\delta_1a_1+\cdots+\delta_n a_n,\quad\varepsilon,\delta=\pm1,
%\end{aligned}
\end{equation}
cut positive octant. Equation of the wall (\ref{wall}) separating
two cubicles can be recast in the form
\begin{equation}\label{wall2}
H_\eta:\eta_1a_1+\eta_2a_2+\cdots+\eta_na_n=0,\quad\eta_i=0,\pm1,
\end{equation}
which tells that  $H_\eta$ is orthogonal to vector $\eta$ pointing
to center of a face of the cube.
%(codimension of the face is equal to the number of nonzero $\eta_i$).

Every extremal edge is intersection of $n-1$ walls, hence given by
an independent system of homogeneous equations with $(n-1)\times
n$ matrix $M$ filled by $0,\pm 1$. The solution is given by a
vector of maximal
%$(n-1)\times (n-1)$
minors taken with alternating signs $(-1)^iM_i$. Their absolute
values
\begin{equation*}
(\abs{M_1},\abs{M_2},\ldots,\abs{M_n})
\end{equation*}
gives a positive solution for another such matrix $M^\prime$ and
spans an extremal edge.
% hence spanned
%by vector
%\begin{equation*}
%(\abs{M_1},\abs{M_2},\ldots,\abs{M_n})
%\end{equation*}
%whose components are absolute values of maximal minors of $(0,\pm
%1)$-matrix $M$ of format $(n-1)\times n$.
%Extremal edges are intersections of the
%walls and can be given by an independent system of $n-1$ equation
%of the form (\ref{wall2}). In other words, every extrimal edge is
%generated by vector
%\begin{equation*}
%(\abs{M_1},\abs{M_2},\ldots,\abs{M_n})
%\end{equation*}
%of absolute values of maximal minors of $0,\pm 1$ matrix $M$ of
%format $(n-1)\times n$.
 Computer implementation of this brute
force approach allows to find all extremal edges up to 6 qubits.
The table below shows the number of extremal edges counted up to a
permutation of qubits, say normalized by condition $a_1\le a_2\le
\ldots\le a_n$.

\begin{center}
\vspace{1mm}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\# qubits &2&3&4&5&6\\
\hline
\# edges&2&4&12&125&11344\\
\hline
\end{tabular}
\end{center}

\begin{Ex} {\it Three qubits.}  Let $0\le a\le b \le c$ be three spectra
in increasing order. Then among 8 quantities $\pm a\pm b
\pm c$ only one inequality is indefinite
\begin{equation*}
a+b+c\ge-a+b+c\ge a-b+c\ge a+b-c\gtrless-a-b+c\ge -a+b-c\ge
a-b-c\ge -a-b-c
\end{equation*}
and depends on whether $a,b,c$ satisfy triangle inequality or not.
In the former case the order is
$$a+b+c\ge-a+b+c\ge a-b+c\ge
a+b-c\ge-a-b+c\ge -a+b-c\ge a-b-c\ge -a-b-c,$$ and in the later
$$a+b+c\ge-a+b+c\ge a-b+c\ge-a-b+c \ge a+b-c\ge -a+b-c\ge
a-b-c\ge -a-b-c.$$ Thus the system has two cubicles up to
permutations of qubits and 12 altogether. The two cubicles
%given by the above inequalities
have 4 extremal edges
\begin{equation}\label{Edges3qbt}
(0,0,1);\quad(0,1,1);\quad(1,1,1);\quad (1,1,2).
\end{equation}
\end{Ex}
\begin{Ex} The 12 extremal edges $a_1\le a_2\le a_3\le a_4$ for 4
qubits are as follows
\begin{align*}
&(0,0,0,1),&&(0,0,1,1),&&(0,1,1,1),&&(0,1,1,2),\\
&(1,1,1,1),&&(1,1,1,2),&&(1,1,1,3),&&(1,1,2,2),\\
&(1,1,2,3),&&(1,1,2,4),&&(1,2,2,3),&&(1,2,3,4).
\end{align*}
\end{Ex}

\begin{rem}
It would be nice to understand better geometry and combinatorics
of the system of hyperplanes $H_\eta$.
%of the cubicles and the extremal edges.
Note that in a configuration of $N$ central hyperplanes in general
position there are ${N\choose n-1}$ lines of intersections. This
gives an {\it upper bound\,} for the number of lines of
intersection in the system of $(3^n-1)/2$ walls (\ref{wall2}) and
leads to estimate
\begin{equation*}
\sum_{\substack{\text{extremal edges}\\a_1\le a_2\le\cdots\le a_n}}
\frac{1}{\mid\text{Stab}\,(a)\mid}
\le \frac{1}{2^{n-1}n!}{(3^n-1)/2\choose n-1},
\end{equation*}
where $\text{Stab}\,(a)$ is stabilizer of edge $a$ in group of
monomial permutations $a_i\mapsto \pm a_j$. In worst case scenario
it may happens that the system $H_\eta$ becomes stochastic
%is close to random one
for $n\gg 1$. Then there may be no reasonable description of its
edges and cubicles, except statistical one.
%Then its edges and cubicles have no reasonable
%combinatorial description.
%there is no universal
%combinatorial description and the system of extremal edges should
%be treated as a stochastic one for $n\gg 1$.

%Note that taking
%Treating equations (\ref{wall2}) modulo 3 one finds that every
%nonzero vector over field of three element $\mathbb{F}_3$ is
%congruent to a solution $(a_1,a_2,\ldots,a_n)$ of independent
%system of $(n-1)$ equations of the form (\ref{wall2}). To make
%$a_i\ge0$ we may have to change signs  $a_i\mapsto \pm a_i$. This
%gives at least $(3^n-1)/2^n$ different extremal edges modulo 3.
%Hence the number of extremal edges increases at least
%exponentially with $n$. However this approach gives only linear
%growth of the number of edges counted up to a permutation of
%qubits.
\end{rem}
\subsubsection{Marginal inequalities} Every vector $a$ in cubicle $T$
gives rise to morphism (\ref{comp_morph}) of {\it complete\,} flag
varieties depending only on $T$
\begin{equation}
\varphi_T:\mathbb{P}(\mathcal{H}_1)\times
\mathbb{P}(\mathcal{H}_2)\times\cdots\times\mathbb{P}(\mathcal{H}_n)\rightarrow
\Flag(\mathcal{H}_1\otimes\mathcal{H}_2\otimes\cdots\otimes\mathcal{H}_n),
\end{equation}
where we identify flag variety of $i$-th qubit
%space $\mathcal{H}_i$
with projective line $\mathbb{P}(\mathcal{H}_i)$. By theorem
\ref{cubicleThm} marginal inequalities come from the induced
cohomology morphism
\begin{equation}
\varphi_T^*:H^*(\Flag(\mathcal{H}_1\otimes\mathcal{H}_2\otimes\cdots\otimes\mathcal{H}_n))
\rightarrow \mathbb{Z}[x_1,x_2,\ldots,x_n]/(x_i^2=0).
\end{equation}
%where we
%identify cohomology of projective line
Here we use isomorphism
$H^*(\mathbb{P}(\mathcal{H}_i))\cong\mathbb{Z}[x_i]/(x_i^2=0)$,
%with truncated polynomial ring $\mathbb{Z}[x_i]/(x_i^2=0)$,
where $x_i=\sigma_{21}$ is Schubert cocycle.
%corresponding to transposition $(2,1)$.
In terms of canonical generators (\ref{canonical}) the morphism
amounts to specialization
\begin{equation}\label{SpecQbt}
\varphi_T^*:z_k\mapsto \pm x_1\pm x_2\pm\cdots \pm x_n,
\end{equation}
where the signs are taken from $k$-th term of the sequence $\pm
a_1\pm a_2\pm\cdots \pm a_n$ arranged  in decreasing order.
Applying this to Schubert cocycle $\sigma_w$ we get decomposition
\begin{equation}
\varphi_T^*(\sigma_w)=\sum c^w_{b_1b_2\ldots
b_n}(T)\cdot x_1^{b_1}x_2^{b_2}\ldots x_n^{b_n}
\end{equation}
into sum of square free monomials with exponents ${b_i=0,1}$. The
theorem \ref{cubicleThm} tells that each time $c^w_{b_1b_2\ldots
b_n}(T)$ is nonzero the following inequality holds for all $a\in
T$
\begin{equation}\label{QubitIneq}
\sum_i (-1)^{b_i}a_i(\rho_1^{(i)}-\rho_2^{(i)})\le \sum_{k}(\pm
a_1\pm a_2\pm\cdots\pm a_n)_k \rho_{w(k)},
\end{equation}
where $\rho$ is  $n$ qubit state with spectrum
$\rho_1\ge\rho_2\ge\cdots\ge \rho_{2^n} $ and margins
$\rho^{(i)}$, $(\pm a_1\pm a_2\pm\cdots\pm a_n)_k$ is $k$-th term
of the sequence $\pm a_1\pm a_2\pm\cdots\pm a_n$ arranged in
decreasing order, and sign $(-1)^{b_i}$ reflects the effect of
transposition of eigenvalues $\rho_1^{(i)}\ge
\rho_2^{(i)}$  of reduced state $\rho^{(i)}$.

Recall that according to Conjecture \ref{conj} all marginal
constraints come from inequalities (\ref{QubitIneq}) with
$c^w_{b_1b_2\ldots b_n}(T)=1$.
\begin{claim}\label{claim} For $n$ qubit system components of multiplicity one
in $\varphi_T^*(\sigma_w)$ can appear only for identical
substitution $w=1$ or for odd transposition $w=(2j-1,2j)$.
\end{claim}
\begin{proof} Indeed $\varphi_T^*(\sigma_w)$ is obtained from
Schubert polynomial $S_w(Z)$ by specialization (\ref{SpecQbt}).
Since $\pm x_1\pm x_2\pm \cdots\pm x_n\equiv x_1+x_2+\cdots+ x_n
\mod 2$, then
\begin{equation}\label{mod2}
\varphi^*_T(\sigma_w)\equiv
S_w(1,1,\ldots,1)(x_1+x_2+\cdots+x_n)^{\ell(w)}\mod 2.
\end{equation}
Let $\ell(w)=\sum_\alpha 2^\alpha$ be binary decomposition of
$\ell(w)$. Then
\begin{equation}
(x_1+x_2+\cdots+x_n)^{\ell(w)}\equiv
\prod_\alpha(x_1^{2^\alpha}+x_2^{2^\alpha}+\cdots+x_n^{2^\alpha})\mod
2.
\end{equation}
Hence multiplicity free monomials with {\it odd\,} coefficient can
appear only for permutation of length $\ell(w)\le 1$, i.e. for
$w=1$ or for transposition $w=(i,i+1)$. Note that
$$S_{(i,i+1)}(Z)=z_1+z_2+\cdots +z_i.$$
Therefore for even transposition $w=(i,i+1)$ coefficient
$S_{w}(1,1,\ldots, 1)=i$ in (\ref{mod2}) is even. So we proved
%a bit more:
that $\varphi_T^*(\sigma_w)$ contains a monomial with odd
coefficient iff $w=(2j-1,2j)$ or $w=1$.
\end{proof}
Recall that identical substitution gives rise to basic marginal
inequality
\begin{equation}\label{basic_qubit}
\sum_i a_i(\rho_1^{(i)}-\rho_2^{(i)})\le \sum_{k}(\pm
a_1\pm a_2\pm\cdots\pm a_n)_k \rho_k
\end{equation}
for every extremal edge $a$. From Claim \ref{claim} it follows
\begin{thm}[modulo Conjecture \ref{conj}]\label{Mod_Qbt} All marginal inequaliteis for $n$
qubits can be obtained from basic ones $(\ref{basic_qubit})$ by
odd transposition $\rho_{2j-1}\leftrightarrow\rho_{2j}$ in the
right hand side,
%and changing
accompanied with sign change
$a_i(\rho_1^{(i)}-\rho_2^{(i)})\mapsto-a_i(\rho_1^{(i)}-\rho_2^{(i)})
$ of one  term in the left hand side.
\end{thm}
\begin{rem}\label{Rem}
Right hand side of basic inequality (\ref{basic_qubit}) is
invariant under permutation of $a_i$. To get a strongest
inequality
% To make left hand side maximal possible
we have to arrange them in the same order as quantities
$\delta_{i}=\rho^{(i)}_1-\rho^{(i)}_2$ to make LHS maximal
possible. Let's assume for certainty
\begin{equation}
\delta_{1}\le\delta_{2}\le\cdots \le
\delta_{n},\qquad
a_1\le a_2\le \cdots\le a_n.
\end{equation}
%Similarly,
To get the strongest modified inequality one have to revert the
sign  of the minimal term $a_1\delta_1\mapsto -a_1\delta_1$ in the
LHS.
\end{rem}
\begin{Ex} {\it Three qubits.\,}
The above procedure being applied to extremal edges
(\ref{Edges3qbt}) for three qubits returns the following list of
marginal inequalities grouped by their extremal edges. The first
inequality in each group is the basic one. The transposed
eigenvalues in modified inequalities typeset in bold face. We
expect $\delta_i=\rho^{(i)}_1-\rho^{(i)}_2$ to be arranded in
increasing order $\delta_1\le\delta_2\le\delta_3$.
\begin{align*}
\delta_3
&\le\rho_1+\rho_2+\rho_3+\rho_4-\rho_5-\rho_6-\rho_7-\rho_8.\\[2mm]
\delta_2+\delta_3 &\le 2\rho_1+ 2\rho_2- 2\rho_7- 2\rho_8.\\[2mm]
\delta_1+\delta_2+\delta_3&\le3\rho_1+\rho_2+\rho_3+\rho_4-\rho_5-\rho_6-\rho_7-3\rho_8,\\
-\delta_1+\delta_2+\delta_3&\le3\boldsymbol{\rho_2}+\boldsymbol{\rho_1}+\rho_3+\rho_4-\rho_5-\rho_6-\rho_7-3\rho_8,\\
-\delta_1+\delta_2+\delta_3&\le3\rho_1+\rho_2+\rho_3+\rho_4-\rho_5-\rho_6-
\boldsymbol{\rho_8}-3\boldsymbol{\rho_7}.
\\[2mm]
\delta_1+\delta_2+2\delta_3&\le4\rho_1+2\rho_2+2\rho_3-2\rho_6-2\rho_7-4\rho_8,\\
-\delta_1+\delta_2+2\delta_3&\le4\boldsymbol{\rho_2}+2\boldsymbol{\rho_1}+2\rho_3-2\rho_6-2\rho_7-4\rho_8,\\
-\delta_1+\delta_2+2\delta_3&\le4\rho_1+2\rho_2+2\boldsymbol{\rho_4}-2\rho_6-2\rho_7-4\rho_8,\\
-\delta_1+\delta_2+2\delta_3&\le4\rho_1+2\rho_2+2\rho_3-2\boldsymbol{\rho_5}-2\rho_7-4\rho_8,\\
-\delta_1+\delta_2+2\delta_3&\le4\rho_1+2\rho_2+2\rho_3-2\rho_6-2\boldsymbol{\rho_8}-
4\boldsymbol{\rho_7}.
\end{align*}

One can check\footnote{Thanks to Maple simplex package.} that the
above  inequalities are independent, and other  inequalities
(\ref{QubitIneq}) follows from these ones, in conformity with
Conjecture \ref{conj}.
%This confirm Conjecture for three qubits.
%It
The later is still valid for four qubits,  however in this case
two modified inequalities are redundant, see Appendix. Actually
there are many ``trivial" redundant inequalities coming from
transposition of eigenvalues $\rho_{2j-1},\rho_{2j}$ entering in
RHS of basic inequality (\ref{basic_qubit}) with the same
coefficient. In settings of theorem \ref{ExtrEdge}, which deals
with extremal edges rather then cubicles, such transpositions are
forbidden and the ``trivial" redundancy never occurs.
%However, we
%removed other redundant inequalities by hand, see Remark
\ref{Rem}.
\end{Ex}




\section{Representation theoretical
interpretation}\label{ReprTheor}

Apparently marginal inequalities give a complete solution of the
quantum marginal problem. This however may be an illusion, since
the number of inequalities increases drastically with rank of the
system. In this section we pursue another approach, based on
reduction
%The main result of this section is a reduction
of the univariant marginal problem to {\it representation
theory\,} of the symmetric group. We recall first the basic facts
of the latter \cite{James,Macdonald,Sagan}.

\subsection{Digest of representation theory}\label{repres}
%Our approach to univariant marginal problem is based on its
%reduction to representation theory of the symmetric group.
%Our main result is a reduction of the univariant marginal problem
%to {\it representation theory\,} of the symmetric group. Here we
%just recall some basic facts of the later
%\cite{James,Macdonald,Sagan}.

Let's start with ensemble of $N$ identical systems with state
space
\begin{equation}\label{tensor}\mathcal{H}^{\otimes
N}=\underbrace{\mathcal{H}\otimes\mathcal{H}\otimes\cdots\otimes\mathcal{H}}_N.
\end{equation}
Unitary $\SU(\mathcal{H})$ and symmetric ${\mathrm{S}}_N$ groups
act on this space from the left and from the right by formulae
\begin{align*}
g:\psi_1\otimes\psi_2\otimes\cdots\otimes\psi_N&\mapsto
g\psi_1\otimes g\psi_2\otimes\cdots\otimes g\psi_N,& g&\in \SU(\mathcal{H}),\\
\sigma:\psi_1\otimes\psi_2\otimes\cdots\otimes\psi_N&\mapsto
\psi_{\sigma(1)}\otimes\psi_{\sigma(2)}\otimes\cdots\otimes\psi_{\sigma(N)},
 &\sigma&\in {\mathrm{S}}_N.
\end{align*}
%These actions commute.
%In 1901 Issai~Schur in his celebrated thesis \cite{Schur} found
Issai~Schur in his celebrated thesis \cite{Schur} of 1901 found
decomposition of tensor space (\ref{tensor}) into irreducible
components with respect to these actions
%Decomposition of tensor space (\ref{tensor}) into
%%$\SU(\mathcal{H})\times {\mathrm{S}}_N$
%irreducible components was
%found in celebrated  Schur's thesis \cite{Schur} in 1901
\begin{equation}\label{Schur}
\mathcal{H}^{\otimes N}=\bigoplus_\lambda
\mathcal{H}_\lambda\otimes \mathcal{S}_\lambda,
\end{equation}
where $\mathcal{H}_\lambda$ and ${\mathcal{S}}_\lambda$ are
irreducible representations of the unitary  and the symmetric
 groups respectively. The components
 %of the decomposition (\ref{Schur})
 describe tensors of different types of symmetry. They are parameterized
 by {\it Young diagrams\,}
 $$\lambda:\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_d\ge0,\qquad
 \lambda_1+\lambda_2+\cdots+\lambda_d=N$$ of $N$ cells and no more then
 $d=\dim\mathcal{H}$ rows of length $\lambda_i$. Representations
 $\mathcal{H}_\lambda$ and ${\mathcal{S}}_\lambda$ can be
 intrinsically characterized as follows.
\begin{itemize}
\item Let $V$ be irreducible representation of $\SU(\mathcal{H})$
and $\lambda$ be maximal diagram in {\it lexicographic order\,}
such that diagonal subgroup has eigenvector $\psi$ of weight
$\lambda$, that is
$$\mathrm{diag}(x_1,x_2,\ldots,x_d)\psi=
x_1^{\lambda_1}x_2^{\lambda_2}\cdots x_d^{\lambda_d}\psi.$$ Then
$V\simeq\mathcal{H}_\lambda$.
\item Let now $M$ be irreducible representation of $\mathrm{S}_N$,
and $\lambda$ be maximal diagram in the {\it majorization order\,}
(see $n^\circ$\ref{restr}) such that  $M$ contains a nonzero
invariant with respect to permutations in {\it rows\,} of diagram
$\lambda$ (filled in arbitrary way by numbers $1,2,\ldots,N$).
Then $M\simeq{\mathcal{S}}_\lambda$.
\end{itemize}
\begin{Ex} One row diagram $\lambda$ corresponds to trivial
representation
%$\mathcal{S}^\lambda=\mathrm{id}$
of the symmetric group. Representation $\mathcal{H}_\lambda$ in
this case consists of {\it symmetric tensors\,} in
$\mathcal{H}^{\otimes N}$ (Bose--Einstein statistics).

Column diagram $\lambda$ defines sign representation
$\sigma\mapsto \mathrm{sgn}(\sigma)$ of $S_N$, while
$\mathcal{H}_\lambda$ is the space of {\it antisymmetric
tensors\,} in $\mathcal{H}^{\otimes N}$ (Fermi--Dirak statistics).
%\marginpar{\footnotesize Add Weyl's determinants for
%$\mathcal{H}_\lambda$ and $\mathcal{S}_\lambda$, see p. 167.}

Other diagrams correspond to more complicate symmetry types of
tensors which some physicists associate with {\it
parastatistics\,}.
\end{Ex}
\subsection{Degression: Hermitian spectral problem}\label{AddSpec}
Here we consider a {\it model example\,} which illustrates
relation between representation theory and spectral problems. As
we have seen at the end of $n^\circ$\ref{margin} univariant
marginal problem falls into this category.
%Univariant marginal problem falls in this
%category (see Problem \ref{spectral}).

Let's start with decomposition of tensor product of irreducible
representations of the unitary group $\SU(\mathcal{H})$
%irreducible representations $\mathcal{H}_\lambda,\mathcal{H}_\mu$
%of unitary group $\SU(\mathcal{H})$ and decomposition of their
%tensor product into irreducible components
\begin{equation}\label{L-R}\mathcal{H}_\lambda\otimes\mathcal{H}_\mu=\sum_\nu
C_{\lambda\mu}^\nu\mathcal{H}_\nu.\end{equation} Representation
$\mathcal{H}_\nu$ enters into the decomposition with multiplicity
 $C_{\lambda\mu}^\nu$
%of the component $\mathcal{H}_\nu$
which physicists  call {\it Clebsch--Gordon\,} and mathematiciens
{\it Littlewood--Richardson\,} coefficient. The last authors
invented in 1934 an efficient algorithm  for calculation
$C_{\lambda,\mu}^\nu$, known as {\it Littlewood--Richardson rule}
\footnote{ {\it ``The Littlewood--Richardson rule helped to get men on the moon,
but it was not proved until after they had got there."}, see
%The first part of this story may be an exaggeration.
 \cite{James87}.}, see \cite{L-R,Macdonald} for details.
Littlewood--Richardson calculator is available at \cite{Buch}.

\begin{thm*}[Klyachko \cite{Kl98}]
The following conditions on Young dyagrams $\lambda,\mu,\nu$ are
equivalent
\begin{enumerate}
\item $C_{\lambda\mu}^\nu\ne0$, i.e.
$\mathcal{H}_\nu\subset\mathcal{H}_\lambda\otimes\mathcal{H}_\mu$.
\item There exist Hermitian operators $A,B,C=A+B$ in $\mathcal{H}$ with spectra
$\lambda,\mu,\nu$.
%\item A recursive system of linear inequalities between
%$\lambda,\mu,\nu$, known as Horn--Klyachko inequalities \cite{??},
%holds.
\end{enumerate}
Here we identify Young diagram $\lambda$ with integeral spectrum
$\lambda_1\ge\lambda_2\ge\lambda_3\ge\cdots$ formed by lengths of
its rows.
\end{thm*}
\begin{rem}\label{H-K}
(i) The conditions of the theorem are actually equivalent to a
system of linear inequalities on $\lambda,\mu,\nu$, known as {\it
Horn--Klyachko inequalities\,} \cite{Kl98,Ful2}.

(ii) The theorem as it stated  can be applied to integral spectra
only. However by scaling one can extend it to rational spectra,
and by limit arguments to real spectra as well. In the last case
we lose connection with representation theory, and compatibility
conditions for spectra of matrices $A,B,C=A+B$ are given by H-K
inequalities.
\end{rem}
%\marginpar{\footnotesize Add picture and explain L-R rule in this
%case and may be Knutson-Tao hive model. Otherwise skip the
%example.}
%\begin{Ex} For matrix  $B$  of rank one Horn--Klyachko
%inequalities amounts to {\it Cauchy interlacing theorem\,}:
%Eigenvalues of $ A$ and $A+B$   interlace.
%\end{Ex}
\subsection{Back to quantum marginal problem} \label{main} Let now
%consider decomposition of
consider tensor product of irreducible representations of the
symmetric group $S_N$
\begin{equation}\label{Kron}
\mathcal{S}_\lambda\otimes\mathcal{S}_\mu=\sum_\nu
g(\lambda,\mu,\nu)\mathcal{S}_\nu.
\end{equation}
The multiplicities $g(\lambda,\mu,\nu)$ in this case called {\it
Kronecker coefficients\,}.
%\footnote{Probably because he invented
%tensor product. Some people call them Clebsch--Gordon coefficients
%as well.}.
In contrast
%difference
with Littlewood--Richardson ones
%coefficients
$C_{\lambda\mu}^\nu$ they
are symmetric in $\lambda,\mu,\nu$ and
\begin{equation}g(\lambda,\mu,\nu)=\dim(\mathcal{S}_\lambda\otimes\mathcal{S}_\mu\otimes
\mathcal{S}_\nu)^{S_N}.\end{equation}
%Here
The superscript refers to space of invariants $V^{S_N}=\{\,x\in
V\mid  x^\sigma=x \quad\forall \sigma\in S_N\,\}$.

Now we can state our main result for univariant marginal problem,
which is similar in form to the previous theorem on spectra of
Hermitian matrices $A$, $B$, and $A+B$.

\begin{thm}\label{mainThm} The following conditions are equivalent
\begin{enumerate}
\item $g(m\lambda,m\nu,m\mu)\ne0$ for some $m>0$, i.e.
$\mathcal{S}_{m\nu}\subset\mathcal{S}_{m\lambda}\otimes\mathcal{S}_{m\mu}$.
\item There exists mixed state of $\rho_{AB}$ of two component
system $\mathcal{H}_A\otimes\mathcal{H}_B$ with spectrum $\nu$ and
margins of spectra $\lambda,\mu$.
\item There exists pure state
$\psi\in\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$
with univariant margins of spectra $\lambda,\mu,\nu$.
\end{enumerate}
\end{thm}

\begin{proof}
The proof is a combination of well known results and runs as
follows.

$\bullet$ As we've yet mentioned in the proof of Theorem
\ref{cubicleThm} the quantum marginal problem amounts to
decomposition of projection of a coadjoint orbit of group
$\SU(\mathcal{H}_A\otimes\mathcal{H}_B)$ into coadjoint orbits of
subgroup $\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)$.

$\bullet$ Combining this with Heckman's theorem \cite{Heck}, see
also
%\cite{Guil-Stern} and
in \cite[Thm 3.4.2]{Ber-Sja}, we arrive at characterization of
quantum margins
%$\rho_A,\rho_B$ of mixed state $\rho_{AB}$
by inclusion
\begin{equation}
\mathcal{H}_A^\lambda\otimes\mathcal{H}_B^\mu
\subset
\mathcal{H}_{AB}^\nu\left.\right|_{\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)}
\end{equation}
for some spectra $\lambda,\mu,\nu$ proportional to
$\Spec\rho_A,\Spec\rho_B,\Spec\rho_{AB}$. The later, without loss
of generality, expected to be rational.

$\bullet$ Finally, the following equation for multiplicities
\begin{equation}\label{mult}\text{Mult. }\mathcal{H}_A^\mu\otimes\mathcal{H}_B^\nu
\text{ in }
\mathcal{H}_{AB}^\lambda\left.\right|_{\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)}=
\text{Mult. }
 \mathcal{S}^\lambda\text{ in }
\mathcal{S}^\mu\otimes\mathcal{S}^\nu
\end{equation}
reduces the problem to Kronecker coefficients of the symmetric
group.
%representations of the symmetric group.

%The result follows from Heckman's theorem \cite{Heck}, see also
%%\cite{Guil-Stern} and
%Theorem 3.4.2 in \cite{Ber-Sja}, which reduces the quantum
%marginal problem to reduce the problem to decomposition of
%restriction of irreducible representations of $G$ onto subgroup
%$H$.
%
%
%A detailed  proof of the theorem is rather involved and will be
%published elsewhere.\footnote{{\it ``Physics is too complicated
%for physicists."\,} (Hilbert)} Here we sketch only the main steps.
%
%%indicate only main ingredients.
%
%$\bullet$ We start with Berenstein--Sjamaar \cite{Ber-Sja}
%description of (co)adjoint orbits of subgroup $H\subset G$ which
%enter into projection of a (co)adjoint orbit of group $G$.
%
%$\bullet$ Applying this to
%%orbit of density matrix $\rho_{ABC}$,
%$G=\SU(\mathcal{H}_A\otimes\mathcal{H}_B)$ and
%$H=\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)$ we can, in
%principle, produce
%%a rather complicate but
%a finite complete system of linear inequalities between spectrum
%of a mixed  state $\rho_{AB}$ and that of its margins $\rho_A$,
%$\rho_B$.
%%decomposition of projection of coadjoint orbit of group
%%$\SU(\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C)$ into
%%orbits of its subgroup
%%$\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)\times\SU(\mathcal{H}_C)$.
%
%$\bullet$ Then we employ Heckman's theorem \cite{Heck}, see also
%\cite{Guil-Stern} and Theorem 3.4.2 in \cite{Ber-Sja}, to reduce
%the problem to decomposition of restriction of irreducible
%representations of $G$ onto subgroup $H$.
%
%$\bullet$ For the above choice of $G$ and $H$ the restriction
%problem amounts to decomposition of tensor product of irreducible
%representations of the symmetric group. Namely the following
%equation for multiplicities holds
%\begin{equation}\label{mult}\text{Mult. }\mathcal{H}_A^\mu\otimes\mathcal{H}_B^\nu
%\text{ in }
%\mathcal{H}_{AB}^\lambda\left.\right|_{\SU(\mathcal{H}_A)\times\SU(\mathcal{H}_B)}=
%\text{Mult. }
% \mathcal{S}^\lambda\text{ in }
%\mathcal{S}^\mu\otimes\mathcal{S}^\nu.
%\end{equation}
%

$\bullet$ Equivalence $(2)\Leftrightarrow(3)$ is strightforward
and independent of representation theory.
\end{proof}

In the rest of this paper we'll use the theorem in both directions
to obtain new results in quantum marginal problem and in
representation theory.

\section{Examples and applications}
%Littlewood--Richardson coefficients (\ref{L-R}) appears in
%mathematics in many different contexts.  This provides a
%comprehensive understanding of their behavior both from conceptual
%and combinatoric points of view.
%
% Information on Kronecker's coefficients (\ref{Kron}), in contrast,
%is very sparse and sporadic. Currently there is no simple way to
%calculate them for big $n$. Here we recall some available
%information.
%\subsection{More on Kronecker's coefficients}
\subsection{Polygonal inequalities revisited}\label{murn}
%Currently we have much better understanding of

%Here we recall some basic facts.

Admitting an abuse of language we'll use the same notation for
Young diagram $\lambda$ and  for the corresponding irreducible
representation of the symmetric group $S_n$. It is convenient to
single out the first row of diagram
$\lambda=(n-|\bar\lambda|,\bar\lambda)$. The remaining part
$\bar\lambda$ is called {\it reduced diagram}, and the number of
its cells $d(\lambda)=|\bar\lambda|$ is said to be {\it depth\,}
of $\lambda$.
%where {\it reduced
%diagram\,} $\bar\lambda$ is obtained from $\lambda$ by removing
%its first row.
We treat $n$ as a parameter and use parentheses to return the
original diagram $
(\bar\lambda)=(n-|\bar\lambda|,\bar\lambda)=\lambda$.

For small values of $n$ entries of $(\mu)=(n-|\mu|,\mu)$ may be
not in decreasing order. Then
%we use
the following rule is applied
\begin{equation}\label{rule}(\ldots,p,q,\ldots)=-(\ldots,q-1,p+1,\ldots)\end{equation}
 to transform
$(\mu)$ into a  Young diagram with sign $\pm$ or into zero (for
$q=p+1$). By the above convention $(\mu)$ is also understood  as a
{\it virtual representation\,} of $S_n$.

%It seems the first paper on tensor product of representations of
%the symmetric group was published by F.~D.~Murnaghan
%\cite{Murnaghan38} in 1938. He observed that
%discovered two properties of the Kronecker coefficients
\begin{thm*}[Murnaghan \cite{Murnaghan38,Murnaghan55}, Littlewood \cite{Littlewood}]
%\begin{enumerate}
$(1)$ Coefficients of decomposition
%Kronecker coefficient
%$g(\lambda,\mu,\nu)$ depends only on {\it reduced diagrams\,}
%$\bar\lambda,\bar\mu,\bar\nu$,  that is
\begin{equation}\label{reduced}(\bar\lambda)\otimes(\bar\mu)=\sum_{\bar\nu}\bar
g(\bar\lambda,\bar\mu,\bar\nu)(\bar\nu).\end{equation} are
independent of $n$, and $g(\lambda,\mu,\nu)=\bar
g(\bar\lambda,\bar\mu,\bar\nu)$ for $n\gg 1$.
%obtained from
%$\lambda,\mu,\nu$ by removing their first rows, but not on the
%degree $n=|\lambda|=|\mu|=|\nu|$ of the symmetric group $S_n$ at
%least for $n\gg 1$. Therefore we'll use reduced notation $\bar
%g(\bar\lambda,\bar\mu,\bar\nu)$ instead of $g(\lambda,\mu,\nu)$.

$(2)$ The coefficient $g(\lambda,\mu,\nu)$  vanishes except depth
of the diagrams
%the reduced diagrams
satisfies triangle inequalities
\begin{equation}\label{tri}d(\lambda)\le d(\mu)+d(\nu),\qquad
d(\mu)\le d(\nu)+d(\lambda),\qquad d(\nu)\le d(\lambda)+d(\mu).
\end{equation}

$(3)$ In the case of
%one of the triangle inequalities degenerates into
equality in (\ref{tri}) the Kronecker coefficient coincides with
Littlewood-Richardson one for reduced dyagrams
\begin{equation}\label{bound} g(\lambda,\mu,\nu)=C_{\bar\mu\bar\nu}^{\bar\lambda},
\qquad d(\lambda)=d(\mu)+d(\nu).
\end{equation}
%\end{enumerate}
\end{thm*}
%\marginpar{\footnotesize Add interpretation of this as a deformation
%of cohomology ring of Grassmannian.}
\begin{rem} Murnaghan \cite{Murnaghan38} gave dozens
%58
examples of decomposition
(\ref{reduced}) like the following one
\begin{equation*}
\begin{split}
(n-2,2)\otimes(n-2,1^2)& =(n-1,1)+(n-2,2)+2(n-2,1^2)+(n-3,3)
\\&  +2(n-3,2,1)+ (n-3,1^3)+(n-4,3,1)+(n-4,2,1^2).
\end{split}
\end{equation*}
This equation literally gives Kronecker coefficients
%of the tensor product
for $n\ge7$. Otherwise rule (\ref{rule}) should be invoked which
may result in cancelation of some virtual components. See
\cite{Vallejo99} for a general stabilization bound for $n$.
\end{rem}
\begin{rem} Murnaghan theorem allows to defined new product of
Young diagrams
\begin{equation}
\bar{\lambda}\circ\bar{\mu}=\sum_{\bar{\nu}}q^{\bar{\lambda}+\bar{\mu}-\bar{\nu}}
g(\bar{\lambda},\bar{\mu},\bar{\nu})\bar{\nu}
\end{equation}
depending on parameter $q$. By (\ref{bound}) the multiplication is
a {\it deformation\,} of  cohomology ring of (infinite)
Grassmannian. The deformation is different from {\it quantum
cohomology\,} of Grassmannian which plays central role in unitary
spectral problem \cite{A-W}. It would be very interesting to find
a geometric interpretation of this deformation.
\end{rem}

Theorem \ref{mainThm} allows recast the second claim of
Murnaghan's theorem into the following result.

\begin{cor} Let
$\psi\in\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$ be
pure state with univariant margins of spectra $\lambda,\mu, \nu$.
Then their depth satisfies triangle inequalities
\begin{equation}\label{triang} d(\lambda)\le d(\mu)+d(\nu),\qquad
d(\mu)\le d(\nu)+d(\lambda),\qquad d(\nu)\le d(\lambda)+d(\mu),
\end{equation}
where the  depth of spectrum
$\lambda_1\ge\lambda_2\ge\lambda_3\ge\cdots$ is\,
 $\lambda_2+\lambda_3+\cdots:=d(\lambda)$.
%is defined by .
\end{cor}

\begin{rem} Extension of the corollary to multicomponent systems
is straightforward. For $N$-qubit it returns polygonal
inequalities (\ref{poly}). The proof given by S.~Bravyi
\cite{Brav} actually works for depth as well.
\end{rem}

\subsection{Quasiclassical limit} Here we analyze the boundary
case (3) of Murnaghan's theorem in terms of the marginal problem.

Let $\rho_{AB}$ be mixed state of composite system
$\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ with margins
$\rho_A,\rho_B$ and spectra
$\lambda_{AB},\lambda_{A},\lambda_{B}$. Suppose triangle
inequalities  for depth (\ref{triang}) degenerate into equality
\begin{equation}\label{deg}
d(\lambda_{AB})=d(\lambda_{A})+d(\lambda_{B}).
\end{equation}
%Let
Using notations
$$\psi_{AB},\qquad\psi_{A},\qquad\psi_{B}$$
%be
for eigenstates of $\rho_{AB},\rho_{A},\rho_{B}$ with maximal
eigenvalues,
$$\overline{\mathcal{H}}_{AB},\qquad\overline{\mathcal{H}}_{A},\qquad
\overline{\mathcal{H}}_{B}$$
%be
for their orthogonal complements, and
$$\overline{\rho}_{AB},\qquad\overline{\rho}_{A},\qquad\overline{\rho}_{B}$$
%be
for restrictions of $\rho_{AB},\rho_A,\rho_{B}$ onto subspaces
$\overline{\mathcal{H}}_{AB},\overline{\mathcal{H}}_{A},\overline{\mathcal{H}}_{B}$,
one can show that equation (\ref{deg}) is equivalent to the
following conditions
\begin{itemize}
\item $\psi_{AB}=\psi_{A}\otimes \psi_{B}$;
\item $\overline{\rho}_{AB}$ has support in subspace
$\overline{\mathcal{H}}_A\otimes\psi_B+\psi_A\otimes\overline{\mathcal{H}}_B
\simeq\overline{\mathcal{H}}_A\oplus\overline{\mathcal{H}}_B$;
\item $\overline{\rho}_A$ and $\overline{\rho}_B$ are just
restrictions of $\overline{\rho}_{AB}$ onto
$\overline{\mathcal{H}}_A\otimes\psi_B$ and
$\psi_A\otimes\overline{\mathcal{H}}_B$,
\begin{equation}
\overline{\rho}_{AB}=\left[
\begin{tabular}{c|c}
$\overline{\rho}_A$& $\ast$\\
\hline
$\ast$&$\overline{\rho}_B$
\end{tabular}\right]
%\overline{\rho}_{AB}=\left[\vcenter{\offinterlineskip\halign{
% \strut\quad\hss$#$\hss\quad\vrule&\quad\hss$#$\hss\quad\cr
% \overline{\rho}_A&\cr\noalign{\hrule}
% *&\overline{\rho}_B\cr}}\right].
\end{equation}
\end{itemize}
We call this {\it quasiclassical limit\,} because all the events
happen in classical direct sum
$\overline{\mathcal{H}}_A\oplus\overline{\mathcal{H}}_B$ rather
then in quantum tensor product. In this case Theorem \ref{mainThm}
combined with equation (\ref{bound}) and Remark \ref{H-K} gives
%amounts to the following
\begin{thm}\label{quasi} The following conditions are equivalent
\begin{enumerate}
\item $C_{\lambda\mu}^\nu\ne 0$, that is
$\mathcal{H}_\nu\subset\mathcal{H}_\lambda\otimes\mathcal{H}_\mu$.
\item There exist Hermitian matrices $H_A$, $H_B$, and
$$
H_{AB}=\left[
\begin{tabular}{c|c}
$H_A$& $\ast$\\
\hline
$\ast$&$H_B$
\end{tabular}\right]
$$
 with spectra $\lambda$, $\mu$, and $\nu$ respectively.
 \item Horn--Klyachko inequalities hold for $\lambda,\mu,\nu$.
\end{enumerate}
\end{thm}
%Generalization of the theorem to arbitrary block diagonal matrices
%is straightforward:
One have to use multiple L-R coefficients
$$\mathcal{H}_\lambda\otimes\mathcal{H}_\mu\otimes\mathcal{H}_\nu\otimes\cdots=
\sum_\sigma
C_{\lambda\mu\nu\cdots}^\sigma\mathcal{H}_\sigma$$ to deal with
arbitrary  number of diagonal blocks
$\mathcal{H}_A,\mathcal{H}_B,\mathcal{H}_C,\ldots$ of spectra
$\lambda,\mu,\nu,\ldots$. For blocks of size one the theorem
amounts to {\it Horn  majorization inequality\,} $$\Spec H \succ
\diag H$$
since for one row diagrams $a,b,c,\ldots$\quad
$C_{abc\cdots}^\sigma\neq0\Leftrightarrow \sigma\succ
(a,b,c,\ldots)$.


\begin{rem} It is more natural and easy deduce Theorem
\ref{quasi} directly from Berensein-Sjamaar theorem
\cite{Ber-Sja}, then deduce it from Murnaghan results. See
\cite{Li-Poon03} for another approach.
%It seems however nobody did this before.
Note also that one can control spectrum of Hermitian matrix
$$
H=\left[
\begin{tabular}{c|c}
$\ast$& $X$\\
\hline
$X^*$&$\ast$
\end{tabular}\right]
$$
by linear inequalities in $\Spec H$ and {\it singular\,} spectrum
of $X$ \cite{Li-Poon,Fom-Ful}. It would be very interesting  to
merge these two results
%with Theorem \ref{quasi}
to gain control over spectrum of Hamiltonian
$$
H_{AB}=\left[
\begin{tabular}{c|c}
$H_A$& $H_\text{int}$\\
\hline
$H_\text{int}^*$&$H_B$
\end{tabular}\right]
$$
of composite classical system in terms of spectra of the
components $A$, $B$, and singular spectrum of $H_\text{int}$,
which measures  the {\it strength\,} of their interaction. This,
however, can't be done by {\it linear\,} inequalities.

\end{rem}
\begin{rem} \label{sat} Note an important {\it saturation property\,} of
Littlewood--Richardson coefficients \cite{Knut-Tao}
%For Littlewood--Richardson coefficients the
%following {\it saturation property\,} holds \cite{Knut-Tao}
$$ C_{m\lambda,m\mu}^{m\nu}\ne 0\Longleftrightarrow C_{\lambda\mu}^\nu\ne
0,$$ which simplifies the statement of Theorem \ref{quasi} (cf.
Theorem \ref{mainThm}).
%However its analogue
%Similar result {\it fails\,}
It has no analogue  for Kronecker coefficients,
%{\it fails\,},
%$g(\lambda,\mu,\nu)$ don't share this property,
e.g. $2^2\subset2^2\otimes2^2$ but $1^2\not\subset1^2\otimes1^2$.
%One can ask
\begin{conj}
Saturation property still holds for reduced Kronecker coefficients
(\ref{reduced}).
\end{conj}
For example, the saturation conjecture is valid for
%It holds for
{\it leading\,} reduced coefficient
$\bar{g}(\bar{\lambda},\bar{\mu},\bar{\nu})=
C_{\bar{\mu}\bar{\nu}}^{\bar{\lambda}}$,
$\bar{\mu}+\bar{\nu}=\bar{\lambda}$, see Murnaghan theorem
$n^\circ$\ref{murn}.
\end{rem}

\subsection{Maximal eigenvalue of a state with given margins}
Here and in the next $n^\circ$\ref{rank} we consider the following
%concerned with the problem:
\begin{prob} How close to a pure state can be mixed state $\rho_{AB}$
with given margins $\rho_A,\rho_B$?
\end{prob}
As we have seen
%Recall from
in $n^\circ$\ref{pure} margins of a pure state are isospectral.
Hence for $\Spec\rho_A\neq\Spec\rho_B$ state $\rho_{AB}$ can't be
pure, and we want to get it as close to a pure state as possible.
Recall that state $\rho$ is pure iff its maximal eigenvalue is
equal to one. Hence the maximal eigenvalue, known as {\it spectral
norm\,} $\norm{\rho}_s$ of operator $\rho$,
%of $\rho$
may be considered as a measure of purity.
%, which is known
%in mathematics
%as {\it spectral norm\,} $\norm{\rho}_s$ of
%operator $\rho$.
\begin{thm} \label{Max} Let $\rho_A,\rho_B$ be marginal states of spectra
$$\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n,\qquad\mu_1\ge\mu_2\ge\cdots\ge\mu_n$$
extend by zeros to make them of the same length. Then maximal
spectral norm of states $\rho_{AB}$ with margins $\rho_A,\rho_B$
is given by equation
\begin{equation}\label{max}
\max\norm{\rho_{AB}}_s=\sum_i\min(\lambda_i,\mu_i)=1-\frac12\sum_i\abs{\lambda_i-\mu_i}.
\end{equation}
\end{thm}
%Then for every mixed state $\rho_{AB}$ with margins
%$\rho_A,\rho_B$ the following inequality holds
%\begin{equation}\label{max}
%\norm{\rho_{AB}}_s\le\sum_i\min(\lambda_i,\mu_i)=1-\frac12\sum_i\abs{\lambda_i-\mu_i},
%\end{equation}
%and
%%the inequality is sharp (i.e.
%equality is attained for some $\rho_{AB}$.

The proof amounts to application of Theorem \ref{mainThm} to the
following representation theoretical counterpart.
% For proof one have to apply Main Theorem
%$n^\circ$\ref{main} to the following representation theoretical
%result.
%This is just a
%translation into marginal problem language of the following result
%This follows from Main Theorem $n^\circ$\ref{main} and
%the representation theoretical counterpart:
%the following result on representations of the symmetric group.
\begin{thm*}[Klemm \cite{Klemm}, Dvir \cite{Dvir}, Clausen \& Meier \cite{Cl-Ma}] Maximal
length of the first row of Young diagrams
$\nu\subset\lambda\otimes\mu$
%s.t.
%$\mathcal{S}_\nu\subset\mathcal{S}_\lambda\otimes\mathcal{S}_\mu$
is equal to $\abs{\lambda\cap\mu}=\sum_i\min(\lambda_i,\mu_i)$.
\end{thm*}

It is  an interesting problem to describe states $\rho_{AB}$ with
maximal spectral norm.
%given by right hand side  of (\ref{max}).
%In principle
%Information on spectra of such states can be extracted from the
%following
There is a simple formula for Kronecker coefficient of
representation $\nu\subset\lambda\otimes\mu$ with maximal first
row
%$\nu_1=\abs{\lambda\cap\mu}$
%is given by formula
\cite{Dvir,Cl-Ma}
\begin{equation}\label{DCM}
g(\lambda,\mu,\nu)=\sum_{\alpha,\beta}C_{\lambda\cap\mu,\alpha}^\lambda
C_{\lambda\cap\mu,\beta}^\mu\,
g(\alpha,\beta,\overline{\nu}),\qquad \nu_1=\abs{\lambda\cap\mu},
\end{equation}
%This equation
which allows, in principle, recursively find maximal component
$\nu\subset\lambda\otimes\mu$ in lexicographic order. For example,
%maximal value of
$\nu_2$ is equal to maximum of $\abs{\alpha\cap\beta}$ s.t. both
Littlewood--Richardson coefficients
$C_{\lambda\cap\mu,\alpha}^\lambda, C_{\lambda\cap\mu,\beta}^\mu$
are nonzero. This combinatorial problem amounts to construction of
a couple of skew standard tableaux of shapes $\lambda\backslash
(\lambda\cap\mu)$ and $\mu\backslash (\lambda\cap\mu)$ with as
close content as possible. I skip transposition of this into the
marginal problem key.
%\marginpar{\footnotesize The problem is about couple of skew
%standard tableaux of shapes $\lambda\backslash (\lambda\cap\mu)$
%and $\mu\backslash (\lambda\cap\mu)$ with as close content as
%possible.} The last condition amounts to a standard system of
%linear inequalities on $\alpha,\beta$.
%I leave to a reader
%translation of this result into the marginal problem setting.
%which corresponds to
%maximal lexicographic spectrum of $\rho_{AB}$.

See \cite{Vallejo03} for construction of other components in
$\lambda\otimes\mu$.


\subsection{Rank of a state with given
margins}\label{rank} Pure state can be also characterized by its
rank
$$\rho \text{ pure}\Longleftrightarrow\rk\rho=1.$$
Hence $\rk\rho$ or better $\delta(\rho)=\log\rk\rho$ is another
measure of purity (or rather impurity). The later is just {\it
informational capacity\,} of support of $\rho$. It follows from
Example \ref{slice} that margins of a pure state satisfy triangle
inequalities
\begin{equation*}\label{triangle}\delta(\rho_A)\le\delta(\rho_B)+\delta(\rho_C),\quad
\delta(\rho_B)\le\delta(\rho_C)+\delta(\rho_A),\quad
\delta(\rho_C)\le\delta(\rho_A)+\delta(\rho_B),
\end{equation*}
%as well as
and the same is true for $\delta(\rho_{A})$, $\delta(\rho_{B})$,
and $\delta(\rho_{AB})$. Representation theoretical counterpart of
this
\begin{equation}\label{schur}
\height(\nu)\le\height(\lambda)\cdot\height(\mu)\quad\text{for}\quad
\nu\subset\lambda\otimes\mu
\end{equation}
%\marginpar{\footnotesize See also A.~Regev, J. Algebra,
%66(1980), 505--510.}
comes back to Schur \cite{Schur}, see also
\cite{Regev80}. Here the {\it height\,} $\height(\nu)$ of diagram
$\nu$ is the number of its rows.
 Note that decomposable state
$\rho_{AB}=\rho_A\otimes\rho_B$ has margins $\rho_A,\rho_B$, and
{\it maximal\,} possible rank $\rk\rho_{AB}=\rk{\rho_A}\rk\rho_B$.


% see $n^\circ$\ref{scalar}.
\begin{thm}\label{Rank}  There exists state $\rho_{AB}$ of two component system with
given margins $\rho_A,\rho_B$ and
\begin{equation}
\rk\rho_{AB}\le\max(\rk\rho_A,\rk\rho_B).
\end{equation}
In addition, such a state can be taken with  maximal possible
spectral norm, given by Theorem \ref{Max}.
%$\abs{\rk\rho_A-\rk\rho_B}\le$
%$$\rk\rho_{AB}\le\max(\rk\rho_A,\rk\rho_B)-\#(\text{\rm common
%eigenvalues $\ne0$ of $\rho_A,\rho_B$})+1.$$
\end{thm}

The proof of the theorem once again is just a translation into
marginal problem language of the following representation
theoretical counterpart.
%result.
%, where {\it height\,} of a diagram is the number of its
%rows.
\begin{thm*}[Berele \& Imbo \cite{Ber-Imb}] Let
$\lambda$ and $\mu$ be Young diagram of height $\le k$.  Then
tensor product $\lambda\otimes\mu$ contains a component $\nu$  of
height $\le k$.
\end{thm*}
%\begin{conj} Let $\rho_{AB}$ be mixed state with margins $\rho_A$,
%$\rho_B$. Then $\delta(\rho_{A})$, $\delta(\rho_{B})$,
%$\delta(\rho_{AB})$ satisfy triangle inequality. Equivalently, let
%$\rho_A$, $\rho_B$, $\rho_C$ be margins of a pure state. Then
%$\delta(\rho_A)$, $\delta(\rho_B)$, $\delta(\rho_C)$ satisfy
%triangle inequalities.
%%$$\delta(\rho_{AB})\ge \abs{\delta(\rho_{A})-\delta(\rho_{B})}.$$
%\end{conj}
Vallejo \cite{Vallejo03} gave an explicit construction of such a
component $\nu\subset\lambda\otimes\mu$
%which in addition has
with maximal possible first row $\nu_1=\abs{\lambda\cap\mu}$.
%${\height}(\nu)\le\max{\height(\lambda)\height(\mu)}$.
\begin{conj}
State $\rho_{AB}$ with maximal lexicographical spectrum has
minimal rank among all states with given margins $\rho_A,\rho_B$.
%and maximal spectrum in lexicographic order has minimal possible
%rank.
%$\le\max(\rk\rho_A,\rk\rho_B)$.
\end{conj}

\subsection{Relation with classical marginal problem}\label{cl}
Triplet of spectra $\lambda,\mu,\nu$ is
%called
said to be {\it quasiclassical\,} iff there exist $p_{ij}\ge0$
such that
\begin{equation}\label{classical}\lambda=\sum_i
p_{ij},\qquad \mu=\sum_j p_{ij},\qquad p\prec\nu,
\end{equation} where the
last condition means that content of matrix $p=[p_{ij}]$ being
arranged in decreasing order is majorized by $\nu$.
 If the content of matrix $p$ coincides with
$\nu$ the triplet is said to be {\it classical\,}, cf.
$n^\circ$\ref{restr}.
%less then $\nu$ in majorization order $n^\circ$\ref{restr}.
We borrow this definition, with a minor modification,  from Sergei
Bravyi \cite{Brav} who proved
%the first half
part $(1)\Leftrightarrow(2)$ of the next theorem.
\begin{thm} The following conditions on spectra $\lambda,\mu,\nu$
are equivalent
\begin{enumerate}
\item Triplet $\lambda,\mu,\nu$ is quasiclassical.
\item There exists mixed state $\rho_{AB}$ such that
$$\Spec\rho_{AB}\prec\nu,\quad \Spec\rho_A=\lambda,\quad
\Spec\rho_B=\mu.$$
\item There exists mixed state $\widetilde{\rho}_{AB}$ such that
$$\Spec\widetilde{\rho}_{AB}=\nu,\quad \Spec\widetilde{\rho}_A\succ\lambda,\quad
\Spec\widetilde{\rho}_B\succ\mu.$$
\end{enumerate}
\end{thm}
%\marginpar{\footnotesize The theorem gives information on
%minimal components of $\lambda\otimes\mu$ in majorization order,
%see $n^\circ$\ref{dual}.}
%The remaining
%Equivalence
The second part $(1)\Leftrightarrow(3)$ comes from Theorem
\ref{mainThm} and  standard facts from representation theory of
the symmetric group:
%listed below.
\begin{itemize}
\item Let $[\lambda]$ be {\it permutation representation\,} induced from Schur
subgroup
$\mathrm{S}_\lambda=\mathrm{S}_{\lambda_1}\times\mathrm{S}_{\lambda_2}\times\cdots
\times\mathrm{S}_{\lambda_m}\subset\mathrm{S}_{n}$. Then
$\widetilde{\lambda}\subset
[\lambda]\Longleftrightarrow\widetilde{\lambda}\succ\lambda$.
\item Hence $\nu\subset\widetilde{\lambda}\otimes \widetilde{\mu}$
for some $\widetilde{\lambda}\succ\lambda$ and
$\widetilde{\nu}\succ\nu$ iff $\nu\subset[\lambda]\otimes[\mu]$.
\item
$\nu\subset[\lambda]\otimes[\mu]\Longleftrightarrow \lambda,\mu
\text{ are margins of integer matrix }p_{ij}\ge0\text{ s.t. }
p\prec\nu.$
\end{itemize}
The first property is actually a source of the definition of
irreducible representation $\mathcal{S}_{\lambda}$ (currently
shortened to $\lambda$) in $n^\circ$\ref{repres}.  The last one
%follows
can be seen from decomposition of tensor product
$[\lambda]\otimes[\mu]$ into direct sum of permutation modules
corresponding to intersection $Z_{ij}=X_i\cap Y_j$ of partitions
\begin{align*}\{\,1,2,\ldots,n\,\}&=\bigsqcup_i X_i,\qquad |X_i|=\lambda_i,\\
\{\,1,2,\ldots,n\,\}&=\bigsqcup_j Y_j,\qquad |Y_j|=\mu_j,
\end{align*}
of types $\lambda$ and $\mu$. It follows that  every component
$\nu\subset[\lambda]\otimes[\mu]$ satisfies (\ref{classical}) with
$p_{ij}=\abs{Z_{ij}}$, and vice versa.
\begin{cor}\label{minimal} Let $\rho_{AB}$ be mixed state with given margins
$\rho_A,\rho_B$ of spectra $\lambda_A,\lambda_B$. Suppose that
$\lambda_{AB}=\Spec\rho_{AB}$ is minimal possible in majorization
order. Then triplet $\lambda_{A},\lambda_{B},\lambda_{AB}$ is
classical.
\end{cor}

\begin{rem}
%There is an elementary proof of
Implication $(3)\Rightarrow(2)$ has a simple analytical proof.
%is elementary.
Indeed, inequality
$\Spec\widetilde{\rho}_A\succ\lambda$ means that
$\rho_A=\sum_ip_iU_i\widetilde{\rho}_AU_i^\dag$ has spectrum
$\lambda$ for some unitary operators $U_i$ and probabilities
$p_i\ge 0$ \cite{Wehrl}. In a similar way there exists
$\rho_B=\sum_jq_jV_j\widetilde{\rho}_BV_j^\dag$ with spectrum
$\mu$. Then
$$\rho_{AB}=\sum_{ij}p_iq_j(U_i\otimes V_j)\widetilde{\rho}_{AB}(U_i^\dag\otimes
V_j^\dag)$$ has margins $\rho_A,\rho_B$ and $\Spec
\rho_{AB}\prec\Spec\widetilde{\rho}_{AB}=\nu$.

%No proof of implication
Neither {\it analytical\,} proof of $(2)\Rightarrow(3)$,
%independent of representation theory, as well as no
nor {\it representation theoretical\,} proof of
$(3)\Rightarrow(2)$ are known, see $n^\circ$\ref{dual} below.
%It would be interesting to find a direct proof of equivalence
%$(2)\Leftrightarrow(3)$. Note that implication is elementary.
\end{rem}

\subsection{Two qubit revisited}\label{QbtRev} Here we return back to two qubit marginal problem
$n^\circ$\ref{two_qubit}
%, which
from representation theoretical
%point of view
perspective. By Theorem \ref{mainThm} it amounts to
%counterpart of two qubit marginal problem is
decomposition of tensor product
$$\lambda\otimes\mu=\sum_\nu g(\lambda,\mu,\nu)\nu
$$ of representations defined by
{\it two row\,} diagrams $\lambda=(\lambda_1,\lambda_2)$ and
$\mu=(\mu_1,\mu_2)$.  By Schur inequality (\ref{schur})
%it includes only
the components $\nu=(\nu_1,\nu_2,\nu_3,\nu_4)$ have at most four
rows.

Currently neither combinatorial description nor efficient
algorithm are available for Kronecker coefficients
$g(\lambda,\mu,\nu)$, in contrast with Littlewood--Richardson
ones. One of the first results in this direction belongs to
A.~Lascoux \cite{Las} who resolved  the case of hook diagrams
$\lambda,\mu$.
%In the simplest case of
For two row diagrams the problem was first addressed by Remmel and
Whitehead in long paper \cite{Rem-Whd}, and later by Rosas
\cite{Rosas}. Unfortunately the results are too complicated to be
reproduced here. Mersedes Rosas
%represented
write down $g(\lambda,\mu,\nu)$ as a {\it difference\,} of the
number of lattice points in two polygons. A close look at her
formula shows that for symmetric group $\mathrm{S}_n$ of {\it odd
degree\,} $n$ one polygon can be moved into another by an affine
transformation respecting the lattice. As result we get
%Note that
%In contrast to Littlrewood--Richardson coefficients
%There is no
%closed formula or
%general
%Currently neither combinatorial description nor effective
%algorithm is known for
%%combinatorial description or simple
%%algorithm for
%%dealing with
%%calculation of
%Kronecker coefficients, in drastic contrast to
%Littlewood--Richardson ones.
%The papers illustrate this
%difficulties
%calculation of Kronecker coefficients even
%in the first nontrivial case.
%However the  formulae obtained in \cite{Rosas,Rem-Whd} are too
%complicated to be reproduced  here.
%, and illustrate the general state of
%affairs.
%They are too complicated to be reproduced  here.
%In \cite{Rosas} Kronecker coefficient $g(\lambda,\mu,\nu)$
%%represented
%is written as {\it difference\,} of numbers of lattice points in
%two polygons.
%It turns out
%However
%A close look shows, however, that for symmetric group
%$\mathrm{S}_n$ of {\it odd degree\,} $n$ one polygon is enough.
%However for symmetric group $\mathrm{S}_n$ of {\it odd degree\,}
%$n$ they can be simplified as followos.
%But in the case of for symmetric group $\mathrm{S}_n$ of {\it odd
%degree\,} $n$ Rosas calculation of $g(\lambda,\mu,\nu)$
%\cite{Rosas} can be interpreted as counting lattice points in a
%convex polygon.
%result \cite{Rosas} can be recasted as follows.
\begin{claim} For odd $n$ Kronecker coefficient
$g(\lambda,\mu,\nu)$ for two row diagrams $\lambda,\mu$ is equal
to the number of integers $a,b$ satisfying the following
conditions
\begin{subequations}\label{septagon}
\begin{align}
\nu_3+\nu_4&\le a \le \nu_2+\nu_4,\\
\nu_2+\nu_4&\le b \le\min(\nu_2+\nu_3,\nu_1+\nu_4),\\
\lambda_2-\mu_2&\le b-a \le\lambda_1-\mu_2,\label{mid}\\
a+b&\le \lambda_2+\mu_2,\\
a+b&\equiv \lambda_2+\mu_2 \pmod 2,
\end{align}
\end{subequations}
where we expect, without loss of generality, $\lambda_2\ge\mu_2$.
\end{claim}
%The conditions imply Bravyi inequalities $n^\circ$\ref{two_qubit}
It follows that
\begin{align*}
\lambda_2+\mu_2&\ge a+b \ge \nu_2+\nu_3+2\nu_4,\\
\abs{\lambda_2-\mu_2}&\le b-a \le \min(\nu_2-\nu_4,\nu_1-\nu_3),\\
\lambda_2 &\ge a\ge \nu_3+\nu_4,\\
\mu_2&\ge a\ge \nu_3+\nu_4,
\end{align*}
which in turn implies Bravyi inequalities (\ref{brav}). One can
easily check that the later are sufficient for
$g(\lambda,\mu,\nu)\ne0$. This gives another proof of Bravyi
theorem $n^\circ$\ref{two_qubit}.
\begin{rem}
For even $n$ a precise description of $g(\lambda,\mu,\nu)$ is more
complicate.
%But with
Admitting an absolute error $\le1$ the Kronecker coefficient  is
equal to {\it weighted\,} number of lattice points in heptagon
\eqref{septagon}, assigning weight $1/2$ to boundary points on
the line $b-a =\lambda_1-\mu_2$.

%assuming boundary  points on the side $b-a =\lambda_1-\mu_2$ have
%weight $1/2$.

Available information on Kronecker coefficients of three row
diagrams seems to be insufficient to deduce Higichi theorem
$n^\circ$\ref{hig} or 397 independent marginal inequalities for
two qutrits in section \ref{3x3}.
\end{rem}

\section{Applications to representation theory} \label{Appl2Repr}
%As we've mentioned above
%available
Unfortunately currently available information on Kronecker
coefficients is sparse and sporadic.\footnote{Arguably this is
{\it ``... the last major problem in ordinary representation
theory of $S_n$"} \cite{Dvir}.}
%This essentially limits
%possible
Therefore direct applications of Theorem \ref{mainThm} to marginal
problem is limited. We believe however that
%close relation
interplay between %Kronecker coefficients
representation theory of $\mathrm{S}_n$ and quantum margins
%can be used
%may be essential
%is crucial, and
eventually may lead to  solution of both problems simultaneously,
as it happens, for example, for Hermitian spectral problem and
representations of $\SU(\mathcal{H})$. We leave this topic for a
future  study, and bound ourselves to few examples of using
Theorem \ref{mainThm} in backward direction.
%This should be a subject of further study. Here we bound ourselves
%to few examples of using Main Theorem $n^\circ$\ref{main} in
%backward direction to
%get new results in representation theory.
%gain new information
%to extend our understanding of Kronecker coefficients.
One obstruction to this is lack of the saturation property (see
Remark \ref{sat}), which bounds rigorous results to
%{\it asymptotic regime}
{\it stable regime} of very long diagrams $m\lambda$, $m\gg1$,
although some of them may still hold
%be valid
in general.
%Our approach
%Here we use Main Theorem $n^\circ$\ref{main} in backward direction
%to shed new light on their behavior.
%However lack of saturation property (see Remark \ref{sat}) bounds
%rigorous results  to {\it asymptotic regime},
% $g(m\lambda,m\mu,m\nu)$, $m\gg1$,
%although  some of them may be still valid without this assumption.
We'll use special notation for {\it stable inclusion\,}
\begin{equation}\label{stable}\nu\underset{stbl}{\subset}
\lambda\otimes\mu \Longleftrightarrow m\nu\subset
m\lambda\otimes m\mu \text{ for some } m>0,
\end{equation}
where the diagrams are multiplied by $m$ row-wise.

%To simplify notations we write $\nu\underset{stbl}{\subset}
%\lambda\otimes\mu$ for {\it stable inclusion\,} $m\nu\subset
%m\lambda\otimes m\mu$, $m>0$.


%Here we deduce some new result apply Main Theorem \ref{main} Our
%approach taps a new source of such information. However lack of
%saturation property (see Remark \ref{sat}) confines  it only to
%{\it asymptotic behavior\,} of
%Kronecker coefficients
%$g(m\lambda,m\mu,m\nu)$ as $m\rightarrow\infty$. Stable support.
\subsection{Stable support of Kronecker coefficients}
Here we are interested in triplets of Young diagrams
$\lambda,\mu,\nu$ such that  $\lambda\subset\mu\otimes\nu$, i.e.
in {\it support\,} of Kronecker coefficients. We have seen, for
example, that for such triplets $\height(\lambda)\le
\height(\mu)\height(\nu)$ ($n^\circ$\ref{rank}), and
$\abs{\overline{\lambda}}\le\abs{\overline{\mu}}+\abs{\overline{\nu}}$
($n^\circ$\ref{murn}).
%The next theorem tells that
It is much easier to describe {\it stable support\,},  defined by
stable inclusion \eqref{stable}.
%inequality $g(m\lambda,m\mu,m\nu)\ne 0$
%for some $m>0$, or by stable inclusion
%$\nu\underset{stbl}{\subset}
%\lambda\otimes\mu$ then


%Let's denote by $\Lambda_r$ the set of Young diagrams with at most
%$r$ rows. {\it Stable support\,} of Kronecker coefficients
%consists of triplets of diagrams from $\Lambda_r$ such that
%$g(m\lambda,m\mu,m\nu)\ne0$ for some $m> 0$, where the diagrams
%are multiplied by $m$ row-wise.
\begin{thm}
Stable inclusion $\nu\underset{stab}{\subset} \lambda\otimes\mu$
for diagrams $\lambda,\mu$ of bounded heights
$$\height\lambda\le m,\qquad\height\mu\le n$$
is given by marginal inequalities for system of format $m\times n$
with $\lambda=\Spec \rho_A$, $\mu=\Spec \rho_B$, $\nu=\Spec
\rho_{AB}$, .
\end{thm}
\begin{proof}
This follows from Theorem \ref{mainThm} combined with marginal
inequalities of section \ref{MargIneq2}.\end{proof}

For example, for two row diagrams the stable inclusion amounts to
Bravyi inequalities, cf. section \ref{QbtRev}.

\begin{cor} Suppose $\nu^{(i)}\subset \lambda^{(i)}\otimes \mu^{(i)}$. Then
\begin{equation}\label{add}(\nu^{(1)}+{\nu}^{(2)})\underset{stab}{\subset}
(\lambda^{(1)}+{\lambda}^{(2)})\otimes(\mu^{(1)}+{\mu}^{(2)})
\end{equation}
where addition of the diagrams is defined row-wise.
\end{cor}
\begin{rem} Numerical experiments suggest that  inclusion \eqref{add}
actually holds in usual sense. This leads to the following
\end{rem}
\begin{conj} Triplets $\lambda,\mu,\nu$ s.t.
$g(\lambda,\mu,\nu)\ne 0$ form a semigroup with respect to
row-wise addition.
\end{conj}
One can expect that the semigroup is finitely generated if heights
of the diagrams are bounded.
\subsection{Rectangular
diagrams} In the case of {\it rectanguler\,} diagrams the previous
problem has very simple answer.
\begin{thm} Let $\lambda,\mu\nu$ be rectangular diagrams of heights
$l,m,n$. Then stable inclusion $\nu\underset{stab}{\subset}
\lambda\otimes\mu$ is equivalent to inequalities
$l\le mn$, $m\le nl$, $n\le lm$.
\end{thm}
\begin{proof} The theorem is about stable inclusion of trivial
representation in $\lambda\otimes\mu\otimes \nu$. By Theorem
\ref{mainThm} trivial representation corresponds to a {\it pure\,}
state $\rho_{ABC}$ of three component system $\mathcal{H}_A\otimes
\mathcal{H}_B\otimes
\mathcal{H}_C$ of formal $l\times m\times n$,
while rectangular diagrams $\lambda\mu\nu$ correspond to {\it
scalar\,} margins $\rho_A,\rho_B,\rho_C$. The result now follows
from criterion of existence of a pure state with scalar margins.
in section \ref{scalar}.
\end{proof}
\begin{rem} The simplest proof of the above criterion comes
from marginal inequalities of Theorem \ref{cubicleThm}.
\end{rem}

%As another example of application quantum marginal problem to
%representation theory consider
%
% Theorem on existence of a completely entangled state in
%section \ref{scalar} implies (stable) criterion for existence of
%invariant in tensor product of rectangular diagrams.
%\subsection{Duality} \label{dual} $\lambda\otimes\mu$ contains a component
%$\rho\prec\nu$ iff $\nu\subset [\lambda]\otimes [\mu]$. See
%$n^\circ$\ref{cl}. In particular minimal components of
%$\lambda\otimes\mu$ coincide with that of $[\lambda]\otimes
%[\mu]$. It seems this resolve a problem in \cite{Vallejo00} about
%minimal components in $\lambda\otimes\nu$ in majorization order,
%cf. Corollary \ref{minimal}.
%
%\subsection{Maximal height of tensor product}
%There is a strange problem raised by Regev \cite{Regev82} about
%equality $\height(\nu)=\height(\lambda)\cdot\height(\mu)$ for
%given heights in the right hand side, and some
%$\nu\subset\lambda\otimes\mu$, and sufficiently big degree of the
%symmetric group $n\gg1$. A simple solution comes from marginal
%problem since $\rho_{AB}=\rho_A\otimes\rho_{B}$ has margins
%$\rho_A,\rho_B$ and maximal rank
%$\rk\rho_{AB}=\rk\rho_A\rk\rho_B$, cf. $n^\circ$\ref{rank}.
%
%\subsection[B-S]{Berenstein-Sjamaar theotrem and
%completely entangled states} Make use of B-S theorem to deduce
%criterion $n^\circ$\ref{scalar} for existence of a completely
%entangled state.
\appendix
\section{Marginal inequalities}

%\marginpar{\footnotesize Add list of extremal edges for every
%system, including 5 qubits, and pictures of the decomposition into
%cubicles. Estimation of the number of marginal inequalities for 5
%qubits: $125\times16=2000$. This number can be handled by Maple,
%if one skip the permutations of qubits. Do this! I've also add
%format $2\times2\times3$ to include all systems of rank $\le4$.}
%complete and independent sets of
The appendix contains marginal inequalities for systems of rank
$\le 4$, i.e. for arrays  up to four qubits and for systems of
formats $2\times3$, $2\times4$, $3\times3$,  and
$2\times2\times3$. It covers all systems with few hundreds (rather
then thousands) marginal inequalities. Other statistical data are
collected in the following table.

\vspace{3mm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
System & Rank&Inequalities&Edges &Permutations\\
\hline
$2\times2$&2&7\;(4)&3&9\\
\hline
$2\times2\times2$&3&40\;(10)&10&111\\
%\hline
$2\times3$&3&41&6&98\\
\hline
$2\times4$&4&234&11&2191\\
%\hline
$3\times3$&4&387\;(197)&17&2298\\
%\hline
$2\times2\times3$&4&442\;(232)&39&3914\\
%\hline
$2\times2\times2\times2$&4&805\;(50)&101&3723\\
\hline
\end{tabular}
\end{center}

For system of format $p\times q\times\cdots$ the last two columns
show the number of extremal edges and the number of permutations
$w\in S_{p\cdot q\cdot\ldots}$ of length $\ell(w)\le
\frac{p(p-1)}{2}+\frac{q(q-1)}{2}+\cdots$.
The number of marginal inequalities produced by permutation $w$
and extremal edge $E$ is equal to the number of components in
$\varphi_E^*(\sigma_w)$. For system of rank 4 this amounts all
together  to hundreds of thousands inequalities.
%Their redundancy check
%\marginpar{\footnotesize Add reference to joint paper with
%I.~Ozen.} Removing redundant ones by a modern PC takes days (to be
compared with seconds for rank $\le 3$). Note that the number of
inequalities in most cases can be essentially reduced using
symmetry w.r. to permutations of equidimensional components. These
numbers are shown in parenthesis. Further reduction is possible
using duality
$$(\lambda_1\ge\lambda_2\ge \cdots\ge\lambda_n)
\mapsto (-\lambda_n\ge-\lambda_{n-1}\ge \cdots\ge-\lambda_1)$$
applied to every spectrum involved in a marginal inequality.

Because of huge number of marginal inequalities for systems of
rank more then three, they provide only an illusion of a complete
solution of the
%quantum marginal
problem and
%and therefore
have limited practical value. I
%keep
reproduce the inequalities here primary as experimental data for
those brave people who may try to understand them better.
%Marginal
%inequalities for systems of rank 4 have limited practical value
%because  The tables primary designed as an experimental basis for
%testing
%general structure of the marginal inequalities
%general conjectures rather then for practical applications.
%\begin{tabular}
%\end{tabular}

Details of the calculation \cite{Kl-Oz} will be published
elsewhere.
\subsection{Arrays of qubits}
In this case all marginal inequalities can be produced
automatically from the list of extremal edges, see $n^o$
\ref{mineq} for details.  All the spectra are arranged in
decreasing order $\rho_1\ge\rho_2\ge\cdots$. By technical reason
we normalize mixed states to trace zero. Hence univariant margins
have spectra $(\lambda,-\lambda),(\mu,-\mu),$ etc. To get marginal
inequalities in standard normalization $\Tr\rho=1$ one have to
change $2\lambda,2\mu,\ldots$ in LHS of the inequalities by
$\lambda_1-\lambda_2,\mu_1-\mu_2,\ldots$. To save space we skip
all the inequalities obtained from another one by a permutation of
qubits.
%The systems of inequalities given below
The reduced systems are complete if
%applied to
marginal spectra arranged in increasing order
$\lambda\le\mu\le\cdots$.
%\marginpar{\footnotesize I've to adjust the inequalities to this
%condition.}
\subsubsection{Two qubits}\label{2qbt} For completeness we reproduce here
Bravyi inequalities (\ref{brav}) for two qubits in current
notations.
\begin{align*}
2\lambda&\le\nu_1+\nu_2-\nu_3-\nu_4,\\
\lambda+\mu&\le \nu_1-\nu_4,\\
\lambda-\mu&\le\nu_2-\nu_4,\\
\lambda-\mu&\le\nu_1-\nu_3.
\end{align*}

\subsubsection{Three qubits}\label{3qbt} Below are 10 marginal inequalities
for three qubits.
%Together with inequalities obtained by
By permutations of qubits they give a complete system of  40
independent constraints.
\begin{align*}
2\lambda &\le\rho_1+\rho_2+\rho_3+\rho_4-\rho_5-\rho_6-\rho_7-\rho_8,\\
\lambda+\mu &\le \rho_1+ \rho_2- \rho_7- \rho_8,\\[2mm]
2\lambda+\mu+\nu&\le2\rho_1+\rho_2+\rho_3-\rho_6-\rho_7-2\rho_8,\\
2\lambda+\mu-\nu&\le2\rho_2+\rho_1+\rho_3-\rho_6-\rho_7-2\rho_8,\\
2\lambda+\mu-\nu&\le2\rho_1+\rho_2+\rho_4-\rho_6-\rho_7-2\rho_8,\\
2\lambda+\mu-\nu&\le2\rho_1+\rho_2+\rho_3-\rho_5-\rho_7-2\rho_8,\\
2\lambda+\mu-\nu&\le2\rho_1+\rho_2+\rho_3-\rho_6-\rho_8-2\rho_7,\\[2mm]
2\lambda+2\mu+2\nu&\le3\rho_1+\rho_2+\rho_3+\rho_4-\rho_5-\rho_6-\rho_7-3\rho_8,\\
2\lambda+2\mu-2\nu&\le3\rho_2+\rho_1+\rho_3+\rho_4-\rho_5-\rho_6-\rho_7-3\rho_8,\\
2\lambda+2\mu-2\nu&\le3\rho_1+\rho_2+\rho_3+\rho_4-\rho_5-\rho_6-\rho_8-3\rho_7.
%2 x[2] <= z[1] + z[2] + z[3] + z[4] - \nu_5 - z[6] - z[7] - z[8]
%2 x[3] <= z[1] + z[2] + z[3] + z[4] - \nu_5 - z[6] - z[7] - z[8]
%2 x[1] + 2 x[3] <= 2 z[1] + 2 z[2] - 2 z[7] - 2 z[8]
%2 x[2] + 2 x[3] <= 2 z[1] + 2 z[2] - 2 z[7] - 2 z[8]
%4 x[1] + 2 x[2] + 2 x[3] <= 4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] -2z[7] - 4 z[8]
%2 x[1] + 4 x[2] + 2 x[3] <= 4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2z[7] - 4 z[8]
%2 x[1] + 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%2 x[1] + 2 x[2] - 2 x[3] <=3 z[2] + z[1] + z[3] + z[4] - \nu_5 - z[6] - z[7] - 3 z[8]
%2 x[1] - 2 x[2] + 2 x[3] <=3 z[2] + z[1] + z[3] + z[4] - \nu_5 - z[6] - z[7] - 3 z[8]
%-2 x[1] + 2 x[2] + 2 x[3] <=3 z[2] + z[1] + z[3] + z[4] - \nu_5 - z[6] - z[7] - 3 z[8]
%4 x[1] + 2 x[2] - 2 x[3] <=4 z[2] + 2 z[1] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%4 x[1] - 2 x[2] + 2 x[3] <=4 z[2] + 2 z[1] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%2 x[1] + 4 x[2] - 2 x[3] <=4 z[2] + 2 z[1] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%-2 x[1] + 4 x[2] + 2 x[3] <=4 z[2] + 2 z[1] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%2 x[1] - 2 x[2] + 4 x[3] <=4 z[2] + 2 z[1] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%-2 x[1] + 2 x[2] + 4 x[3] <=4 z[2] + 2 z[1] + 2 z[3] - 2 z[6] - 2 z[7] - 4 z[8]
%4 x[1] + 2 x[2] - 2 x[3] <=4 z[1] + 2 z[2] + 2 z[4] - 2 z[6] - 2 z[7] - 4 z[8]
%4 x[1] - 2 x[2] + 2 x[3] <=4 z[1] + 2 z[2] + 2 z[4] - 2 z[6] - 2 z[7] - 4 z[8]
%2 x[1] + 4 x[2] - 2 x[3] <=4 z[1] + 2 z[2] + 2 z[4] - 2 z[6] - 2 z[7] - 4 z[8]
%-2 x[1] + 4 x[2] + 2 x[3] <=4 z[1] + 2 z[2] + 2 z[4] - 2 z[6] - 2 z[7] - 4 z[8]
%2 x[1] - 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[4] - 2 z[6] - 2 z[7] - 4 z[8]
%-2 x[1] + 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[4] - 2 z[6] - 2 z[7] - 4 z[8]
%4 x[1] + 2 x[2] - 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 \nu_5 - 2 z[7] - 4 z[8]
%4 x[1] - 2 x[2] + 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 \nu_5 - 2 z[7] - 4 z[8]
%2 x[1] + 4 x[2] - 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 \nu_5 - 2 z[7] - 4 z[8]
%-2 x[1] + 4 x[2] + 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 \nu_5 - 2 z[7] - 4 z[8]
%2 x[1] - 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 \nu_5 - 2 z[7] - 4 z[8]
%-2 x[1] + 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 \nu_5 - 2 z[7] - 4 z[8]
%2 x[1] + 2 x[2] - 2 x[3] <=3 z[1] + z[2] + z[3] + z[4] - \nu_5 - z[6] - z[8] - 3 z[7]
%2 x[1] - 2 x[2] + 2 x[3] <=3 z[1] + z[2] + z[3] + z[4] - \nu_5 - z[6] - z[8] - 3 z[7]
%-2 x[1] + 2 x[2] + 2 x[3] <=3 z[1] + z[2] + z[3] + z[4] - \nu_5 - z[6] - z[8] - 3 z[7]
%4 x[1] + 2 x[2] - 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[8] - 4 z[7]
%4 x[1] - 2 x[2] + 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[8] - 4 z[7]
%2 x[1] + 4 x[2] - 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[8] - 4 z[7]
%-2 x[1] + 4 x[2] + 2 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[8] - 4 z[7]
%2 x[1] - 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[8] - 4 z[7]
%-2 x[1] + 2 x[2] + 4 x[3] <=4 z[1] + 2 z[2] + 2 z[3] - 2 z[6] - 2 z[8] - 4 z[7]
\end{align*}

\subsubsection{Four qubits}\label{4qbt}
In this case there are 805 independent marginal inequalities. To
save space we skip inequalities obtained from a preceding one by a
permutation of qubits. The resulting 50 inequalities are grouped
by the extremal edges. The basic inequality stands first, and the
remaining ones in the group are obtained by transposition of
eigenvalues $\tau_{2i-1},\tau_{2i}$ typeset in bold face and
change sign of the first coefficient in LHS, see Theorem
\ref{Mod_Qbt} for details. Note that in the fifth group odd
transpositions $\tau_{5},\tau_{6}$ and $\tau_{11},\tau_{12}$ give
redundant inequalities. Arguably this is the last system of
marginal constraints which can be published. For 5 qubits there
are more then thousand independent inequalities counted up to a
permutation of qubits.


{\footnotesize
\begin{align*}
2\rho &\le
\tau_1+\tau_2+\tau_3+\tau_4+\tau_5+\tau_6+\tau_7+\tau_8-\tau_9-\tau_{10}-
\tau_{11}-\tau_{12}-\tau_{13}-\tau_{14}-\tau_{15}-\tau_{16},\\[2mm]
2 \nu + 2 \rho &\le 2 \tau_1 + 2 \tau_2 + 2 \tau_3 + 2
\tau_4 - 2
\tau_{13} -
2 \tau_{14} - 2 \tau_{15} - 2 \tau_{16},\\[2mm]
2\mu+2\nu+2\rho &\le
3\tau_1+3\tau_2+\tau_3+\tau_4+\tau_5+\tau_6+\tau_7+\tau_8-\tau_9-\tau_{10}-\tau_{11}-
\tau_{12}-\tau_{13}-\tau_{14}-3\tau_{15}-3\tau_{16},\\[2mm]
2 \mu + 2 \nu + 4 \rho &\le 4 \tau_1 + 4 \tau_2 + 2 \tau_3 + 2
\tau_4 + 2
\tau_5 + 2 \tau_6 - 2 \tau_{11} - 2 \tau_{12} - 2 \tau_{13} - 2 \tau_{14} - 4 \tau_{15} -
4 \tau_{16},
\displaybreak[0]\\[2mm]
2 \lambda + 2 \mu + 2 \nu + 2 \rho&\le 4
\tau_1 + 2\tau_2 + 2\tau_3 + 2 \tau_4 + 2 \tau_5 - 2 \tau_{12} - 2
\tau_{13} - 2\tau_{14} - 2 \tau_{15} -
4\tau_{16},\\
2 \lambda + 2 \mu + 2 \nu - 2 \rho&\le 4 \boldsymbol{\tau_2} + 2
\boldsymbol{\tau_1} + 2
\tau_3 + 2 \tau_4 + 2 \tau_5 - 2 \tau_{12} - 2 \tau_{13} - 2
\tau_{14} - 2 \tau_{15} - 4
\tau_{16},\\
2 \lambda + 2 \mu + 2 \nu - 2 \rho &\le 4 \tau_1 + 2 \tau_2 + 2
\tau_3 +
2 \tau_4 + 2 \tau_5 - 2 \tau_{12} - 2 \tau_{13} - 2 \tau_{14} - 2
\boldsymbol{\tau_{16}} - 4\boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 2 \mu + 2 \nu + 4 \rho &\le 5 \tau_1 + 3 \tau_2 + 3
\tau_3 + 3 \tau_4 + \tau_5 + \tau_6 + \tau_7 +\tau_8-\tau_9 - \tau_{10} - \tau_{11} -
\tau_{12}
- 3 \tau_{13}- 3 \tau_{14} - 3 \tau_{15} - 5 \tau_{16},\\
2 \lambda + 2 \mu - 2 \nu + 4 \rho &\le 5 \boldsymbol{\tau_2} + 3
\boldsymbol{\tau_1} + 3
\tau_3 + 3 \tau_4 + \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - \tau_{12}
- 3 \tau_{13} - 3 \tau_{14} - 3 \tau_{15} - 5 \tau_{16},\\
2 \lambda + 2 \mu - 2 \nu + 4 \rho &\le5 \tau_1 + 3 \tau_2 + 3
\tau_3 + 3 \tau_4 + \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - \tau_{12}
- 3 \tau_{13} - 3 \tau_{14} - 3 \boldsymbol{\tau_{16}} - 5
\boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 2 \mu + 2 \nu + 6 \rho&\le 6 \tau_1 + 4 \tau_2 + 4
\tau_3 + 4
\tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} - 2 \tau_{11} - 2 \tau_{12} - 4
\tau_{13} - 4 \tau_{14} - 4 \tau_{15} - 6 \tau_{16},\\
2 \lambda + 2 \mu - 2 \nu + 6 \rho &\le 6 \boldsymbol{\tau_2} + 4
\boldsymbol{\tau_1} + 4
\tau_3 + 4
\tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} - 2 \tau_{11} - 2 \tau_{12} - 4
\tau_{13} - 4 \tau_{14} - 4 \tau_{15} - 6 \tau_{16},\\
2 \lambda + 2 \mu - 2 \nu + 6 \rho &\le 6 \tau_1 + 4 \tau_2 + 4
\tau_3 +
4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \boldsymbol{\tau_8} - 2
\tau_{10} - 2
\tau_{11}
 - 2 \tau_{12} -4 \tau_{13} - 4 \tau_{14} - 4 \tau_{15} - 6 \tau_{16},\\
 2 \lambda + 2 \mu - 2 \nu + 6 \rho&\le 6 \tau_1 + 4 \tau_2 + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \boldsymbol{\tau_9} - 2
\tau_{11} - 2 \tau_{12} - 4
\tau_{13} - 4 \tau_{14} - 4 \tau_{15} - 6 \tau_{16},\\
2 \lambda + 2 \mu - 2 \nu + 6 \rho &\le6 \tau_1 + 4 \tau_2 + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 2 \tau_{12} - 4 \tau_{13} - 4 \tau_{14} - 4
\boldsymbol{\tau_{16}} - 6 \boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 2 \mu + 4 \nu + 4 \rho &\le 6 \tau_1 + 4 \tau_2 + 4
\tau_3 + 2 \tau_4 + 2 \tau_5 + 2 \tau_6 - 2 \tau_{11} - 2 \tau_{12} - 2 \tau_{13} - 4
\tau_{14} - 4 \tau_{15} - 6 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 4 \rho&\le 6 \boldsymbol{\tau_2} + 4
\boldsymbol{\tau_1} + 4
\tau_3 + 2 \tau_4 + 2 \tau_5 + 2 \tau_6 - 2 \tau_{11} - 2
\tau_{12} - 2 \tau_{13} - 4 \tau_{14} -
4 \tau_{15} - 6 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 4 \rho &\le 6 \tau_1 + 4 \tau_2 + 4
\boldsymbol{\tau_4} + 2 \boldsymbol{\tau_3} + 2 \tau_5 + 2 \tau_6 - 2 \tau_{11} - 2
\tau_{12} - 2 \tau_{13} - 4 \tau_{14} -
4 \tau_{15} - 6 \tau_{16},\\
2\lambda - 2 \mu + 4 \nu + 4 \rho &\le6 \tau_1 + 4 \tau_2 + 4
\tau_3 + 2
\tau_4 + 2 \tau_5 + 2 \tau_6 - 2 \tau_{11} - 2 \tau_{12} - 2 \boldsymbol{\tau_{14}} -
4 \boldsymbol{\tau_{13}} - 4
\tau_{15} - 6 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 4 \rho &\le 6 \tau_1 + 4 \tau_2 + 4
\tau_3 + 2
\tau_4 + 2 \tau_5 + 2 \tau_6 - 2 \tau_{11} - 2 \tau_{12} - 2 \tau_{13} - 4 \tau_{14} - 4
\boldsymbol{\tau_{16}} - 6 \boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 2 \mu + 4 \nu + 6 \rho &\le 7 \tau_1 + 5 \tau_2 + 5
\tau_3 + 3 \tau_4 + 3 \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - 3
\tau_{12} - 3 \tau_{13} - 5 \tau_{14} - 5 \tau_{15} - 7 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 6 \rho &\le 7 \boldsymbol{\tau_2} + 5
\boldsymbol{\tau_1} + 5
\tau_3 + 3 \tau_4 + 3 \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - 3
\tau_{12} - 3 \tau_{13} - 5 \tau_{14} - 5 \tau_{15} - 7 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 6 \rho &\le7 \tau_1 + 5 \tau_2 + 5
\boldsymbol{\tau_4} + 3 \boldsymbol{\tau_3} + 3 \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - 3
\tau_{12} - 3 \tau_{13} - 5 \tau_{14} - 5 \tau_{15} - 7 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 6 \rho &\le7 \tau_1 + 5 \tau_2 + 5
\tau_3 + 3 \tau_4 + 3 \boldsymbol{\tau_6} + \boldsymbol{\tau_5} + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - 3
\tau_{12} - 3 \tau_{13} - 5 \tau_{14} - 5 \tau_{15} - 7 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 6 \rho &\le7 \tau_1 + 5 \tau_2 + 5
\tau_3 + 3 \tau_4 + 3 \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \boldsymbol{\tau_{12}} - 3
\boldsymbol{\tau_{11}} - 3 \tau_{13} - 5 \tau_{14} - 5 \tau_{15} - 7 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 6 \rho &\le7 \tau_1 + 5 \tau_2 + 5
\tau_3 + 3 \tau_4 + 3 \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - 3
\tau_{12} - 3 \boldsymbol{\tau_{14}} - 5 \boldsymbol{\tau_{13}} - 5 \tau_{15} - 7 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 6 \rho &\le 7 \tau_1 + 5 \tau_2 + 5
\tau_3 + 3 \tau_4 + 3 \tau_5 + \tau_6 + \tau_7 + \tau_8 - \tau_9 -
\tau_{10} - \tau_{11} - 3
\tau_{12} - 3 \tau_{13} - 5 \tau_{14} - 5 \boldsymbol{\tau_{16}} - 7 \boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 4 \mu + 4 \nu + 6 \rho&\le 8 \tau_1 + 6 \tau_2 + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 2 \tau_{12} -
4 \tau_{13} - 4 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
-2 \lambda + 4 \mu + 4 \nu + 6 \rho&\le 8 \boldsymbol{\tau_2} + 6
\boldsymbol{\tau_1} + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 2 \tau_{12} -
4 \tau_{13} - 4 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
-2 \lambda + 4 \mu + 4 \nu + 6 \rho &\le8 \tau_1 + 6 \tau_2 + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \boldsymbol{\tau_8} - 2 \tau_{10} -
2 \tau_{11} - 2 \tau_{12} -
4 \tau_{13} - 4 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
-2 \lambda + 4 \mu + 4 \nu + 6 \rho&\le 8 \tau_1 + 6 \tau_2 + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \boldsymbol{\tau_9} - 2
\tau_{11} - 2 \tau_{12} - 4
\tau_{13} - 4 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
 -2 \lambda + 4 \mu + 4 \nu + 6 \rho &\le8
\tau_1 + 6 \tau_2 + 4
\tau_3 + 4 \tau_4 + 2 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 2 \tau_{12} - 4 \tau_{13} - 4 \tau_{14} - 6
\boldsymbol{\tau_{16}} - 8 \boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 2 \mu + 4 \nu + 8 \rho&\le 8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} -
4 \tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho&\le 8 \boldsymbol{\tau_2} + 6
\boldsymbol{\tau_1} + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} -
4 \tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho &\le 8 \tau_1 + 6 \tau_2 + 6
\boldsymbol{\tau_4} + 4
\boldsymbol{\tau_3} + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} - 2 \tau_{11} - 4 \tau_{12} - 4
\tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho &\le 8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \boldsymbol{\tau_6} + 2 \boldsymbol{\tau_5} + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} -
4 \tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho&\le 8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \boldsymbol{\tau_8} - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} -
4 \tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho&\le 8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \boldsymbol{\tau_9} - 2
\tau_{11} - 4 \tau_{12} - 4
\tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho&\le 8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \boldsymbol{\tau_{12}} - 4 \boldsymbol{\tau_{11}} -
4 \tau_{13} - 6 \tau_{14} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho &\le8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} -
4 \boldsymbol{\tau_{14}} - 6 \boldsymbol{\tau_{13}} - 6 \tau_{15} - 8 \tau_{16},\\
2 \lambda - 2 \mu + 4 \nu + 8 \rho &\le8 \tau_1 + 6 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} - 4 \tau_{13} - 6 \tau_{14} - 6
\boldsymbol{\tau_{16}} - 8
\boldsymbol{\tau_{15}},
\displaybreak[0]\\[2mm]
2 \lambda + 4 \mu + 6 \nu + 8 \rho&\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} -
4 \tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho &\le 10 \boldsymbol{\tau_2} +
8
\boldsymbol{\tau_1} + 6
\tau_3 + 4
\tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} - 2 \tau_{11} - 4 \tau_{12} - 4
\tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho &\le 10 \tau_1 + 8 \tau_2 + 6
\boldsymbol{\tau_4} + 4 \boldsymbol{\tau_3} + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12}
- 4 \tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho &\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4
\tau_4 + 4 \boldsymbol{\tau_6} + 2 \boldsymbol{\tau_5} + 2 \tau_7 - 2 \tau_{10} - 2 \tau_{11} - 4 \tau_{12} - 4
\tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho&\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \boldsymbol{\tau_8} - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12}
- 4 \tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho &\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4
\tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \boldsymbol{\tau_9} - 2 \tau_{11} - 4 \tau_{12} - 4
\tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho&\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \boldsymbol{\tau_{12}} - 4 \boldsymbol{\tau_{11}}
- 4 \tau_{13} - 6 \tau_{14} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho &\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12}
- 4 \boldsymbol{\tau_{14}} - 6 \boldsymbol{\tau_{13}} - 8 \tau_{15} - 10 \tau_{16},\\
-2 \lambda + 4 \mu + 6 \nu + 8 \rho &\le 10 \tau_1 + 8 \tau_2 + 6
\tau_3 + 4 \tau_4 + 4 \tau_5 + 2 \tau_6 + 2 \tau_7 - 2 \tau_{10} -
2 \tau_{11} - 4 \tau_{12} - 4 \tau_{13} - 6 \tau_{14} - 8
\boldsymbol{\tau_{16}} - 10 \boldsymbol{\tau_{15}}.
\end{align*}}
\subsection{Two qutrits}\label{3x3}
Below are $197$ marginal inequalities for system
 of two qutrits $\mathcal{H}\otimes
\mathcal{H}$, $\dim\mathcal{H}=3$. Together with inequalities obtained
by transposition of qutrits they form a complete and independent
system of 387 marginal constraints. One can check that for mixed
state of rank three (i.e. for $\nu_i=0,i>3$) the system
%is equivalent
amounts to Higuchi inequalities $n^o$ \ref{hig}. {\footnotesize
\begin{align*}
%2\mu_1-\mu_2-\mu_3 &\le
%2\nu_1+2\nu_2+2\nu_3-\nu_4-\nu_5-\nu_6-\nu_7-\nu_8-\nu_9,\\
2\lambda_1-\lambda_2-\lambda_3 &\le
2\nu_1+2\nu_2+2\nu_3-\nu_4-\nu_5-\nu_6-\nu_7-\nu_8-\nu_9,\\
%\mu_1+\mu_2-2\mu_3 &\le
%\nu_1+\nu_2+\nu_3+\nu_4+\nu_5+\nu_6-2\nu_7-2\nu_8-2\nu_9,\\
\lambda_1+\lambda_2-2\lambda_3 &\le
\nu_1+\nu_2+\nu_3+\nu_4+\nu_5+\nu_6-2\nu_7-2\nu_8-2\nu_9,
\displaybreak[0]\\[2mm]
\lambda_1+\lambda_2-2\lambda_3+\mu_1+\mu_2-2\mu_3 &\le
2\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-\nu_8-4\nu_9,\\
\lambda_1+\lambda_2-2\lambda_3+\mu_1+\mu_3-2\mu_2 &\le
2\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-\nu_9-4\nu_8,\\
%\lambda_1+\lambda_3-2\lambda_2+\mu_1+\mu_2-2\mu_3 &\le
%2\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-\nu_9-4\nu_8,\\
\lambda_1+\lambda_2-2\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
2\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_7-\nu_8-4\nu_9,\\
%\lambda_2+\lambda_3-2\lambda_1+\mu_1+\mu_2-2\mu_3 &\le
%2\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_7-\nu_8-4\nu_9,\\
\lambda_1+\lambda_2-2\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
2\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-\nu_9-4\nu_7,
%\lambda_2+\lambda_3-2\lambda_1+\mu_1+\mu_2-2\mu_3 &\le
%2\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-\nu_9-4\nu_7.
\displaybreak[0]\\[2mm]
2\lambda_1-\lambda_2-\lambda_3+2\mu_1-\mu_2-\mu_3 &\le
4\nu_1+\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-2\nu_9,\\
2\lambda_1-\lambda_2-\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
4\nu_2+\nu_1+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-2\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+2\mu_1-\mu_2-\mu_3 &\le
%4\nu_2+\nu_1+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-2\nu_9,\\
2\lambda_1-\lambda_2-\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
4\nu_3+\nu_1+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-2\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+2\mu_1-\mu_2-\mu_3 &\le
%4\nu_3+\nu_1+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-2\nu_9,\\
2\lambda_1-\lambda_2-\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
4\nu_1+\nu_2+\nu_3+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-2\nu_9.
%2\lambda_3-\lambda_1-\lambda_2+2\mu_1-\mu_2-\mu_3 &\le
%4\nu_1+\nu_2+\nu_3+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-2\nu_9.
\displaybreak[0]\\[2mm]
%2\lambda_1-\lambda_2-\lambda_3+\mu_1+\mu_2-2\mu_3 &\le 3\nu_1+3\nu_2-3\nu_8-3\nu_9,\\
\lambda_1+\lambda_2-2\lambda_3+2\mu_1-\mu_2-\mu_3 &\le
3\nu_1+3\nu_2-3\nu_8-3\nu_9,\\
%2\lambda_1-\lambda_2-\lambda_3+\mu_1+\mu_3-2\mu_2 &\le 3\nu_1+3\nu_3-3\nu_8-3\nu_9,\\
\lambda_1+\lambda_3-2\lambda_2+2\mu_1-\mu_2-\mu_3 &\le
3\nu_1+3\nu_3-3\nu_8-3\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+\mu_1+\mu_2-2\mu_3 &\le 3\nu_1+3\nu_2-3\nu_7-3\nu_9,\\
\lambda_1+\lambda_2-2\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
3\nu_1+3\nu_2-3\nu_7-3\nu_9,\\
%2\lambda_1-\lambda_2-\lambda_3+\mu_2+\mu_3-2\mu_1 &\le 3\nu_2+3\nu_3-3\nu_8-3\nu_9,\\
\lambda_2+\lambda_3-2\lambda_1+2\mu_1-\mu_2-\mu_3 &\le
3\nu_2+3\nu_3-3\nu_8-3\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+\mu_1+\mu_3-2\mu_2 &\le 3\nu_2+3\nu_3-3\nu_8-3\nu_9,\\
\lambda_1+\lambda_3-2\lambda_2+2\mu_2-\mu_1-\mu_3 &\le
3\nu_2+3\nu_3-3\nu_8-3\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+\mu_1+\mu_2-2\mu_3 &\le 3\nu_2+3\nu_3-3\nu_8-3\nu_9,\\
\lambda_1+\lambda_2-2\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
3\nu_2+3\nu_3-3\nu_8-3\nu_9,\\
%2\lambda_1-\lambda_2-\lambda_3+\mu_2+\mu_3-2\mu_1 &\le 3\nu_1+3\nu_2-3\nu_7-3\nu_8,\\
\lambda_2+\lambda_3-2\lambda_1+2\mu_1-\mu_2-\mu_3 &\le
3\nu_1+3\nu_2-3\nu_7-3\nu_8,\\
%2\lambda_2-\lambda_1-\lambda_3+\mu_1+\mu_3-2\mu_2 &\le 3\nu_1+3\nu_2-3\nu_7-3\nu_8,\\
\lambda_1+\lambda_3-2\lambda_2+2\mu_2-\mu_1-\mu_3 &\le
3\nu_1+3\nu_2-3\nu_7-3\nu_8,\\
%2\lambda_3-\lambda_1-\lambda_2+\mu_1+\mu_2-2\mu_3 &\le 3\nu_1+3\nu_2-3\nu_7-3\nu_8,
\lambda_1+\lambda_2-2\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
3\nu_1+3\nu_2-3\nu_7-3\nu_8,
\displaybreak[0]\\[2mm]
\lambda_1-\lambda_3+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_1+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_5-\nu_7-\nu_8-2\nu_9,\\
\lambda_2-\lambda_3+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_5-\nu_8-2\nu_9,\\
\lambda_2-\lambda_3+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_9,\\
\lambda_1-\lambda_3+\mu_2-\mu_3 &\le 2\nu_2+\nu_1+\nu_3-\nu_7-\nu_8-2\nu_9,\\
%\lambda_2-\lambda_3+\mu_1-\mu_3 &\le 2\nu_2+\nu_1+\nu_3-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_2-\lambda_3+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_9,\\
%\lambda_2-\lambda_3+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_1-\lambda_2+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_9,\\
%\lambda_1-\lambda_2+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_3-\nu_7-\nu_9-2\nu_8,\\
%\lambda_1-\lambda_2+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_7-\nu_9-2\nu_8,\\
\lambda_1-\lambda_2+\mu_2-\mu_3 &\le 2\nu_2+\nu_1+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_2+\nu_1+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_2-\mu_3 &\le 2\nu_2+\nu_1+\nu_3-\nu_6-\nu_8-2\nu_9,\\
%\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_2+\nu_1+\nu_3-\nu_6-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_9-2\nu_8,\\
%\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_9-2\nu_8,\\
\lambda_1-\lambda_2+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_4-\nu_7-\nu_9-2\nu_8,\\
%\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_4-\nu_7-\nu_9-2\nu_8,\\
\lambda_1-\lambda_2+\mu_2-\mu_3 &\le 2\nu_2+\nu_1+\nu_3-\nu_7-\nu_9-2\nu_8,\\
%\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_2+\nu_1+\nu_3-\nu_7-\nu_9-2\nu_8,\\
\lambda_1-\lambda_2+\mu_2-\mu_1 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_2-\lambda_1+\mu_1-\mu_2 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_2-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_8,\\
%\lambda_2-\lambda_1+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_8,\\
\lambda_2-\lambda_3+\mu_3-\mu_2 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_2+\mu_2-\mu_3 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_2-\lambda_3+\mu_3-\mu_2 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_8,\\
%\lambda_3-\lambda_2+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_8,\\
\lambda_1-\lambda_3+\mu_3-\mu_1 &\le 2\nu_3+\nu_1+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_1+\mu_1-\mu_3 &\le 2\nu_3+\nu_1+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_3-\mu_1 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_1+\mu_1-\mu_3 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_3+\mu_3-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_8,\\
%\lambda_3-\lambda_1+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_7-2\nu_8,\\
\lambda_1-\lambda_3+\mu_3-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_9-2\nu_7,\\
%\lambda_3-\lambda_1+\mu_1-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_9-2\nu_7,\\
\lambda_1-\lambda_2+\mu_3-\mu_2 &\le 2\nu_3+\nu_1+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_2+\mu_1-\mu_2 &\le 2\nu_3+\nu_1+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_2-\lambda_3+\mu_2-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_9-2\nu_7,\\
%\lambda_2-\lambda_1+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_9-2\nu_7,\\
\lambda_1-\lambda_2+\mu_3-\mu_1 &\le 2\nu_3+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_1+\mu_1-\mu_2 &\le 2\nu_3+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_3-\mu_1 &\le 2\nu_3+\nu_1+\nu_5-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_1+\mu_1-\mu_2 &\le 2\nu_3+\nu_1+\nu_5-\nu_7-\nu_8-2\nu_9,\\
\lambda_1-\lambda_2+\mu_3-\mu_1 &\le 2\nu_2+\nu_1+\nu_3-\nu_6-\nu_7-2\nu_8,\\
%\lambda_3-\lambda_1+\mu_1-\mu_2 &\le 2\nu_2+\nu_1+\nu_3-\nu_6-\nu_7-2\nu_8,\\
\lambda_1-\lambda_2+\mu_3-\mu_1 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_9-2\nu_8,\\
%\lambda_3-\lambda_1+\mu_1-\mu_2 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_9-2\nu_8,\\
\lambda_1-\lambda_2+\mu_3-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_7,\\
%\lambda_3-\lambda_1+\mu_1-\mu_2 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_7,\\
\lambda_2-\lambda_3+\mu_3-\mu_1 &\le 2\nu_3+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
%\lambda_3-\lambda_1+\mu_2-\mu_3 &\le 2\nu_3+\nu_2+\nu_4-\nu_7-\nu_8-2\nu_9,\\
\lambda_2-\lambda_3+\mu_3-\mu_1 &\le 2\nu_2+\nu_1+\nu_3-\nu_6-\nu_7-2\nu_8,\\
%\lambda_3-\lambda_1+\mu_2-\mu_3 &\le 2\nu_2+\nu_1+\nu_3-\nu_6-\nu_7-2\nu_8,\\
\lambda_2-\lambda_3+\mu_3-\mu_1 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_9-2\nu_8,\\
%\lambda_3-\lambda_1+\mu_2-\mu_3 &\le 2\nu_2+\nu_3+\nu_4-\nu_7-\nu_9-2\nu_8,\\
\lambda_2-\lambda_3+\mu_3-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_5-\nu_9-2\nu_7,\\
%\lambda_3-\lambda_1+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_5-\nu_9-2\nu_7,\\
\lambda_2-\lambda_3+\mu_3-\mu_1 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_7.
%\lambda_3-\lambda_1+\mu_2-\mu_3 &\le 2\nu_1+\nu_2+\nu_3-\nu_6-\nu_8-2\nu_7.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_3+\mu_1+\mu_2-2\mu_3 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_1+\lambda_2-2\lambda_3+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_3+\mu_1+\mu_3-2\mu_2 &\le
4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_3+\mu_1+\mu_3-2\mu_2 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_2-3\lambda_3+\mu_1+\mu_2-2\mu_3 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_1+\lambda_2-2\lambda_3+3\mu_2-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_3+\mu_1+\mu_3-2\mu_2 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_1-3\lambda_2+\mu_1+\mu_2-2\mu_3 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_1+\lambda_2-2\lambda_3+3\mu_1-3\mu_2 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_1-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_3 &\le
%4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_2-3\lambda_3+\mu_1+\mu_3-2\mu_2 &\le
4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_2-3\mu_3 &\le
%4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_2+\nu_3+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_2+\mu_1+\mu_3-2\mu_2 &\le
4\nu_1+4\nu_2+\nu_3+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_1-3\mu_2 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_2-3\lambda_3+\mu_1+\mu_3-2\mu_2 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_2-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_1-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_1-3\lambda_2+\mu_1+\mu_3-2\mu_2 &\le
4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_1+\lambda_3-2\lambda_2+3\mu_1-3\mu_2 &\le
%4\nu_1+4\nu_3+\nu_2+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_1-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-2\nu_9-5\nu_7,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-2\nu_9-5\nu_7,\\
3\lambda_1-3\lambda_2+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_3+\nu_2+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-5\nu_9,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_2 &\le
%4\nu_1+4\nu_3+\nu_2+\nu_5+\nu_6-2\nu_4-2\nu_7-2\nu_8-5\nu_9,\\
3\lambda_1-3\lambda_2+\mu_2+\mu_3-2\mu_1 &\le
4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_1-3\mu_2 &\le
%4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_2-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
%\lambda_2+\lambda_3-2\lambda_1+3\mu_2-3\mu_3 &\le
%4\nu_2+4\nu_3+\nu_1+\nu_4+\nu_5-2\nu_6-2\nu_7-2\nu_9-5\nu_8,\\
3\lambda_2-3\lambda_3+\mu_2+\mu_3-2\mu_1 &\le
4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-2\nu_9-5\nu_7.
%\lambda_2+\lambda_3-2\lambda_1+3\mu_2-3\mu_3 &\le
%4\nu_1+4\nu_2+\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-2\nu_9-5\nu_7.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
3\lambda_1-3\lambda_2+2\mu_2-\mu_1-\mu_3 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_1-3\mu_2 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
3\lambda_2-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_8-4\nu_7-4\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_2-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_8-4\nu_7-4\nu_9,\\
3\lambda_1-3\lambda_2+2\mu_3-\mu_1-\mu_2 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_2 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
3\lambda_2-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_2-3\mu_3 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_9-4\nu_7-4\nu_8,\\
3\lambda_1-3\lambda_3+2\mu_1-\mu_2-\mu_3 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_1-\lambda_2-\lambda_3+3\mu_1-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_1-3\mu_3 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_2-3\lambda_3+2\mu_1-\mu_2-\mu_3 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_1-\lambda_2-\lambda_3+3\mu_2-3\mu_3 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_1-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_2+2\mu_1-\mu_2-\mu_3 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_1-\lambda_2-\lambda_3+3\mu_1-3\mu_2 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_1-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_3+2\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_3 &\le
%5\nu_3+2\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_3 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_2+2\mu_2-\mu_1-\mu_3 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_1-3\mu_2 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_2-3\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_7-4\nu_8-4\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_2-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_5-\nu_7-4\nu_8-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_1+2\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_3 &\le
%5\nu_1+2\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
3\lambda_1-3\lambda_3+2\mu_3-\mu_1-\mu_2 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_3 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
3\lambda_2-3\lambda_3+2\mu_2-\mu_1-\mu_3 &\le
5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
%2\lambda_2-\lambda_1-\lambda_3+3\mu_2-3\mu_3 &\le
%5\nu_2+2\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-\nu_8-4\nu_7-4\nu_9,\\
3\lambda_1-3\lambda_2+2\mu_3-\mu_1-\mu_2 &\le
5\nu_3+2\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9.
%2\lambda_3-\lambda_1-\lambda_2+3\mu_1-3\mu_2 &\le
%5\nu_3+2\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_6-\nu_7-4\nu_8-4\nu_9.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_3+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_1-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_1-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_3 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_1-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_1-3\lambda_2+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_1-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_1-3\lambda_2+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_7-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_7-8\nu_9,\\
3\lambda_1-3\lambda_2+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
3\lambda_1-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_2+4\nu_1+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_1-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-5\nu_7-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-5\nu_7-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_8-5\nu_7-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_8-5\nu_7-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_7-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_3 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_7-8\nu_9,\\
3\lambda_2-3\lambda_3+4\mu_1+\mu_3-5\mu_2 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_1+\lambda_3-5\lambda_2+3\mu_2-3\mu_3 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_2+4\mu_1+\mu_2-5\mu_3  &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_1-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_9-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_7-5\nu_9-8\nu_8,\\
3\lambda_1-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
3\lambda_1-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_1-3\mu_2 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
3\lambda_2-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_2-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_3+4\nu_1+4\nu_4+\nu_2+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_3+4\nu_1+4\nu_4+\nu_2+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_3+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_2 &\le
%7\nu_2+4\nu_3+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_2+4\nu_3+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_3+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_2-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_4-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_4-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_4-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_4-2\nu_5-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_5+\nu_3+\nu_6-2\nu_4-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_5+\nu_3+\nu_6-2\nu_4-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_2 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_2+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_2 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_2-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
%3\mu_2-3\mu_1+4\lambda_2+\lambda_1-5\lambda_3 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_2-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_2-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_2-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_1+\mu_2-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_8-8\nu_7,\\
%4\lambda_1+\lambda_2-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_8-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_3+4\nu_2+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_3+4\nu_2+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_5-2\nu_4-2\nu_7-5\nu_8-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_5-2\nu_4-2\nu_7-5\nu_8-8\nu_9,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_4-2\nu_5-2\nu_8-5\nu_7-8\nu_9,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_6+\nu_3+\nu_4-2\nu_5-2\nu_8-5\nu_7-8\nu_9,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_1+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_7-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_7-8\nu_8,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_3+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_3+4\nu_4+\nu_1+\nu_5-2\nu_6-2\nu_7-5\nu_9-8\nu_8,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_6-2\nu_5-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_6-2\nu_5-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-5\nu_9-8\nu_7,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_8-5\nu_9-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_8-8\nu_7,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_3+\nu_4+\nu_6-2\nu_5-2\nu_9-5\nu_8-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_8-8\nu_7,\\
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_1+4\nu_2+4\nu_4+\nu_3+\nu_5-2\nu_6-2\nu_9-5\nu_8-8\nu_7,\\
3\lambda_3-3\lambda_1+4\mu_2+\mu_1-5\mu_3 &\le
7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_8-8\nu_7.
%4\lambda_2+\lambda_1-5\lambda_3+3\mu_3-3\mu_1 &\le
%7\nu_2+4\nu_1+4\nu_3+\nu_4+\nu_5-2\nu_6-2\nu_9-5\nu_8-8\nu_7.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_3+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_1-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_2-3\mu_3 &\le
%8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_3+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_3+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_2-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_2-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_2+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_2 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_2+5\mu_2-\mu_1-4\mu_3 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_2-\lambda_1-4\lambda_3+3\mu_1-3\mu_2 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_2-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_3+2\nu_2+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_3+2\nu_2+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_1+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_3 &\le
%8\nu_2+5\nu_1+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_3+2\nu_2+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_3+2\nu_2+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_3 &\le
%8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_9-7\nu_8,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_3 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_1-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_3+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_1-3\mu_2 &\le
%8\nu_1+5\nu_3+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_2-3\lambda_3+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_3 &\le
%8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_3+5\nu_2+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_2+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_2 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_2 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_7-4\nu_5-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_6-\nu_4-\nu_7-4\nu_5-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_6-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_2 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_6-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_6-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_5-\nu_6-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
3\lambda_2-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_7-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_7-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_7-7\nu_8,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_7-7\nu_8,\\
3\lambda_2-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_2-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_2+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_2 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_2-4\mu_3 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_8-4\nu_6-4\nu_9-7\nu_7,\\
%5\lambda_1-\lambda_2-4\lambda_3+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_8-4\nu_6-4\nu_9-7\nu_7,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_2+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_2+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_2+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_2+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_6-\nu_5-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_6-\nu_5-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_3+2\nu_2+2\nu_5-\nu_6-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_3+2\nu_2+2\nu_5-\nu_6-\nu_7-4\nu_4-4\nu_8-7\nu_9,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_7-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+ 3\mu_3-3\mu_1 &\le
%8\nu_2+5\nu_1+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_7-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+ 3\mu_3-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_4-\nu_5-\nu_7-4\nu_6-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_1+2\nu_2+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_2+5\nu_3+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_2+5\nu_3+2\nu_1+2\nu_5-\nu_4-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_3+5\nu_2+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_3+5\nu_2+2\nu_1+2\nu_4-\nu_5-\nu_6-4\nu_7-4\nu_9-7\nu_8,\\
3\lambda_3-3\lambda_1+5\mu_1-\mu_3-4\mu_2 &\le
8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_8-7\nu_7.
%5\lambda_1-\lambda_3-4\lambda_2+3\mu_3-3\mu_1 &\le
%8\nu_1+5\nu_2+2\nu_3+2\nu_4-\nu_5-\nu_9-4\nu_6-4\nu_8-7\nu_7.
\end{align*}
}

\subsection{Systems of format $2\times n$}
In this case all cubicles and extremal edges are explicitly known,
see Example \ref{2xn}.  The tables below give marginal
inequalities for $n=3,4$.
%two component
%systems $\mathcal{H}_{AB}=\mathcal{H}_A\otimes
%\mathcal{H}_B$ with $\dim\mathcal{H}_A=2$ and $\dim\mathcal{H}_B=3,4$.
For $n>4$
%$\dim\mathcal{H}_B>4$
the number of marginal inequalities increase  to thousands and
can't be published.
\subsubsection{Format $2\times3$}\label{2x3} In this case there are 41 independent
marginal inequalities. {\footnotesize
\begin{align*}
\mu_1-\mu_2 &\le \nu_1+\nu_2+\nu_3-\nu_4-\nu_5-\nu_6.
\displaybreak[0]\\[2mm]
\lambda_1+\lambda_2-2\lambda_3 &\le \nu_1+\nu_2+\nu_3+\nu_4-2\nu_5-2\nu_6,\\
\lambda_2+\lambda_3-2\lambda_1 &\le \nu_1+\nu_2+\nu_3+\nu_6-2\nu_4-2\nu_5.
\displaybreak[0]\\[2mm]
2\lambda_1-\lambda_2-\lambda_3 &\le 2\nu_1+2\nu_2-\nu_3-\nu_4-\nu_5-\nu_6,\\
2\lambda_3-\lambda_1-\lambda_2 &\le
2\nu_2+2\nu_3-\nu_1-\nu_4-\nu_5-\nu_6.
\displaybreak[0]\\[2mm]
2\lambda_1-2\lambda_3-\mu_2+\mu_1 &\le 3\nu_1+\nu_2+\nu_3-\nu_4-\nu_5-3\nu_6,\\
2\lambda_1-2\lambda_3+\mu_2-\mu_1 &\le 3\nu_2+\nu_1+\nu_3-\nu_4-\nu_5-3\nu_6,\\
2\lambda_2-2\lambda_3-\mu_2+\mu_1 &\le 3\nu_2+\nu_1+\nu_3-\nu_4-\nu_5-3\nu_6,\\
2\lambda_1-2\lambda_2+\mu_2-\mu_1 &\le 3\nu_2+\nu_1+\nu_4-\nu_3-\nu_5-3\nu_6,\\
2\lambda_1-2\lambda_3+\mu_2-\mu_1 &\le 3\nu_1+\nu_2+\nu_4-\nu_3-\nu_5-3\nu_6,\\
2\lambda_1-2\lambda_2-\mu_2+\mu_1 &\le 3\nu_1+\nu_2+\nu_4-\nu_3-\nu_5-3\nu_6,\\
2\lambda_2-2\lambda_3-\mu_2+\mu_1 &\le 3\nu_1+\nu_2+\nu_4-\nu_3-\nu_5-3\nu_6,\\
2\lambda_3-2\lambda_1-\mu_2+\mu_1 &\le 3\nu_3+\nu_1+\nu_4-\nu_2-\nu_5-3\nu_6,\\
2\lambda_3-2\lambda_1+\mu_2-\mu_1 &\le 3\nu_3+\nu_2+\nu_4-\nu_1-\nu_5-3\nu_6,\\
2\lambda_3-2\lambda_2+\mu_2-\mu_1 &\le 3\nu_2+\nu_3+\nu_4-\nu_1-\nu_5-3\nu_6,\\
2\lambda_3-2\lambda_1-\mu_2+\mu_1 &\le 3\nu_2+\nu_3+\nu_4-\nu_1-\nu_5-3\nu_6,\\
2\lambda_2-2\lambda_1+\mu_2-\mu_1 &\le 3\nu_1+\nu_2+\nu_6-\nu_4-\nu_3-3\nu_5,\\
2\lambda_3-2\lambda_1-\mu_2+\mu_1 &\le 3\nu_1+\nu_2+\nu_6-\nu_4-\nu_3-3\nu_5,\\
2\lambda_2-2\lambda_3+\mu_2-\mu_1 &\le 3\nu_1+\nu_2+\nu_4-\nu_3-\nu_6-3\nu_5,\\
2\lambda_1-2\lambda_2+\mu_2-\mu_1 &\le 3\nu_2+\nu_1+\nu_3-\nu_4-\nu_6-3\nu_5,\\
2\lambda_2-2\lambda_3+\mu_2-\mu_1 &\le 3\nu_2+\nu_1+\nu_3-\nu_4-\nu_6-3\nu_5,\\
2\lambda_1-2\lambda_3+\mu_2-\mu_1 &\le 3\nu_1+\nu_2+\nu_3-\nu_6-\nu_4-3\nu_5,\\
2\lambda_1-2\lambda_2-\mu_2+\mu_1 &\le 3\nu_1+\nu_2+\nu_3-\nu_6-\nu_4-3\nu_5,\\
2\lambda_3-2\lambda_1-\mu_2+\mu_1 &\le 3\nu_1+\nu_2+\nu_5-\nu_3-\nu_6-3\nu_4,\\
2\lambda_3-2\lambda_1+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_6-\nu_3-\nu_5-3\nu_4.
\displaybreak[0]\\[2mm]
2\lambda_1+2\lambda_2-4\lambda_3+3\mu_1-3\mu_2 &\le
5\nu_1+5\nu_2-\nu_3-\nu_4-\nu_5-7\nu_6,\\
2\lambda_3+2\lambda_1-4\lambda_2+3\mu_1-3\mu_2 &\le
5\nu_3+5\nu_1-\nu_2-\nu_4-\nu_5-7\nu_6,\\
2\lambda_1+2\lambda_3-4\lambda_2+3\mu_2-3\mu_1 &\le
5\nu_2+5\nu_3-\nu_1-\nu_4-\nu_5-7\nu_6,\\
2\lambda_2+2\lambda_3-4\lambda_1+3\mu_1-3\mu_2 &\le
5\nu_2+5\nu_3-\nu_1-\nu_4-\nu_5-7\nu_6,\\
2\lambda_2+2\lambda_1-4\lambda_3+3\mu_2-3\mu_1 &\le
5\nu_1+5\nu_2-\nu_3-\nu_6-\nu_4-7\nu_5,\\
2\lambda_3+2\lambda_1-4\lambda_2+3\mu_1-3\mu_2 &\le
5\nu_1+5\nu_2-\nu_3-\nu_6-\nu_4-7\nu_5,\\
2\lambda_2+2\lambda_3-4\lambda_1+3\mu_1-3\mu_2 &\le
5\nu_1+5\nu_3-\nu_2-\nu_6-\nu_4-7\nu_5,\\
2\lambda_2+2\lambda_3-4\lambda_1+3\mu_1-3\mu_2 &\le
5\nu_1+5\nu_2-\nu_3-\nu_5-\nu_6-7\nu_4.
\displaybreak[0]\\[2mm]
4\lambda_1-2\lambda_2-2\lambda_3+3\mu_1-3\mu_2 &\le
7\nu_1+\nu_2+\nu_3+\nu_4-5\nu_5-5\nu_6,\\
4\lambda_1-2\lambda_2-2\lambda_3+3\mu_2-3\mu_1 &\le
7\nu_2+\nu_1+\nu_3+\nu_4-5\nu_5-5\nu_6,\\
4\lambda_2-2\lambda_1-2\lambda_3+3\mu_1-3\mu_2 &\le
7\nu_2+\nu_1+\nu_3+\nu_4-5\nu_5-5\nu_6,\\
4\lambda_2-2\lambda_1-2\lambda_3+3\mu_1-3\mu_2 &\le
7\nu_1+\nu_2+\nu_3+\nu_5-5\nu_4-5\nu_6,\\
4\lambda_3-2\lambda_1-2\lambda_2+3\mu_1-3\mu_2 &\le
7\nu_3+\nu_1+\nu_2+\nu_4-5\nu_5-5\nu_6,\\
4\lambda_3-2\lambda_1-2\lambda_2+3\mu_1-3\mu_2 &\le
7\nu_2+\nu_1+\nu_3+\nu_5-5\nu_4-5\nu_6,\\
4\lambda_2-2\lambda_1-2\lambda_3+3\mu_2-3\mu_1 &\le
7\nu_1+\nu_2+\nu_3+\nu_6-5\nu_4-5\nu_5,\\
4\lambda_3-2\lambda_1-2\lambda_2+3\mu_1-3\mu_2 &\le
7\nu_1+\nu_2+\nu_3+\nu_6-5\nu_4-5\nu_5.
\end{align*}}

\subsubsection{Format $2\times4$}\label{2x4}
Marginal constraints in this case are given by the following 234
independent inequalities. Note that three qubit case can be
reduced to this one.
%marginal inequalities for three qubits can be
%deduced from this system and Bravyi inequalities for two qubits.
{\footnotesize
\begin{align*}
\mu_1-\mu_2 &\le \nu_1+\nu_2+\nu_3+\nu_4-\nu_5-\nu_6-\nu_7-\nu_8.
\\[2mm]
\lambda_1+\lambda_2-\lambda_3-\lambda_4 &\le \nu_1+\nu_2+\nu_3+\nu_4-\nu_5-\nu_6-\nu_7-\nu_8,\\
\lambda_1+\lambda_4-\lambda_2-\lambda_3 &\le \nu_1+\nu_2+\nu_4+\nu_5-\nu_3-\nu_6-\nu_7-\nu_8,\\
\lambda_2+\lambda_3-\lambda_1-\lambda_4 &\le \nu_1+\nu_2+\nu_3+\nu_6-\nu_4-\nu_5-\nu_7-\nu_8,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2 &\le \nu_2+\nu_3+\nu_4+\nu_5-\nu_1-\nu_6-\nu_7-\nu_8,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2 &\le
\nu_1+\nu_2+\nu_3+\nu_8-\nu_4-\nu_5-\nu_6-\nu_7.
\displaybreak[0]\\[2mm]
\lambda_1+\lambda_2+\lambda_3-3\lambda_4 &\le
\nu_1+\nu_2+\nu_3+\nu_4+\nu_5+\nu_6-3\nu_7-3\nu_8,\\
\lambda_1+\lambda_3+\lambda_4-3\lambda_2 &\le
\nu_1+\nu_2+\nu_3+\nu_4+\nu_5+\nu_8-3\nu_6-3\nu_7.
\displaybreak[0]\\[2mm]
3\lambda_1-\lambda_2-\lambda_3-\lambda_4 &\le
3\nu_1+3\nu_2-\nu_3-\nu_4-\nu_5-\nu_6-\nu_7-\nu_8,\\
3\lambda_3-\lambda_1-\lambda_2-\lambda_4 &\le
3\nu_2+3\nu_3-\nu_1-\nu_4-\nu_5-\nu_6-\nu_7-\nu_8.
\displaybreak[0]\\[2mm]
\lambda_1+\lambda_2+\lambda_3-3\lambda_4+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_6-\nu_7-5\nu_8,\\
\lambda_1+\lambda_2+\lambda_4-3\lambda_3+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_6-\nu_7-5\nu_8,\\
\lambda_1+\lambda_2+\lambda_3-3\lambda_4+2\mu_2-2\mu_1 &\le
3\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_6-\nu_8-5\nu_7,\\
\lambda_1+\lambda_2+\lambda_4-3\lambda_3+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_6-\nu_8-5\nu_7,\\
\lambda_1+\lambda_3+\lambda_4-3\lambda_2+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_3+3\nu_4-\nu_2-\nu_5-\nu_6-\nu_7-5\nu_8,\\
\lambda_1+\lambda_3+\lambda_4-3\lambda_2+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_6-\nu_8-5\nu_7,\\
\lambda_1+\lambda_3+\lambda_4-3\lambda_2+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_7-\nu_8-5\nu_6,\\
\lambda_1+\lambda_3+\lambda_4-3\lambda_2+2\mu_2-2\mu_1 &\le
3\nu_2+3\nu_3+3\nu_4-\nu_1-\nu_5-\nu_6-\nu_7-5\nu_8,\\
\lambda_2+\lambda_3+\lambda_4-3\lambda_1+2\mu_1-2\mu_2 &\le
3\nu_2+3\nu_3+3\nu_4-\nu_1-\nu_5-\nu_6-\nu_7-5\nu_8,\\
\lambda_2+\lambda_3+\lambda_4-3\lambda_1+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_3+3\nu_4-\nu_2-\nu_5-\nu_6-\nu_8-5\nu_7,\\
\lambda_2+\lambda_3+\lambda_4-3\lambda_1+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_7-\nu_8-5\nu_6,\\
\lambda_2+\lambda_3+\lambda_4-3\lambda_1+2\mu_1-2\mu_2 &\le
3\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_6-\nu_7-\nu_8-5\nu_5.
\displaybreak[0]\\[2mm]
3\lambda_1-\lambda_2-\lambda_3-\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+\nu_2+\nu_3+\nu_4+\nu_5-3\nu_6-3\nu_7-3\nu_8,\\
3\lambda_1-\lambda_2-\lambda_3-\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_2+\nu_1+\nu_3+\nu_4+\nu_5-3\nu_6-3\nu_7-3\nu_8,\\
3\lambda_2-\lambda_1-\lambda_3-\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_2+\nu_1+\nu_3+\nu_4+\nu_5-3\nu_6-3\nu_7-3\nu_8,\\
3\lambda_2-\lambda_1-\lambda_3-\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+\nu_2+\nu_3+\nu_4+\nu_6-3\nu_5-3\nu_7-3\nu_8,\\
3\lambda_3-\lambda_1-\lambda_2-\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_3+\nu_1+\nu_2+\nu_4+\nu_5-3\nu_6-3\nu_7-3\nu_8,\\
3\lambda_3-\lambda_1-\lambda_2-\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_2+\nu_1+\nu_3+\nu_4+\nu_6-3\nu_5-3\nu_7-3\nu_8,\\
3\lambda_3-\lambda_1-\lambda_2-\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+\nu_2+\nu_3+\nu_4+\nu_7-3\nu_5-3\nu_6-3\nu_8,\\
3\lambda_4-\lambda_1-\lambda_2-\lambda_3+2\mu_1-2\mu_2 &\le
5\nu_4+\nu_1+\nu_2+\nu_3+\nu_5-3\nu_6-3\nu_7-3\nu_8,\\
3\lambda_4-\lambda_1-\lambda_2-\lambda_3+2\mu_1-2\mu_2 &\le
5\nu_3+\nu_1+\nu_2+\nu_4+\nu_6-3\nu_5-3\nu_7-3\nu_8,\\
3\lambda_4-\lambda_1-\lambda_2-\lambda_3+2\mu_1-2\mu_2 &\le
5\nu_2+\nu_1+\nu_3+\nu_4+\nu_7-3\nu_5-3\nu_6-3\nu_8,\\
3\lambda_3-\lambda_1-\lambda_2-\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+\nu_2+\nu_3+\nu_4+\nu_8-3\nu_5-3\nu_6-3\nu_7,\\
3\lambda_4-\lambda_1-\lambda_2-\lambda_3+2\mu_1-2\mu_2 &\le
5\nu_1+\nu_2+\nu_3+\nu_4+\nu_8-3\nu_5-3\nu_6-3\nu_7.
\displaybreak[0]\\[2mm]
\lambda_1+\lambda_2-\lambda_3-\lambda_4+\mu_1-\mu_2 &\le 2\nu_1+2\nu_2-2\nu_7-2\nu_8,\\
\lambda_1+\lambda_3-\lambda_2-\lambda_4+\mu_1-\mu_2 &\le 2\nu_1+2\nu_3-2\nu_7-2\nu_8,\\
\lambda_1+\lambda_3-\lambda_2-\lambda_4+\mu_1-\mu_2 &\le 2\nu_1+2\nu_2-2\nu_6-2\nu_8,\\
\lambda_1+\lambda_3-\lambda_2-\lambda_4+\mu_2-\mu_1 &\le 2\nu_2+2\nu_3-2\nu_7-2\nu_8,\\
\lambda_2+\lambda_3-\lambda_1-\lambda_4+\mu_1-\mu_2 &\le 2\nu_2+2\nu_3-2\nu_7-2\nu_8,\\
\lambda_1+\lambda_4-\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_4-2\nu_7-2\nu_8,\\
\lambda_1+\lambda_4-\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_3-2\nu_6-2\nu_8,\\
\lambda_2+\lambda_3-\lambda_1-\lambda_4+\mu_1-\mu_2 &\le 2\nu_1+2\nu_3-2\nu_6-2\nu_8,\\
\lambda_2+\lambda_3-\lambda_1-\lambda_4+\mu_1-\mu_2 &\le 2\nu_1+2\nu_2-2\nu_5-2\nu_8,\\
\lambda_1+\lambda_3-\lambda_2-\lambda_4+\mu_2-\mu_1 &\le 2\nu_1+2\nu_2-2\nu_6-2\nu_7,\\
\lambda_1+\lambda_4-\lambda_2-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_2-2\nu_6-2\nu_7,\\
\lambda_1+\lambda_4-\lambda_2-\lambda_3+\mu_2-\mu_1 &\le 2\nu_2+2\nu_4-2\nu_7-2\nu_8,\\
\lambda_2+\lambda_4-\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_2+2\nu_4-2\nu_7-2\nu_8,\\
\lambda_2+\lambda_4-\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_4-2\nu_6-2\nu_8,\\
\lambda_1+\lambda_4-\lambda_2-\lambda_3+\mu_2-\mu_1 &\le 2\nu_2+2\nu_3-2\nu_6-2\nu_8,\\
\lambda_2+\lambda_4-\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_2+2\nu_3-2\nu_6-2\nu_8,\\
\lambda_2+\lambda_4-\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_3-2\nu_5-2\nu_8,\\
\lambda_2+\lambda_3-\lambda_1-\lambda_4+\mu_2-\mu_1 &\le 2\nu_1+2\nu_2-2\nu_5-2\nu_7,\\
\lambda_2+\lambda_4-\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_2-2\nu_5-2\nu_7,\\
\lambda_2+\lambda_3-\lambda_1-\lambda_4+\mu_2-\mu_1 &\le 2\nu_1+2\nu_3-2\nu_6-2\nu_7,\\
\lambda_2+\lambda_4-\lambda_1-\lambda_3+\mu_1-\mu_2 &\le 2\nu_1+2\nu_3-2\nu_6-2\nu_7,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_3+2\nu_4-2\nu_7-2\nu_8,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_2+2\nu_4-2\nu_6-2\nu_8,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_2+2\nu_3-2\nu_5-2\nu_8,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_1+2\nu_3-2\nu_5-2\nu_7,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_1+2\nu_4-2\nu_6-2\nu_7,\\
\lambda_3+\lambda_4-\lambda_1-\lambda_2+\mu_1-\mu_2 &\le 2\nu_1+2\nu_2-2\nu_5-2\nu_6.
\displaybreak[0]\\[2mm]
5\lambda_1+\lambda_2-3\lambda_3-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_2-3\lambda_3-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_1+3\nu_3-\nu_4-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_2+\lambda_1-3\lambda_3-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_2+3\nu_1+3\nu_3-\nu_4-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_2-3\lambda_3-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_3-3\lambda_2-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_2+\lambda_1-3\lambda_3-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_3-3\lambda_2-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_7-5\nu_6-5\nu_8,\\
5\lambda_1+\lambda_3-3\lambda_2-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_1+3\nu_4-\nu_3-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_4-3\lambda_2-3\lambda_3+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_5-\nu_3-\nu_4-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_4-3\lambda_2-3\lambda_3+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_4-\nu_3-\nu_5-\nu_7-5\nu_6-5\nu_8,\\
5\lambda_1+\lambda_3-3\lambda_2-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_1+3\nu_3-\nu_4-\nu_5-\nu_7-5\nu_6-5\nu_8,\\
5\lambda_1+\lambda_3-3\lambda_2-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_8-5\nu_6-5\nu_7,\\
5\lambda_1+\lambda_4-3\lambda_2-3\lambda_3+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_3-\nu_4-\nu_5-\nu_8-5\nu_6-5\nu_7,\\
5\lambda_3+\lambda_2-3\lambda_1-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_3+3\nu_1+3\nu_4-\nu_2-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_3+\lambda_1-3\lambda_2-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_3+3\nu_4-\nu_1-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_3+\lambda_2-3\lambda_1-3\lambda_4+2\mu_1-2\mu_2 &\le
7\nu_2+3\nu_3+3\nu_4-\nu_1-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_4-3\lambda_2-3\lambda_3+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_1+3\nu_5-\nu_3-\nu_4-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_1+\lambda_4-3\lambda_2-3\lambda_3+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_1+3\nu_4-\nu_3-\nu_5-\nu_7-5\nu_6-5\nu_8,\\
5\lambda_1+\lambda_4-3\lambda_2-3\lambda_3+2\mu_2-2\mu_1 &\le
7\nu_2+3\nu_1+3\nu_3-\nu_4-\nu_5-\nu_8-5\nu_6-5\nu_7,\\
5\lambda_3+\lambda_2-3\lambda_1-3\lambda_4+2\mu_2-2\mu_1 &\le
7\nu_3+3\nu_2+3\nu_4-\nu_1-\nu_5-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_4+\lambda_3-3\lambda_1-3\lambda_2+2\mu_1-2\mu_2 &\le
7\nu_4+3\nu_1+3\nu_5-\nu_2-\nu_3-\nu_6-5\nu_7-5\nu_8,\\
5\lambda_3+\lambda_4-3\lambda_1-3\lambda_2+2\mu_2-2\mu_1 &\le
7\nu_1+3\nu_2+3\nu_8-\nu_3-\nu_4-\nu_5-5\nu_6-5\nu_7,\\
5\lambda_4+\lambda_3-3\lambda_1-3\lambda_2+2\mu_1-2\mu_2 &\le
7\nu_1+3\nu_2+3\nu_8-\nu_3-\nu_4-\nu_5-5\nu_6-5\nu_7.
\displaybreak[0]\\[2mm]
3\lambda_1+3\lambda_2-\lambda_3-5\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_5-3\nu_6-3\nu_7-7\nu_8,\\
3\lambda_1+3\lambda_3-\lambda_2-5\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_3+\nu_2+\nu_4+\nu_5-3\nu_6-3\nu_7-7\nu_8,\\
3\lambda_1+3\lambda_2-\lambda_3-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_6-3\nu_5-3\nu_7-7\nu_8,\\
3\lambda_1+3\lambda_2-\lambda_4-5\lambda_3+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_6-3\nu_5-3\nu_7-7\nu_8,\\
3\lambda_1+3\lambda_3-\lambda_2-5\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_6-3\nu_5-3\nu_7-7\nu_8,\\
3\lambda_1+3\lambda_2-\lambda_3-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_5-3\nu_6-3\nu_8-7\nu_7,\\
3\lambda_1+3\lambda_2-\lambda_4-5\lambda_3+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_5-3\nu_6-3\nu_8-7\nu_7,\\
3\lambda_1+3\lambda_3-\lambda_2-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_2+5\nu_3+\nu_1+\nu_4+\nu_5-3\nu_6-3\nu_7-7\nu_8,\\
3\lambda_2+3\lambda_3-\lambda_1-5\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_2+5\nu_3+\nu_1+\nu_4+\nu_5-3\nu_6-3\nu_7-7\nu_8,\\
3\lambda_2+3\lambda_3-\lambda_1-5\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_3+\nu_2+\nu_4+\nu_6-3\nu_5-3\nu_7-7\nu_8,\\
3\lambda_2+3\lambda_3-\lambda_1-5\lambda_4+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_5+\nu_6-3\nu_4-3\nu_7-7\nu_8,\\
3\lambda_1+3\lambda_3-\lambda_2-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_6-3\nu_5-3\nu_8-7\nu_7,\\
3\lambda_1+3\lambda_3-\lambda_2-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_3+\nu_2+\nu_4+\nu_5-3\nu_6-3\nu_8-7\nu_7,\\
3\lambda_1+3\lambda_3-\lambda_4-5\lambda_2+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_8-3\nu_5-3\nu_6-7\nu_7,\\
3\lambda_1+3\lambda_4-\lambda_3-5\lambda_2+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_8-3\nu_5-3\nu_6-7\nu_7,\\
3\lambda_2+3\lambda_3-\lambda_1-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_2+\nu_3+\nu_5+\nu_6-3\nu_4-3\nu_8-7\nu_7,\\
3\lambda_2+3\lambda_3-\lambda_1-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_3+\nu_2+\nu_4+\nu_6-3\nu_5-3\nu_8-7\nu_7,\\
3\lambda_2+3\lambda_3-\lambda_1-5\lambda_4+2\mu_2-2\mu_1 &\le
5\nu_2+5\nu_3+\nu_1+\nu_4+\nu_5-3\nu_6-3\nu_8-7\nu_7,\\
3\lambda_1+3\lambda_4-\lambda_3-5\lambda_2+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_7-3\nu_5-3\nu_8-7\nu_6,\\
3\lambda_1+3\lambda_4-\lambda_3-5\lambda_2+2\mu_2-2\mu_1 &\le
5\nu_1+5\nu_2+\nu_3+\nu_4+\nu_8-3\nu_5-3\nu_7-7\nu_6,\\
3\lambda_3+3\lambda_4-\lambda_1-5\lambda_2+2\mu_2-2\mu_1 &\le
5\nu_2+5\nu_3+\nu_4+\nu_5+\nu_6-3\nu_1-3\nu_7-7\nu_8,\\
3\lambda_3+3\lambda_4-\lambda_2-5\lambda_1+2\mu_1-2\mu_2 &\le
5\nu_2+5\nu_3+\nu_4+\nu_5+\nu_6-3\nu_1-3\nu_7-7\nu_8,\\
3\lambda_3+3\lambda_4-\lambda_2-5\lambda_1+2\mu_1-2\mu_2 &\le
5\nu_1+5\nu_2+\nu_3+\nu_6+\nu_7-3\nu_4-3\nu_8-7\nu_5.
\displaybreak[0]\\[2mm]
2\lambda_1-2\lambda_4+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_4-\nu_5-\nu_6-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_4-\nu_5-\nu_6-\nu_7-3\nu_8,\\
2\lambda_2-2\lambda_4+\mu_1-\mu_2 &\le
3\nu_2+\nu_1+\nu_3+\nu_4-\nu_5-\nu_6-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_5-\nu_4-\nu_6-\nu_7-3\nu_8,\\
2\lambda_2-2\lambda_4+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_5-\nu_4-\nu_6-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_4-\nu_5-\nu_6-\nu_8-3\nu_7,\\
2\lambda_1-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_4-\nu_5-\nu_6-\nu_8-3\nu_7,\\
2\lambda_3-2\lambda_4+\mu_1-\mu_2 &\le
3\nu_3+\nu_1+\nu_2+\nu_4-\nu_5-\nu_6-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_3+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_5-\nu_4-\nu_6-\nu_7-3\nu_8,\\
2\lambda_2-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_2+\nu_1+\nu_3+\nu_5-\nu_4-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_4+\mu_1-\mu_2 &\le
3\nu_2+\nu_1+\nu_3+\nu_5-\nu_4-\nu_6-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_3+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_4+\nu_5-\nu_3-\nu_6-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_4+\nu_5-\nu_3-\nu_6-\nu_7-3\nu_8,\\
2\lambda_2-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_4+\nu_5-\nu_3-\nu_6-\nu_7-3\nu_8,\\
2\lambda_2-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_6-\nu_4-\nu_5-\nu_7-3\nu_8,\\
2\lambda_2-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_6-\nu_4-\nu_5-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_4+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_6-\nu_4-\nu_5-\nu_7-3\nu_8,\\
2\lambda_1-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_5-\nu_4-\nu_6-\nu_8-3\nu_7,\\
2\lambda_2-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_5-\nu_4-\nu_6-\nu_8-3\nu_7,\\
2\lambda_2-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_5-\nu_4-\nu_6-\nu_8-3\nu_7,\\
2\lambda_1-2\lambda_3+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_4-\nu_5-\nu_6-\nu_8-3\nu_7,\\
2\lambda_2-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_4-\nu_5-\nu_6-\nu_8-3\nu_7,\\
2\lambda_2-2\lambda_3+\mu_1-\mu_2 &\le
3\nu_2+\nu_1+\nu_3+\nu_4-\nu_5-\nu_6-\nu_8-3\nu_7,\\
2\lambda_1-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_4-\nu_5-\nu_7-\nu_8-3\nu_6,\\
2\lambda_1-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_4+\nu_5-\nu_3-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_6-\nu_4-\nu_5-\nu_8-3\nu_7,\\
2\lambda_1-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_5-\nu_4-\nu_6-\nu_8-3\nu_7,\\
2\lambda_3-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_5-\nu_4-\nu_6-\nu_8-3\nu_7,\\
2\lambda_3-2\lambda_4+\mu_2-\mu_1 &\le
3\nu_3+\nu_1+\nu_2+\nu_4-\nu_5-\nu_6-\nu_8-3\nu_7,\\
\displaybreak[0]\\[2mm]
2\lambda_1-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_2+\nu_1+\nu_3+\nu_4-\nu_5-\nu_7-\nu_8-3\nu_6,\\
2\lambda_4-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_4+\nu_1+\nu_2+\nu_5-\nu_3-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_3+\nu_1+\nu_4+\nu_5-\nu_2-\nu_6-\nu_7-3\nu_8,\\
2\lambda_4-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_3+\nu_1+\nu_4+\nu_5-\nu_2-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_2+\nu_3+\nu_4+\nu_5-\nu_1-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_2+\nu_3+\nu_4+\nu_5-\nu_1-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_8-\nu_4-\nu_5-\nu_6-3\nu_7,\\
2\lambda_4-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_8-\nu_4-\nu_5-\nu_6-3\nu_7,\\
2\lambda_3-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_7-\nu_4-\nu_5-\nu_8-3\nu_6,\\
2\lambda_4-2\lambda_2+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_7-\nu_4-\nu_5-\nu_8-3\nu_6,\\
2\lambda_3-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_6-\nu_4-\nu_7-\nu_8-3\nu_5,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_4+\nu_1+\nu_3+\nu_5-\nu_2-\nu_6-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_1+\mu_2-\mu_1 &\le
3\nu_3+\nu_2+\nu_4+\nu_5-\nu_1-\nu_6-\nu_7-3\nu_8,\\
2\lambda_4-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_3+\nu_2+\nu_4+\nu_5-\nu_1-\nu_6-\nu_7-3\nu_8,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_3+\nu_2+\nu_4+\nu_5-\nu_1-\nu_6-\nu_7-3\nu_8,\\
2\lambda_4-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_2+\nu_3+\nu_4+\nu_6-\nu_1-\nu_5-\nu_7-3\nu_8,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_2+\nu_3+\nu_4+\nu_6-\nu_1-\nu_5-\nu_7-3\nu_8,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_3+\nu_1+\nu_4+\nu_6-\nu_2-\nu_5-\nu_7-3\nu_8,\\
2\lambda_3-2\lambda_1+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_4+\nu_8-\nu_3-\nu_5-\nu_6-3\nu_7,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_4+\nu_8-\nu_3-\nu_5-\nu_6-3\nu_7,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_3+\nu_1+\nu_4+\nu_5-\nu_2-\nu_6-\nu_8-3\nu_7,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_4+\nu_1+\nu_2+\nu_5-\nu_3-\nu_6-\nu_8-3\nu_7,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_4+\nu_7-\nu_3-\nu_5-\nu_8-3\nu_6,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_2+\nu_1+\nu_3+\nu_7-\nu_4-\nu_5-\nu_8-3\nu_6,\\
2\lambda_3-2\lambda_1+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_8-\nu_4-\nu_5-\nu_7-3\nu_6,\\
2\lambda_4-2\lambda_2+\mu_2-\mu_1 &\le
3\nu_1+\nu_2+\nu_3+\nu_8-\nu_4-\nu_5-\nu_7-3\nu_6,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_8-\nu_4-\nu_5-\nu_7-3\nu_6,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_1+\nu_2+\nu_3+\nu_7-\nu_4-\nu_6-\nu_8-3\nu_5,\\
2\lambda_4-2\lambda_1+\mu_1-\mu_2 &\le
3\nu_2+\nu_1+\nu_3+\nu_6-\nu_4-\nu_7-\nu_8-3\nu_5.
\displaybreak[0]\\[2mm]
3\lambda_1+\lambda_2-\lambda_3-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_1-\lambda_3-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_3-\lambda_2-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_1-\lambda_3-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_4-3\lambda_3+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_3-\lambda_2-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_1+\lambda_2-\lambda_4-3\lambda_3+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_1+\lambda_3-\lambda_2-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_4-3\lambda_3+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_1-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_1-\lambda_4-3\lambda_3+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_4-3\lambda_3+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_3-\lambda_2-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_1-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_1-\lambda_4-3\lambda_3+\mu_1-\mu_2 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_2-\lambda_4-3\lambda_3+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_1+\lambda_3-\lambda_2-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_8-4\nu_7,\\
3\lambda_1+\lambda_3-\lambda_2-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_2+\lambda_1-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_2+\lambda_1-\lambda_4-3\lambda_3+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_1+\lambda_2-\lambda_4-3\lambda_3+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_2+\lambda_1-\lambda_3-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_2+\lambda_1-\lambda_4-3\lambda_3+\mu_1-\mu_2 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_3+2\nu_1+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_1-\lambda_2-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_3+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_2+2\nu_3+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_2+\lambda_3-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_6-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_6-2\nu_5-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_5-2\nu_4-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_5-2\nu_4-2\nu_7-4\nu_8,\\
\displaybreak[0]\\[2mm]
3\lambda_1+\lambda_4-\lambda_2-3\lambda_3+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_3-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_4-2\nu_3-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_3-\lambda_4-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_6-4\nu_7,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_6-4\nu_7,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_8-4\nu_6,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_3+2\nu_2+2\nu_4-2\nu_6-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_5-2\nu_4-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_6-2\nu_4-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_5-2\nu_3-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_4-2\nu_3-2\nu_7-4\nu_8,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_5-2\nu_6-4\nu_7,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_5-2\nu_4-2\nu_8-4\nu_7,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_6-2\nu_5-2\nu_8-4\nu_7,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_2+2\nu_3+2\nu_4-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_3+\lambda_2-\lambda_1-3\lambda_4+\mu_2-\mu_1 &\le
4\nu_3+2\nu_1+2\nu_4-2\nu_6-2\nu_8-4\nu_7,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_2+2\nu_1+2\nu_3-2\nu_5-2\nu_8-4\nu_6,\\
3\lambda_1+\lambda_4-\lambda_3-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_3-2\nu_5-2\nu_7-4\nu_6,\\
3\lambda_3+\lambda_4-\lambda_1-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_2+2\nu_3+2\nu_4-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_4-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_2+2\nu_3+2\nu_4-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_4-\lambda_1-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_5-2\nu_6-4\nu_7,\\
3\lambda_4+\lambda_3-\lambda_1-3\lambda_2+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_5-2\nu_6-4\nu_7,\\
3\lambda_4+\lambda_3-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_4+2\nu_1+2\nu_5-2\nu_3-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_4-\lambda_2-3\lambda_1+\mu_2-\mu_1 &\le
4\nu_2+2\nu_3+2\nu_5-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_4+\lambda_3-\lambda_1-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_2+2\nu_3+2\nu_5-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_4+\lambda_3-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_2+2\nu_3+2\nu_5-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_4-\lambda_2-3\lambda_1+\mu_2-\mu_1 &\le
4\nu_3+2\nu_2+2\nu_4-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_4+\lambda_3-\lambda_1-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_3+2\nu_2+2\nu_4-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_4+\lambda_3-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_3+2\nu_2+2\nu_4-2\nu_1-2\nu_7-4\nu_8,\\
3\lambda_3+\lambda_4-\lambda_2-3\lambda_1+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_4-2\nu_6-4\nu_7,\\
3\lambda_4+\lambda_3-\lambda_1-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_4-2\nu_6-4\nu_7,\\
3\lambda_4+\lambda_3-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_4-2\nu_6-4\nu_7,\\
3\lambda_3+\lambda_4-\lambda_2-3\lambda_1+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_5-2\nu_7-4\nu_6,\\
3\lambda_4+\lambda_3-\lambda_1-3\lambda_2+\mu_2-\mu_1 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_5-2\nu_7-4\nu_6,\\
3\lambda_4+\lambda_3-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_8-2\nu_5-2\nu_7-4\nu_6,\\
3\lambda_4+\lambda_3-\lambda_2-3\lambda_1+\mu_1-\mu_2 &\le
4\nu_1+2\nu_2+2\nu_6-2\nu_4-2\nu_8-4\nu_5.
\end{align*}}

\subsection{System of format $2\times2\times3$}\label{2x2x3}
In contrast with other systems of rank $\le4$ in this case there
are $9$ {\it redundant extremal edges\,} for which all the
associated inequalities are redundant.
%In this case there are $39$
%extremal edges but $9$ of them are redundant, i.e. all the
%corresponding marginal inequalities are redundant.
%In particular there are $9$ redundant basic inequalities (
For all the other systems basic inequalities are essential and
independent.

As usual we skip inequalities obtained from another one by a
permutation of qubits. This reduces the number of marginal
inequalities to $232$ (instead of $442$).
%\marginpar{\footnotesize
%I've to switch $\lambda \leftrightarrow\mu$ in the last two groups
%for this to hold.}
Under additional constraint $\lambda_1-\lambda_2\ge
\mu_1-\mu_2$ they form a complete and independent system. {\footnotesize
\begin{align*}
\lambda_1-\lambda_2 &\le
\rho_1+\rho_2+\rho_3+\rho_4+\rho_5+\rho_6-\rho_7-\rho_8-\rho_9-\rho_{10}-\rho_{11}-\rho_{12}.
\displaybreak[0]\\[2mm]
2\nu_1-\nu_2-\nu_3 &\le
2\rho_1+2\rho_2+2\rho_3+2\rho_4-\rho_5-\rho_6-\rho_7-\rho_8-\rho_9-\rho_{10}-\rho_{11}-\rho_{12}.
\displaybreak[0]\\[2mm]
\nu_1+\nu_2-2\nu_3 &\le
\rho_1+\rho_2+\rho_3+\rho_4+\rho_5+\rho_6+\rho_7+\rho_8-2\rho_9-2\rho_{10}-2\rho_{11}-2\rho_{12}.
\displaybreak[0]\\[2mm]
\lambda_1-\lambda_2+\mu_1-\mu_2 &\le
2\rho_1+2\rho_2+2\rho_3-2\rho_{10}-2\rho_{11}-2\rho_{12}.
\displaybreak[0]\\[2mm]
\lambda_1-\lambda_2+\nu_1-\nu_3 &\le
2\rho_1+2\rho_2+\rho_3+\rho_4-\rho_9-\rho_{10}-2\rho_{11}-2\rho_{12}.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_2+2\nu_1-\nu_2-\nu_3 &\le
5\rho_1+5\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6-\rho_7-\rho_8-4\rho_9-4\rho_{10}-4\rho_{11}-4\rho_{12},\\
3\lambda_1-3\lambda_2+2\nu_3-\nu_1-\nu_2 &\le
5\rho_2+5\rho_3+2\rho_1+2\rho_4+2\rho_5+2\rho_6-\rho_7-\rho_8-4\rho_9-4\rho_{10}-4\rho_{11}-4\rho_{12},\\
3\lambda_1-3\lambda_2+2\nu_3-\nu_1-\nu_2 &\le
5\rho_1+5\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6-\rho_8-\rho_9-4\rho_7-4\rho_{10}-4\rho_{11}-4\rho_{12}.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_2+\nu_1+\nu_2-2\nu_3 &\le
4\rho_1+4\rho_2+4\rho_3+4\rho_4+\rho_5+\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-5\rho_{11}-5\rho_{12},\\
3\lambda_1-3\lambda_2+\nu_2+\nu_3-2\nu_1 &\le
4\rho_1+4\rho_2+4\rho_3+4\rho_6+\rho_4+\rho_5-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-5\rho_{11}-5\rho_{12},\\
3\lambda_1-3\lambda_2+\nu_2+\nu_3-2\nu_1 &\le
4\rho_1+4\rho_2+4\rho_3+4\rho_4+\rho_5+\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{12}-5\rho_{10}-5\rho_{11}.
%\displaybreak[0]\\[2mm]
\end{align*}
\begin{align*}
\lambda_1-\lambda_2+\mu_1-\mu_2+2\nu_1-2\nu_3 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_1-\mu_2+2\nu_2-2\nu_3 &\le
4\rho_2+2\rho_1+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_1-2\nu_3 &\le
4\rho_2+2\rho_1+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_1-\mu_2+2\nu_1-2\nu_2 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_5-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_1-\mu_2+2\nu_2-2\nu_3 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_8-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_1-\mu_2+2\nu_1-2\nu_2 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_1-2\nu_3 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_2-2\nu_3 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_8-2\rho_{10}-2\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_1-2\nu_2 &\le
4\rho_2+2\rho_1+2\rho_3+2\rho_5-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_1-2\nu_2 &\le
4\rho_2+2\rho_1+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_2-2\nu_3 &\le
4\rho_2+2\rho_1+2\rho_3+2\rho_4-2\rho_9-2\rho_{10}-2\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_1-2\nu_2 &\le
4\rho_1+2\rho_2+2\rho_4+2\rho_5-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_2-2\nu_3 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_8-2\rho_9-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_3-2\nu_1 &\le
4\rho_2+2\rho_3+2\rho_4+2\rho_5-2\rho_9-2\rho_{10}-2\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+\mu_2-\mu_1+2\nu_3-2\nu_1 &\le
4\rho_1+2\rho_2+2\rho_3+2\rho_4-2\rho_8-2\rho_9-2\rho_{10}-4\rho_{11}.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_1-2\nu_2-2\nu_3 &\le
10\rho_1+4\rho_2+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-8\rho_{11}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_2-2\nu_1-2\nu_3 &\le
10\rho_2+4\rho_1+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-8\rho_{11}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_2-3\mu_1+4\nu_1-2\nu_2-2\nu_3 &\le
10\rho_2+4\rho_1+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-8\rho_{11}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_2-2\nu_1-2\nu_3 &\le
10\rho_1+4\rho_2+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{11}-8\rho_{10}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_3-2\nu_1-2\nu_2 &\le
10\rho_3+4\rho_1+4\rho_2+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-8\rho_{11}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_3-2\nu_1-2\nu_2 &\le
10\rho_2+4\rho_1+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{11}-8\rho_{10}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_3-2\nu_1-2\nu_2 &\le
10\rho_1+4\rho_2+4\rho_3+4\rho_5+4\rho_6-2\rho_4-2\rho_7-2\rho_8-2\rho_9-2\rho_{10}-8\rho_{11}-8\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+4\nu_3-2\nu_1-2\nu_2 &\le
10\rho_1+4\rho_2+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{12}-8\rho_{10}-8\rho_{11},\\
3\lambda_1-3\lambda_2+3\mu_2-3\mu_1+4\nu_2-2\nu_1-2\nu_3 &\le
10\rho_1+4\rho_2+4\rho_3+4\rho_4+4\rho_5-2\rho_6-2\rho_7-2\rho_8-2\rho_9-2\rho_{12}-8\rho_{10}-8\rho_{11}.
\displaybreak[0]\\[2mm]
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_1+2\nu_2-4\nu_3 &\le
8\rho_1+8\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{11}-10\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_1+2\nu_3-4\nu_2 &\le
8\rho_1+8\rho_3+2\rho_2+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{11}-10\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_1+2\nu_3-4\nu_2 &\le
8\rho_1+8\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{12}-10\rho_{11},\\
3\lambda_1-3\lambda_2+3\mu_2-3\mu_1+2\nu_1+2\nu_2-4\nu_3 &\le
8\rho_1+8\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{12}-10\rho_{11},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_2+2\nu_3-4\nu_1 &\le
8\rho_1+8\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6+2\rho_9-4\rho_7-4\rho_8-4\rho_{10}-4\rho_{11}-10\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_2+2\nu_3-4\nu_1 &\le
8\rho_2+8\rho_3+2\rho_1+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{11}-10\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_2-3\mu_1+2\nu_1+2\nu_3-4\nu_2 &\le
8\rho_2+8\rho_3+2\rho_1+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{11}-10\rho_{12},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_2+2\nu_3-4\nu_1 &\le
8\rho_1+8\rho_3+2\rho_2+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{10}-4\rho_{12}-10\rho_{11},\\
3\lambda_1-3\lambda_2+3\mu_1-3\mu_2+2\nu_2+2\nu_3-4\nu_1 &\le
8\rho_1+8\rho_2+2\rho_3+2\rho_4+2\rho_5+2\rho_6+2\rho_7-4\rho_8-4\rho_9-4\rho_{11}-4\rho_{12}-10\rho_{10}.
\displaybreak[0]\\[2mm]
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+2\nu_1-\nu_2-\nu_3 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+2\nu_2-\nu_1-\nu_3 &\le
11\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+2\nu_2-\nu_1-\nu_3 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_5+2\rho_4+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+2\nu_2-\nu_1-\nu_3 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_8-4\rho_7-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+2\nu_2-\nu_1-\nu_3 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{11}-10\rho_{10}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_9-4\rho_7-4\rho_8-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_3+8\rho_1+8\rho_2+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_2+8\rho_1+8\rho_3+5\rho_5+2\rho_4+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_8-4\rho_7-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{11}-10\rho_{10}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_6+2\rho_4+2\rho_5-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_5+2\rho_4+2\rho_6-\rho_8-4\rho_7-4\rho_9-7\rho_{10}-10\rho_{11}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_5+2\rho_4+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{11}-10\rho_{10}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_8-4\rho_7-4\rho_9-7\rho_{11}-10\rho_{10}-10\rho_{12},\\
6\lambda_1-6\lambda_2-3\mu_2+3\mu_1+2\nu_3-\nu_1-\nu_2 &\le
11\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{12}-10\rho_{10}-10\rho_{11}.
\displaybreak[0]\\[2mm]
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_1+\nu_2-2\nu_3 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_1+\nu_3-2\nu_2 &\le
10\rho_1+10\rho_3+7\rho_2+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_1+\nu_3-2\nu_2 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_6+\rho_5-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_1+\nu_3-2\nu_2 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_9-5\rho_8-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_1+\nu_3-2\nu_2 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-11\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_9-5\rho_8-8\rho_{10}-8\rho_{12}-11\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_2+10\rho_3+7\rho_1+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_3+7\rho_2+4\rho_4+4\rho_6+\rho_5-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_3+7\rho_2+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_9-5\rho_8-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_3+7\rho_2+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-11\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_5+4\rho_6+\rho_4-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_6+\rho_5-2\rho_7-2\rho_9-5\rho_8-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_6+\rho_5-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-11\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_8-2\rho_9-5\rho_7-8\rho_{10}-8\rho_{11}-11\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+\nu_2+\nu_3-2\nu_1 &\le
10\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{11}-8\rho_{12}-11\rho_{10}.
\displaybreak[0]\\[2mm]
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_1+\nu_2-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_2+10\rho_1+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_4+4\rho_3+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_7-2\rho_6-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_{10}-8\rho_9-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_1+\nu_3-5\nu_2 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_1+\nu_2-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_2+10\rho_1+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_4+4\rho_3+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_6+\rho_7-2\rho_5-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_7-2\rho_6-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_2+\nu_1-5\nu_3 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_{10}-8\rho_9-8\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_3+\nu_2-5\nu_1 &\le
13\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_{11}-8\rho_9-8\rho_{12}-14\rho_{10},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+4\nu_3+\nu_2-5\nu_1 &\le
13\rho_2+10\rho_3+7\rho_4+4\rho_1+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+4\nu_3+\nu_1-5\nu_2 &\le
13\rho_2+10\rho_3+7\rho_4+4\rho_1+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-8\rho_{11}-14\rho_{12}.
\displaybreak[0]\\[2mm]
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_1-\nu_2-4\nu_3 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_2-\nu_1-4\nu_3 &\le
14\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1-\nu_2-4\nu_3 &\le
14\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_1+8\rho_2+8\rho_4+5\rho_3+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_7-\rho_6-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{10}-7\rho_9-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{12}-13\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_2+8\rho_1+8\rho_4+5\rho_3+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_7-\rho_6-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{10}-7\rho_9-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_2+8\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{12}-13\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1-\nu_3-4\nu_2 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_8-\rho_6-4\rho_7-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_3-\nu_2-4\nu_1 &\le
14\rho_3+8\rho_1+8\rho_4+5\rho_2+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-13\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_3-\nu_2-4\nu_1 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{12}-7\rho_9-10\rho_{10}-13\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_2-\nu_3-4\nu_1 &\le
14\rho_1+8\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{12}-7\rho_9-10\rho_{10}-13\rho_{11}.
\displaybreak[0]\\[2mm]
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+7\nu_1-2\nu_2-5\nu_3 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_2-5\nu_3 &\le
16\rho_2+10\rho_1+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_1+10\rho_2+7\rho_4+4\rho_3+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_2-5\nu_3 &\le
16\rho_1+10\rho_2+7\rho_4+4\rho_3+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_7-2\rho_6-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_{10}-8\rho_9-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_2-5\nu_3 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_{10}-8\rho_9-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_2-5\nu_3 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_2+10\rho_1+7\rho_4+4\rho_3+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_2+10\rho_1+7\rho_3+4\rho_4+4\rho_5+\rho_7-2\rho_6-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_2+10\rho_1+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_{10}-8\rho_9-11\rho_{11}-14\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_2+10\rho_1+7\rho_3+4\rho_4+4\rho_5+\rho_6-2\rho_7-2\rho_8-5\rho_9-8\rho_{10}-11\rho_{12}-14\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+7\nu_1-2\nu_3-5\nu_2 &\le
16\rho_1+10\rho_2+7\rho_3+4\rho_4+4\rho_5+\rho_8-2\rho_6-2\rho_7-5\rho_9-8\rho_{10}-11\rho_{11}-14\rho_{12}.
\displaybreak[0]\\[2mm]
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_1+2\nu_2-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_2+11\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1+2\nu_2-7\nu_3 &\le
14\rho_2+11\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_4+5\rho_3+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1+2\nu_2-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_4+5\rho_3+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_7-\rho_6-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_1-3\mu_2+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{10}-7\rho_9-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1+2\nu_2-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{10}-7\rho_9-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_1+2\nu_2-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{12}-16\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_2+11\rho_1+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{12}-16\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_4+5\rho_3+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{12}-16\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_6+2\rho_7-\rho_5-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{11}-16\rho_{12},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_7-\rho_6-4\rho_8-4\rho_9-7\rho_{10}-10\rho_{12}-16\rho_{11},\\
6\lambda_1-6\lambda_2+3\mu_2-3\mu_1+5\nu_2+2\nu_1-7\nu_3 &\le
14\rho_1+11\rho_2+8\rho_3+5\rho_4+2\rho_5+2\rho_6-\rho_7-4\rho_8-4\rho_{10}-7\rho_9-10\rho_{12}-16\rho_{11}.
\displaybreak[0]\\[2mm]
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_1-\nu_3 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_3 &\le
4\rho_2+3\rho_1+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_1-\nu_2 &\le
4\rho_1+3\rho_3+2\rho_2+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_3 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_5+\rho_4-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_1-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_6-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_3 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_7-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_1-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_9-2\rho_8-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_3 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{11}-3\rho_{10}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_1-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2-2\mu_2+2\mu_1+\nu_3-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_7-2\rho_8-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2-2\mu_2+2\mu_1+\nu_3-\nu_2 &\le
4\rho_3+3\rho_1+2\rho_2+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_1 &\le
4\rho_2+3\rho_3+2\rho_1+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_2-\lambda_1+2\mu_1-2\mu_2+\nu_1-\nu_2 &\le
4\rho_2+3\rho_3+2\rho_1+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2-2\mu_2+2\mu_1+\nu_3-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_6+\rho_4-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_2-\lambda_1+2\mu_1-2\mu_2+\nu_2-\nu_3 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_6+\rho_4-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_5+\rho_6-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_9-2\rho_7-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_2-\lambda_1+2\mu_1-2\mu_2+\nu_1-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_9-2\rho_7-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2-2\mu_2+2\mu_1+\nu_3-\nu_2 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{12}-3\rho_{10}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{12}-3\rho_{11}-4\rho_{10},\\
\lambda_2-\lambda_1+2\mu_1-2\mu_2+\nu_2-\nu_3 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{12}-3\rho_{10}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_2-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{11}-3\rho_{12}-4\rho_{10},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_7-2\rho_9-2\rho_{11}-3\rho_{12}-4\rho_{10},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_5+\rho_6-\rho_8-2\rho_9-2\rho_{11}-3\rho_{10}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_6-\rho_7-2\rho_8-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_5+\rho_6-\rho_7-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_9-2\rho_7-2\rho_{11}-3\rho_{10}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_9-2\rho_8-2\rho_{12}-3\rho_{10}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_6+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_6+\rho_4-\rho_9-2\rho_8-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_6-\rho_8-2\rho_9-2\rho_{12}-3\rho_{10}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_2+3\rho_3+2\rho_1+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{11}-3\rho_{10}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_6+\rho_4-\rho_8-2\rho_9-2\rho_{10}-3\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_7-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_5+\rho_4-\rho_9-2\rho_7-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_3+3\rho_2+2\rho_1+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_3+3\rho_1+2\rho_2+2\rho_4+\rho_6-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_3+3\rho_1+2\rho_2+2\rho_4+\rho_5-\rho_9-2\rho_8-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_3+3\rho_1+2\rho_2+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{10}-3\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_2+3\rho_1+2\rho_3+2\rho_5+\rho_6-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_2+3\rho_1+2\rho_3+2\rho_4+\rho_5-\rho_9-2\rho_7-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_2+3\rho_1+2\rho_3+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{11}-3\rho_{12}-4\rho_{10},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_2+3\rho_3+2\rho_1+2\rho_5+\rho_4-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_2+3\rho_3+2\rho_1+2\rho_4+\rho_5-\rho_7-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_3+2\rho_2+2\rho_6+\rho_4-\rho_8-2\rho_9-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_5+\rho_4-\rho_8-2\rho_9-2\rho_{11}-3\rho_{12}-4\rho_{10},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_3+2\rho_2+2\rho_4+\rho_5-\rho_7-2\rho_8-2\rho_{10}-3\rho_{11}-4\rho_{12},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_2+2\rho_3+2\rho_4+\rho_5-\rho_7-2\rho_8-2\rho_{10}-3\rho_{12}-4\rho_{11},\\
\lambda_1-\lambda_2+2\mu_1-2\mu_2+\nu_3-\nu_1 &\le
4\rho_1+3\rho_3+2\rho_2+2\rho_4+\rho_5-\rho_8-2\rho_9-2\rho_{12}-3\rho_{10}-4\rho_{11}.
\displaybreak[0]\\[2mm]
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_4+2\rho_3+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_4+2\rho_3+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_4+2\rho_3+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_6-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_6-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{10}-4\rho_9-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{10}-4\rho_9-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{10}-4\rho_9-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_2+4\rho_1+4\rho_4+2\rho_3+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_6-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_6-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{10}-4\rho_9-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_4+2\rho_3+2\rho_6-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_4+2\rho_3+2\rho_5-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_4+2\rho_3+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_6-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_6-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_6-2\rho_8-2\rho_{10}-4\rho_9-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_7-2\rho_{10}-4\rho_9-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_1-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1+3\mu_1-3\mu_2+2\nu_2-2\nu_3 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{10}-4\rho_9-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{12}-4\rho_9-4\rho_{11}-6\rho_{10},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_7-2\rho_{10}-4\rho_8-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_6+2\rho_3+2\rho_4-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_2-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_6+2\rho_3+2\rho_4-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_5+2\rho_3+2\rho_6-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_3+4\rho_1+4\rho_4+2\rho_2+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_9-2\rho_{10}-4\rho_7-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_2 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_9-2\rho_{10}-4\rho_7-4\rho_{11}-6\rho_{12},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{12}-4\rho_9-4\rho_{10}-6\rho_{11},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_2-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{12}-4\rho_9-4\rho_{10}-6\rho_{11},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{11}-4\rho_9-4\rho_{12}-6\rho_{10},\\
\lambda_1-\lambda_2+3\mu_1-3\mu_2+2\nu_3-2\nu_1 &\le
6\rho_2+4\rho_3+4\rho_4+2\rho_1+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_2 &\le
6\rho_2+4\rho_3+4\rho_4+2\rho_1+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_6-2\rho_8-2\rho_{12}-4\rho_9-4\rho_{10}-6\rho_{11},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_2+4\rho_1+4\rho_3+2\rho_4+2\rho_5-2\rho_9-2\rho_{10}-4\rho_7-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_6+2\rho_3+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_6+2\rho_3+2\rho_4-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{12}-6\rho_{11},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_2+4\rho_3+4\rho_4+2\rho_1+2\rho_5-2\rho_7-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_1+4\rho_2+4\rho_3+2\rho_4+2\rho_5-2\rho_8-2\rho_{10}-4\rho_7-4\rho_{11}-6\rho_{12},\\
\lambda_2-\lambda_1-3\mu_2+3\mu_1+2\nu_3-2\nu_1 &\le
6\rho_3+4\rho_2+4\rho_4+2\rho_1+2\rho_5-2\rho_8-2\rho_9-4\rho_{10}-4\rho_{11}-6\rho_{12}.
\end{align*}}

%and Here we consider few examples when marginal problem results
%imply no

%\subsection{Miscellanea}
%\subsubsection{Two row diagrams and Bravyi theorem}
%\subsubsection{Application to representation theory}
%In contrast with Littlrewood--Richardson coefficients there is no
%%closed formula or
%combinatorial description of Kronecker coefficients.



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\bibliographystyle{amsplain}
\bibliography{Q_Margins}
\end{document}
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Q_Margins.bbl
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
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