Extension-orthogonal components of nilpotent varieties
Christof Gei{\ss}
Let $Q$ be a Dynkin quiver, and let $\Lambda$ be the corresponding preprojective algebra. Let ${\mathcal I} = \{ C_i \mid i \in I \}$ be a set of pairwise different indecomposable irreducible components of varieties of $\Lambda$-modules such that generically there are no extensions between $C_i$ and $C_j$ for all $i,j$. We show that the number of elements in ${\mathcal I}$ is at most the number of positive roots of $Q$. Furthermore, we give a module theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.
math0208209.tex
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\begin{document}
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\title{Extension-orthogonal components of nilpotent varieties}
\author{Christof Gei{\ss}}
\address{Christof Gei{\ss}\newline
Instituto de Matem\'aticas, UNAM\newline
Ciudad Universitaria\newline
04510 Mexico D.F.\newline
Mexico}
\email{christof@matem.unam.mx}

\author{Jan Schr\"oer}
\address{Jan Schr\"oer\newline
Department of Pure Mathematics\newline
University of Leeds\newline
Leeds LS2 9JT\newline
UK}
\email{jschroer@maths.leeds.ac.uk}



\thanks{Mathematics Subject Classification (2000): 14M99, 
16D70, 16G20, 17B37
}

\begin{abstract}
Let $Q$ be a 
Dynkin quiver, and let $\Lambda$ be the corresponding 
preprojective algebra.
Let ${\mathcal I} = \{ C_i \mid i \in I \}$ be a set of pairwise different
indecomposable irreducible components of varieties of
$\Lambda$-modules such that generically there are no extensions
between $C_i$ and $C_j$ for all $i,j$.
We show that the number of elements in ${\mathcal I}$ is at most
the number of positive roots of $Q$. 
Furthermore, we give a module theoretic interpretation of
Leclerc's counterexample to a conjecture of Berenstein and
Zelevinsky.
\end{abstract}

%%%%%%%%%%
\maketitle
%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let $\field$ be an algebraically closed field.
For a finitely generated $\field$-algebra $A$ let
$\md_A({\bf d})$ be the affine variety of $A$-modules with dimension
vector ${\bf d}$.
For irreducible components
$C_1 \subseteq \md_A({\bf d}_1)$ and
$C_2 \subseteq \md_A({\bf d}_2)$ 
define
\[
{\rm ext}^1_A(C_1,C_2) = \min \{ {\rm dim}\, \Ext^1_A(M_1,M_2) 
\mid (M_1,M_2) \in C_1 \times C_2 \}.
\]
An irreducible component $C \subseteq \md_A({\bf d})$ is {\it indecomposable}
if it contains a dense subset of indecomposable $A$-modules.
A general theory of irreducible components and their decomposition
into indecomposable irreducible components was developed in \cite{CBSc}.
Our aim is to apply this to Lusztig's nilpotent varieties.

If not mentioned otherwise, we always assume that $Q$ is a Dynkin quiver of
type $\A_n$, $\D_n$ or $\E_{6,7,8}$.
By  $R^+$ we denote the set of positive roots of $Q$, and by 
$\Lambda$ we denote the preprojective algebra associated to $Q$,
see \cite{Ri2}.
Let $n$ be the number of vertices of $Q$, and let
$\Lambda({\bf d}) = \md_{\Lambda}({\bf d})$, ${\bf d} \in \N^n$,
be the variety of $\Lambda$-modules with dimension vector ${\bf d}$.
The varieties $\Lambda({\bf d})$ are called 
{\it nilpotent varieties}.
We refer to \cite[Section 12]{L2} for basic properties.
Throughout, we only consider finite-dimensional modules.
Our main result is the following:



\vspace{0.2cm}\begin{Thm}\label{mainresult}
Assume that $\{ C_i \subseteq \Lambda({\bf d}_i) \mid i \in I \}$
is a set of pairwise different indecomposable irreducible components 
such that
${\rm ext}_{\Lambda}^1(C_i,C_j) = 0$ for all $i,j \in I$.
Then $|I| \leq |R^+|$.
\end{Thm}\vspace{0.2cm}



As a consequence we get the following result on $\Lambda$-modules without
self-extensions:



\vspace{0.2cm}\begin{Cor}\label{maincor}
Let $M$ be a $\Lambda$-module
with ${\rm Ext}_{\Lambda}^1(M,M) = 0$.
Then the number of pairwise non-isomorphic 
indecomposable direct summands of $M$ is at most $|R^+|$. 
\end{Cor}\vspace{0.2cm}



Let $U_v^-$ be the negative part of the 
quantized enveloping algebra of the Lie algebra corresponding to 
$Q$.
We regard $U_v^-$ as a $\Q(v)$-algebra.
Let ${\mathcal B}$ be the canonical basis and
${\mathcal B}^*$ the dual canonical basis of $U_v^-$,
see \cite{BZ}, \cite{LNT}, \cite{L2} or \cite{R} for definitions.
By \cite[Section 5]{KS}, \cite{L1} the elements of ${\mathcal B}$ (and thus
of ${\mathcal B}^*$) 
correspond to the irreducible components of the nilpotent varieties
$\Lambda({\bf d})$, ${\bf d} \in \N^n$.
Let $b^*(C)$ be the dual canonical basis vector corresponding to
an irreducible component $C$.
We denote the structure constants of $U_v^-$ with respect to the basis
${\mathcal B}^*$ by $\lambda_{C,D}^E$, i.e.
\[
b^*(C)b^*(D) = \sum_E \lambda_{C,D}^E b^*(E).
\]
Following the terminology in \cite{BZ}, two
dual canonical basis vectors
$b^*(C)$ and $b^*(D)$ are called 
{\it multiplicative} if 
\[
b^*(C)b^*(D) = \lambda_{C,D}^E b^*(E)
\] 
for some irreducible component $E$,
and they are {\it quasi-commutative} if
\[
b^*(C)b^*(D) = \lambda b^*(D)b^*(C)
\]
for some $\lambda \in \Q(v)$.
The following conjecture was stated in \cite[Section 1]{BZ}:


\vspace{0.2cm}\begin{Conj}[Berenstein, Zelevinsky]\label{bzconj}
Two dual canonical basis vectors are multiplicative
if and only if they are quasi-commutative.
\end{Conj}\vspace{0.2cm}


One direction of this conjecture was proved by Reineke 
\cite[Corollary 4.5]{R}.
The other direction turned out to be wrong.
Namely,
Leclerc \cite{Le} constructed examples of quasi-commutative 
elements in ${\mathcal B}^*$
which are not multiplicative.
Using preprojective algebras, we give a module theoretic interpretation
of one of his examples. 

Marsh and Reineke \cite{MR} conjectured
that the multiplicative behaviour of
dual canonical basis vectors
should be related to sets of irreducible components with ${\rm Ext}^1$ 
vanishing generically between them.
This was the principal motivation for our work.

The paper is organized as follows: In Section \ref{section2} we review
the main results from \cite{CBSc}.
In Section \ref{section3} we recall known results for the case
that $\Lambda$ is an algebra of finite or tame
representation type.
The proof of Theorem \ref{mainresult} and its corollary can be found in
Section \ref{section4}.
Finally, Section \ref{section5} is devoted to the interpretation of 
Leclerc's example.

\vspace{0.5cm}

{\bf Acknowledgements.} 
The second author is greatful to the IM UNAM, Mexico City, where
most of this work was done. 
We thank Robert Marsh and Markus Reineke 
for helpful and interesting discussions. 






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\section{Varieties of modules - definitions and known results}
\label{section2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we work with arbitrary finite quivers.

\subsection{}
Let
$Q = (Q_0,Q_1)$ be a finite quiver, 
where  $Q_0$ denotes the set of 
vertices and $Q_1$ the set of arrows of $Q$.
Assume that $|Q_0| = n$.
For an arrow $\alpha$ let $s \alpha$ be its starting vertex and
$e \alpha$ its end vertex.
An element ${\bf d} = (d_i)_{i \in Q_0} \in \N^n$ is called
a {\it dimension vector} for $Q$.
A {\it representation} of $Q$ with dimension vector ${\bf d}$ is a matrix
tuple $M = (M_\alpha)_{\alpha \in Q_1}$ with $M_\alpha \in
\M_{d_{s\alpha} \times d_{e\alpha}}(\field)$.
A {\it path} of length $l \geq 1$ in $Q$ is a 
sequence $p = \alpha_1 \cdots \alpha_l$ of arrows in $Q_1$ 
such that $e\alpha_i  = s\alpha_{i+1}$ for $1 \leq i \leq l-1$.
Define $sp = s\alpha_1$ and $ep = e\alpha_l$.
For a representation $M$ and a path $p = \alpha_1 \cdots \alpha_l$
define $M_p = M_{\alpha_1} \cdots M_{\alpha_l}$ which is a matrix
in $\M_{d_{sp} \times d_{ep}}(\field)$.
A {\it relation} for $Q$ is a $\field$-linear combination 
$\sum_{i=1}^t \lambda_i p_i$ of paths $p_i$ of length
at least two such that $sp_i = sp_j$ and $ep_i = ep_j$ for all
$1 \leq i,j \leq t$.
A representation $M$ satisfies such a relation if
$\sum_{i=1}^t \lambda_i M_{p_i} = 0$.
Given a set $\rho$ of relations for $Q$ let $\rep_{(Q,\rho)}({\bf d})$
be the affine variety of representations of $Q$ with dimension
vector ${\bf d}$ which satisfy all relations in $\rho$.

\subsection{}
One can interpret this construction in a module theoretic way.
Namely, let $\field Q$ be the path algebra of $Q$, and let
$A = \field Q/(\rho)$, where $(\rho)$ is the ideal generated by
the elements in $\rho$.
Then $\md_A({\bf d}) = \rep_{(Q,\rho)}({\bf d})$ is the affine 
{\it variety of
$A$-modules} with dimension vector ${\bf d}$.
If $A = \field Q/(\rho)$ is finite-dimensional, then 
$A$ is called a {\it basic algebra}.
In this case, the vertices of $Q$ correspond to the isomorphism
classes of simple $A$-modules, and the entry $d_i$, $i \in Q_0$, 
of ${\bf d}$ is the multiplicity of the simple module corresponding to
$i$ in a composition series of any $M \in \md_A({\bf d})$.
The group $\GL({\bf d}) = \prod_{i \in Q_0} \GL_{d_i}(\field)$ acts
on $\md_A({\bf d})$ by conjugation, i.e. 
\[
g \cdot M = (g_{s\alpha} M_\alpha g_{e\alpha}^{-1})_{\alpha \in Q_1}.
\]
The orbit of $M$ under this action is denoted by $\Or(M)$.
There is a 1-1 correspondence between the set of orbits in $\md_A({\bf d})$ 
and the set of isomorphism classes of $A$-modules with dimension vector 
${\bf d}$.


\subsection{}
Given 
irreducible components $C_i \subseteq \md_A({\bf d}_i)$, $1 \leq i \leq t$,
we consider all $A$-modules with dimension vector ${\bf d} = 
{\bf d}_1 + \cdots + {\bf d}_t$,
which are of the form $M_1 \oplus \cdots \oplus M_t$ with 
the $M_i$ in $C_i$, and we denote by 
$C_1 \oplus \cdots \oplus C_t$ the corresponding
subset of $\md_A({\bf d})$. 
This is the image of the map
\[
\GL({\bf d}) \times C_1 \times \cdots \times C_t
\longrightarrow \md_A({\bf d}) 
\]
\[
(g,M_1,\cdots,M_t) \mapsto g \cdot \left(
\bigoplus_{i=1}^t M_i \right).
\]
We call $C_1 \oplus \cdots \oplus C_t$ the {\it direct sum} of the 
components $C_i$.
It follows that the
closure $\overline{C_1 \oplus \cdots \oplus C_t}$ is irreducible.
For an irreducible component $C$ define $C^n = \bigoplus_{i=1}^n C$.
We call $C$ {\it indecomposable} if $C$ contains a dense subset of
indecomposable $A$-modules.
The following result from \cite{CBSc} is
an analogue of the Krull-Remak-Schmidt
Theorem.

\vspace{0.2cm}\begin{Thm}\label{thm1}
If $C \subseteq \md_A({\bf d})$ is an irreducible component,
then 
\[
C = \overline{C_1 \oplus \cdots \oplus C_t}
\]
for some indecomposable 
irreducible components $C_i \subseteq \md_A({\bf d}_i)$,
$1 \leq i \leq t$, and
$C_1, \cdots, C_t$
are uniquely determined by this, up to reordering.
The above direct sum is called the canonical decomposition of $C$.
\end{Thm}\vspace{0.2cm}


However, the closure of a direct sum of irreducible
components is not in general an irreducible component.
The next result is also proved in \cite{CBSc}.


\vspace{0.2cm}\begin{Thm}\label{thm2}
If $C_i \subseteq \md_A({\bf d}_i)$, $1 \leq i \leq t$, are irreducible
components, and ${\bf d} = {\bf d}_1 + \cdots + {\bf d}_t$,
then $\overline{C_1 \oplus \cdots \oplus C_t}$
is an irreducible component of $\md_A({\bf d})$
if and only if ${\rm ext}_A^1(C_i,C_j) = 0$ for all $i \not= j$.
\end{Thm}\vspace{0.2cm}


Instead of taking direct sums of the modules in two irreducible
components, one can take extensions. 
Let ${\bf d} = {\bf d}_1 + {\bf d}_2$ be dimension vectors, let
$G = \GL({\bf d}_1) \times \GL({\bf d}_2)$,
and let $S$ be a $G$-stable subset of 
$\md_A({\bf d}_1) \times \md_A({\bf d}_2)$.
We denote by ${\mathcal E}(S)$
the $\GL({\bf d})$-stable subset of $\md_A({\bf d})$
corresponding to all modules $M$ which belong to
a short exact sequence
\[
0 \longrightarrow M_2 \longrightarrow M \longrightarrow M_1 
\longrightarrow 0
\]
with $(M_1,M_2) \in S$, see \cite{CBSc} for more details.

For an irreducible component $C \subseteq \md_A({\bf d})$ let
\[
\mu_g(C) = {\rm dim}\, C - \max \{ {\rm dim}\, \Or(M) \mid M \in C \}
\]
be the {\it generic number of parameters of} $C$.
Thus $\mu_g(C) = 0$ if and only if $C$ contains a dense orbit $\Or(M)$.
For example, if $P$ is a projective $A$-module, then 
$\Ext_A^1(P,P) = 0$.
This implies that the closure of the orbit $\Or(P)$ is an irreducible
component, and we get $\mu_g(\overline{\Or(P)}) = 0$.
Also, if $C = \overline{C_1 \oplus \cdots \oplus C_t}$ with
${\rm ext}^1_A(C_i,C_j) = 0$ for all $i \not= j$, then
\[
\mu_g(C) = \sum_{i=1}^t \mu_g(C_i).
\]

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\section{The finite and tame cases}
\label{section3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As in the introduction let $Q$ be a Dynkin quiver.
Then $\Lambda$ is of finite representation 
type if and only if
$Q$ is of type $\A_i$, $i \leq 4$.
In this case, 
if
$\{ C_i \subseteq \Lambda({\bf d}_i) \mid i \in I \}$ is a maximal set of 
pairwise different indecomposable irreducible
components such that ${\rm ext}_{\Lambda}^1(C_i,C_j) = 0$ for all $i,j$,
then $|I| = |R^+|$.
This follows from \cite{BZ} for $i \leq 3$, and the case $i = 4$ was
done by Marsh and Reineke.

Recall that for a tame algebra $A$ one has
$\mu_g(C) \leq 1$ for any indecomposable irreducible
component $C \subseteq \md_A({\bf d})$.
It is known that $\Lambda$ is of tame representation type
if and only if $Q$ is of type $\A_5$ or $\D_4$.
In this case, a complete classification of the indecomposable
irreducible components, and a necessary and sufficient condition
for ${\rm ext}^1_{\Lambda}(C,D) = 0$ for any two irreducible components
$C$ and $D$ was obtained in \cite{GSc}.
In particular, this implies the following:


\vspace{0.2cm}\begin{Thm}
Assume that $Q$ is of type $\A_5$ or $\D_4$.
Then the following hold:
%
\begin{itemize}

\item[(1)] For any irreducible component $C \subseteq \Lambda({\bf d})$
we have ${\rm ext}_{\Lambda}^1(C,C) = 0$;

\item[(2)] If $C \subseteq \Lambda({\bf d})$ is an indecomposable
irreducible component, then we have $\mu_g(C) = 0$ or
$\mu_g(C) = 1$.
For suitable ${\bf d}$ there exists an indecomposable irreducible
component $C \subseteq \Lambda({\bf d})$ with
$\mu_g(C) = 1$;

\item[(3)] 
Let $\{ C_i \subseteq \Lambda({\bf d}_i) 
\mid i \in I \}$ 
be a maximal set of pairwise different indecomposable irreducible 
components such that
${\rm ext}_{\Lambda}^1(C_i,C_j) = 0$ for all $i,j$.
Then there is at most one $C_i$ with $\mu_g(C_i) = 1$.
In this case, we have
$|I| = |R^+| - 1$, 
and we get
$|I| =  |R^+|$, otherwise.

\end{itemize}
%
\end{Thm}\vspace{0.2cm}


This leads us to the following conjecture for arbitrary Dynkin quivers of 
type $\A_n$, $\D_n$ or $\E_{6,7,8}$:


\vspace{0.2cm}\begin{Conj}\label{Conj1}
If
$\{ C_i \subseteq \Lambda({\bf d}_i) \mid i \in I \}$ is a maximal set of 
pairwise different indecomposable irreducible
components such that ${\rm ext}_{\Lambda}^1(C_i,C_j) = 0$ for all $i,j$,
then
\[
|I| = |R^+| - \sum_{i \in I} \mu_g(C_i).
\] 
\end{Conj}\vspace{0.2cm}


In all remaining cases the algebra $\Lambda$ is
of wild representation type.
So one should expect 
irreducible components $C$ with ${\rm ext}_{\Lambda}^1(C,C) \not= 0$.
Thus, maybe one should study sets 
$\{ C_i \subseteq \Lambda({\bf d}_i) \mid i \in I \}$ of
irreducible components with the weaker condition 
${\rm ext}_{\Lambda}^1(C_i,C_j) = 0$
for all $i \not= j$.
However, we do not know how to generalize Theorem \ref{mainresult} to
this case.


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\section{Proof of Theorem \ref{mainresult}}
\label{section4}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As before let $Q$ be a Dynkin quiver, and 
let $R^+ = \{ {\bf a}_i \mid 1 \leq i \leq N \}$
be the set of positive roots of $Q$.
By Gabriel's Theorem there is a 1-1 correspondence between the
isomorphism classes of indecomposable $\field Q$-modules and
the elements in $R^+$.
This correspondence associates to a root ${\bf a}_i$ the
isomorphism class $[M({\bf a}_i)]$ of an
indecomposable $\field Q$-module $M({\bf a}_i)$ with dimension
vector ${\bf a}_i$.
By the Theorem of Krull-Remak-Schmidt each $\field Q$-module is 
isomorphic to a (up to reordering)
unique direct sum of the indecomposable modules $M({\bf a}_i)$.
The maps
\[
\alpha = (\alpha_1, \cdots, \alpha_N) \mapsto
\Or(M_\alpha) \mapsto C_\alpha \mapsto b^*(C_\alpha)
\]
define 1-1 correspondences between $\N^N$, the set of orbits
$\Or(M) \subseteq \md_{\field Q}({\bf d})$, ${\bf d} \in \N^n$,
the set of irreducible components of 
$\Lambda({\bf d})$, ${\bf d} \in \N^n$,
and the set ${\mathcal B}^*$ of dual canonical basis vectors,
where
\begin{eqnarray*}
M_\alpha &=& \bigoplus_{i=1}^N M({\bf a}_i)^{\alpha_i},\\
C_\alpha &=&  \overline{\pi^{-1}(\Or(M_\alpha))}
\end{eqnarray*} 
with
\[
\pi: \Lambda({\bf d}) \to \md_{\field Q}({\bf d})
\]
the canonical projection map.

Let $\alpha, \beta \in \N^N$.
By Theorem \ref{thm2} the closure
$\overline{C_\alpha \oplus C_\beta}$ is an irreducible
component if and only if ${\rm ext}^1_{\Lambda}(C_\alpha,C_\beta)
= {\rm ext}^1_{\Lambda}(C_\beta,C_\alpha) = 0$.
In this case, we have $\overline{C_\alpha \oplus C_\beta}
= C_{\alpha + \beta}$.

Let $\alpha_i = (\alpha_{1i}, \cdots, \alpha_{Ni})$, 
$1 \leq i \leq N+1$, be non-zero pairwise different elements in $\N^N$
such that $C_{\alpha_i}$ is an indecomposable irreducible component
for all $i$.
To get a contradiction, we assume that
${\rm ext}^1_{\Lambda}(C_{\alpha_i},C_{\alpha_j}) = 0$ for all
$1 \leq i,j \leq N+1$.
For all ${\bf m} = (m_1, \cdots, m_{N+1}) \in \N^{N+1}$ define
\[
C_{\bf m} = C_{\sum_{i=1}^{N+1} m_i \alpha_i}.
\]
We get
\[
C_{\bf m} = 
\overline{C_{\alpha_1}^{m_1} \oplus \cdots \oplus 
C_{\alpha_{N+1}}^{m_{N+1}}}.
\]
Since the $C_{\alpha_i}$ are indecomposable, the above is the canonical
decomposition of the irreducible component 
$C_{\bf m}$.

We claim that there exist some elements 
${\bf m} = (m_1, \cdots, m_{N+1})
\not= {\bf l} = (l_1, \cdots, l_{N+1})$ in $\N^{N+1}$
such that
\[
\sum_{i=1}^{N+1} m_i \alpha_i = \sum_{i=1}^{N+1} l_i \alpha_i.
\]
This implies $C_{\bf m} = C_{\bf l}$.
Thus, we get a contradiction to the unicity 
of the canonical decomposition of irreducible components, 
see Theorem \ref{thm1}.

Let
$\Delta = (\alpha_{ij})$ be the $N \times (N+1)$-matrix where the 
$j$th column
is just the vector $\alpha_j$.
Thus we have to find some ${\bf d} \in \N^N$ and some
${\bf m} \not= {\bf l}$ in $\N^{N+1}$ such that
\[
\Delta{\bf m} = \Delta{\bf l} = {\bf d}.
\]
Since all entries in $\Delta$ are in $\N$, this would imply 
that ${\bf m}$, ${\bf l}$ and ${\bf d}$ are all non-zero.
Furthermore, there must be a non-zero element 
${\bf z} = (z_1, \cdots, z_{N+1}) 
\in \Z^{N+1}$ such that $\Delta{\bf z} = 0$.

First, we consider the case ${\bf z} \in \N^{N+1}$.
Let ${\bf m} = (1, \cdots, 1) \in \N^{N+1}$, ${\bf d} = \Delta{\bf m}$ and
${\bf l} = {\bf m} + {\bf z}$.
Obviously, ${\bf d} \in \N^N$.
We get $\Delta{\bf m} = \Delta{\bf l} = {\bf d}$ with 
${\bf m} \not= {\bf l}$ in $\N^{N+1}$.

Next, assume that ${\bf z} \notin \N^{N+1}$.
Define $\lambda = - \min\{ z_i \mid 1 \leq i \leq N+1 \}$.
Let ${\bf m} = (\lambda, \cdots, \lambda) \in \N^{N+1}$,
${\bf d} = \Delta{\bf m}$ and
${\bf l} = {\bf m} + {\bf z}$.
Again, we get  ${\bf d} \in \N^N$ and
$\Delta{\bf m} = \Delta{\bf l} = {\bf d}$ with
${\bf m} \not= {\bf l}$ in $\N^{N+1}$.
This finishes the proof of Theorem \ref{mainresult}.

Corollary \ref{maincor} follows immediately from
the fact that an orbit ${\mathcal O}(N) \subseteq \md_A({\bf d})$
of an $A$-module $N$ is open provided ${\rm Ext}_A^1(N,N) = 0$.
Clearly, ${\mathcal O}(N)$ is open if and only if the closure
$\overline{{\mathcal O}(N)}$ is an irreducible component.
Then we use Theorems \ref{mainresult} and \ref{thm2}.









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Interpretation of Leclerc's example}\label{section5}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the following, we use the notation introduced at the beginning
of Section \ref{section4}.

Reineke proved in \cite[Lemma 4.6]{R} that the multiplicativity of 
$b^*(C_\alpha)$ and $b^*(C_\beta)$ implies that 
\[
b^*(C_\alpha)b^*(C_\beta) = v^mb^*(C_{\alpha+\beta})
\]
for some $m \in \Z$.
He also showed that $\lambda_{C,D}^E \not= 0$ if and only if
$\lambda_{D,C}^E \not= 0$.
This follows from \cite[Proposition 4.4]{R}.
Thus one direction of Conjecture \ref{bzconj} holds,
namely if two dual canonical basis vectors are multiplicative, then
they are quasi-commutative.
The following related problem should be of interest:


\vspace{0.2cm}\begin{Prob}
Describe the elements $\alpha, \beta \in \N^N$ such that
\[
C_{\alpha+\beta} = \overline{C_\alpha \oplus C_\beta}.
\]
\end{Prob}\vspace{0.2cm}


As mentioned in the introduction, 
Leclerc recently constructed in \cite{Le} counterexamples for the
other direction of the Berenstein-Zelevinsky Conjecture.
We give a module theoretic interpretation of one of his examples:

Let $Q$ be the quiver of type $\A_5$
with arrows $a_i: i+1 \to i$, $1 \leq i \leq 4$.
Thus $\Lambda$ is given by the quiver 
\[
\xymatrix{
1\ar@<1ex>[r]^{\bar{a}_1}&\ar@<1ex>[l]^{a_1} 2 \ar@<1ex>[r]^{\bar{a}_2}&
\ar@<1ex>[l]^{a_2} 3\ar@<1ex>[r]^{\bar{a}_3}&\ar@<1ex>[l]^{a_3} 4 
\ar@<1ex>[r]^{\bar{a}_4}&
\ar@<1ex>[l]^{a_4} 5
}
\]
and the following set of relations
\[
\{  \bar{a}_1a_1, a_1\bar{a}_1-\bar{a}_2a_2,a_2\bar{a}_2-\bar{a}_3a_3,
a_3\bar{a}_3-\bar{a}_4a_4,a_4\bar{a}_4 \}.
\]
Now $R^+$ contains exactly 15 elements, namely for each
$1 \leq i \leq j \leq 5$ there is a positive root 
$[i,j] = (d_l)_{1 \leq l \leq 5}$ with $d_l = 1$ for
$i \leq l \leq j$, and $d_l = 0$, else.
We identify $\N R^+$ with 
$\N^{15}$ by fixing a linear ordering on
$R^+$, namely let
\begin{eqnarray*}
& & [1,1] < [1,2] < [1,3]<[1,4]<[1,5]<[2,2]<[2,3]<[2,4]<[2,5]\\
& & <[3,3]<[3,4]<[3,5]<[4,4]<[4,5]<[5,5].
\end{eqnarray*}
Define
%
\begin{eqnarray*} 
\alpha &=& [1,2] + [2,4] + [3,3] + [4,5],\\
\beta &=& [1,2] + [1,4] +  [2,3] + [2,5] + [3,4] + 
[4,5].
\end{eqnarray*}
%
Thus, regarded as elements in $\N^{15}$ we have
\begin{eqnarray*}
\alpha &=& (0,1,0,0,0,0,0,1,0,1,0,0,0,1,0),\\
\beta &=&  (0,1,0,1,0,0,1,0,1,0,1,0,0,1,0).
\end{eqnarray*}
In \cite{Le}
Leclerc showed that
\[
b^*(C_\alpha)^2 = v^{-2} \left( b^*(C_{\alpha+\alpha})
+ b^*(C_\beta) \right).
\]
This is obviously a counterexample to the Berenstein-Zelevinsky Conjecture.
Now define  
%
\begin{eqnarray*} 
\beta_1 &=& [1,2] + [2,3] + [3,4] + [4,5],\\ 
\beta_2 &=& [1,4] + [2,5].
\end{eqnarray*}
%
Thus we have $\beta = \beta_1 + \beta_2$.

\vspace{0.2cm}\begin{Prop}
Let $\alpha, \beta, \beta_1,\beta_2$ be as above.
Then the following hold:
\begin{itemize}

\item[(1)]
The irreducible components $C_\alpha$, $C_{\beta_1}$ and
$C_{\beta_2}$ are indecomposable with
$\mu_g(C_\alpha) = 1$ and $\mu_g(C_{\beta_1}) = \mu_g(C_{\beta_2}) = 0$;

\item[(2)]
We have
$C_{\alpha+\alpha} = \overline{C_\alpha \oplus C_\alpha}$ and
$C_\beta = \overline{C_{\beta_1} \oplus C_{\beta_2}}$.
Thus 
\[
b^*(C_\alpha)^2 = 
v^{-2} \left( b^*(\overline{C_{\alpha} \oplus 
C_{\alpha}})
+ b^*(\overline{C_{\beta_1} \oplus C_{\beta_2}}) \right).
\]
\end{itemize}
\end{Prop}

\begin{proof}
For $\lambda \in \field \setminus \{ 0,1 \}$ let $M_\lambda$ be 
the $8$-dimensional $\Lambda$-module 
where the arrows of $\Lambda$ operate on a basis $\{ 1, \cdots, 8 \}$
as in the following picture:

\begin{center}
\unitlength0.5cm
\begin{picture}(11,6)

\put(3,5){$2$}
\put(2.8,4.8){\vector(-1,-1){1.3}}\put(1.6,4.5){$a_1$}
\put(3.5,4.8){\vector(1,-1){1.3}}\put(3.4,3.8){$\bar{a}_2$}
\put(3.6,4.8){\vector(2,-1){3}}\put(4.5,4.4){$\lambda$}
\put(1,3){$1$}
\put(1.5,2.8){\vector(1,-1){1.3}}\put(1.4,1.9){$\bar{a}_1$}
\put(3,1){$3$}
\put(5,3){$4$}
\put(4.8,2.8){\vector(-1,-1){1.3}}\put(4.4,1.9){$a_2$}

\put(8.8,5){$6$}
\put(8.6,4.8){\vector(-1,-1){1.3}}\put(8.2,3.9){$a_3$}
\put(9.3,4.8){\vector(1,-1){1.3}}\put(10,4.2){$\bar{a}_4$}
\put(8.5,4.8){\vector(-2,-1){3}}
\put(6.8,3){$5$}
\put(7.3,2.8){\vector(1,-1){1.3}}\put(7.2,1.9){$\bar{a}_3$}
\put(8.8,1){$7$}
\put(10.8,3){$8$}
\put(10.6,2.8){\vector(-1,-1){1.3}}\put(10.2,1.9){$a_4$}

\end{picture}
\end{center}

Thus, for example $2 \cdot a_1 = 1$,
$2 \cdot \bar{a}_2 = 4 + \lambda 5$, $6 \cdot a_3 = 4 + 5$,
$1 \cdot \bar{a}_1 = 3$, etc.
Note that $M_\lambda$ lies in $C_\alpha$.

The modules $M_\lambda$ are indecomposable and
$
{\rm dim}\, {\rm End}_{\Lambda}(M_\lambda) = 3.
$
From a well-known general fact we know that
each irreducible component of $\Lambda(1,2,2,2,1)$
has dimension $2+4+4+2 = 12$, see for example \cite[Section 12]{L2}.
The group $\GL(1,2,2,2,1)$ acts as described in Section \ref{section2}
on $\Lambda(1,2,2,2,1)$ and has dimension $14$.
Thus we get
$
{\rm dim}\, {\mathcal O}(M_\lambda) = 14-3 = 11.
$
One checks easily that $M_\lambda$ and $M_\mu$ are isomorphic if and
only if $\lambda = \mu$.
This implies
\[
{\rm dim}\, \overline{ \{ {\mathcal O}(M_\lambda) \mid
\lambda \in \field \setminus \{ 0,1 \} \} } = 11+1 = 12.
\]
We get
\[
C_\alpha =  \overline{\{ {\mathcal O}(M_\lambda) \mid
\lambda \in \field \setminus \{ 0,1 \} \}}.
\]
Thus $C_\alpha$ is an indecomposable irreducible component
with $\mu_g(C_\alpha) = 1$.

Next, let $P_2$ and $P_4$ be the indecomposable projective 
$\Lambda$-modules corresponding to the vertices $2$ and $4$, respectively.
These modules are both $8$-dimensional and 
look as in the following picture:

\begin{center}
\unitlength0.5cm
\begin{picture}(22,10)

\put(3,9){$2$}
\put(2.8,8.8){\vector(-1,-1){1.3}}\put(2.3,7.8){$a_1$}
\put(3.5,8.8){\vector(1,-1){1.3}}\put(4.2,8.2){$\bar{a}_2$}
\put(1,7){$1$}
\put(1.5,6.8){\vector(1,-1){1.3}}\put(1.4,5.9){$\bar{a}_1$}
\put(3,5){$3$}
\put(3.5,4.8){\vector(1,-1){1.3}}\put(3.4,3.9){$\bar{a}_2$}
\put(3,1){$P_2$}
\put(5,7){$4$}
\put(4.8,6.8){\vector(-1,-1){1.3}}\put(4.4,5.9){$a_2$}
\put(5.5,6.8){\vector(1,-1){1.3}}\put(6.2,6.2){$\bar{a}_3$}
\put(5,3){$5$}
\put(5.5,2.8){\vector(1,-1){1.3}}\put(5.4,1.9){$\bar{a}_3$}
\put(7,5){$6$}
\put(6.8,4.8){\vector(-1,-1){1.3}}\put(6.4,3.9){$a_3$}
\put(7.5,4.8){\vector(1,-1){1.3}}\put(8.2,4.2){$\bar{a}_4$}
\put(7,1){$7$}
\put(9,3){$8$}
\put(8.8,2.8){\vector(-1,-1){1.3}}\put(8.4,1.9){$a_4$}

\put(13,3){$1$}
\put(13.5,2.8){\vector(1,-1){1.3}}\put(13.4,1.9){$\bar{a}_1$}
\put(15,5){$2$}
\put(14.8,4.8){\vector(-1,-1){1.3}}\put(13.6,4.5){$a_1$}
\put(15.5,4.8){\vector(1,-1){1.3}}\put(15.4,3.9){$\bar{a}_2$}
\put(17,7){$4$}
\put(16.8,6.8){\vector(-1,-1){1.3}}\put(15.6,6.5){$a_2$}
\put(17.5,6.8){\vector(1,-1){1.3}}\put(17.4,5.9){$\bar{a}_3$}
\put(19,9){$6$}
\put(18.8,8.8){\vector(-1,-1){1.3}}\put(17.6,8.5){$a_3$}
\put(19.5,8.8){\vector(1,-1){1.3}}\put(19.4,7.9){$\bar{a}_4$}
\put(15,1){$3$}
\put(17,3){$5$}
\put(16.8,2.8){\vector(-1,-1){1.3}}\put(16.4,1.9){$a_2$}
\put(19,5){$7$}
\put(18.8,4.8){\vector(-1,-1){1.3}}\put(18.4,3.9){$a_3$}
\put(19,1){$P_4$}
\put(21,7){$8$}
\put(20.8,6.8){\vector(-1,-1){1.3}}\put(20.4,5.9){$a_4$}

\end{picture}
\end{center}

We have ${\rm Ext}_{\Lambda}^1(P_i,P_j) = 0$ for
all $i,j \in \{ 2,4 \}$.
This follows directly from the projectivity of both modules.
From this and the above pictures we get
%
\begin{eqnarray*}
C_{\beta_1} &=& \overline{{\mathcal O}(P_2)},\\ 
C_{\beta_2} &=& \overline{{\mathcal O}(P_4)},\\
C_\beta &=& \overline{C_{\beta_1} \oplus C_{\beta_2}}.
\end{eqnarray*}
%
In particular, $C_{\beta_1}$ and $C_{\beta_2}$ are indecomposable
irreducible components with $\mu_g(C_{\beta_1}) = \mu_g(C_{\beta_2}) = 0$.
This finishes the proof.
\end{proof}


For irreducible components $C \subseteq \Lambda({\bf d})$ and 
$D \subseteq \Lambda({\bf e})$ define
\[
{\mathcal V}(C,D) = 
\bigcap_{U \subseteq C, V \subseteq D} 
\left\{ E \subseteq \Lambda({\bf d}+{\bf e}) \text{ irred. comp.} \mid 
E \subseteq 
\overline{{\mathcal E}(U \times V)} \right\},
\]
where $U$ (resp. $V$) runs through all non-empty $\GL({\bf d})$-stable 
(resp. $\GL({\bf e})$-stable) open subsets of
$C$ (resp. $D$),
see Section \ref{section2} for the definition of ${\mathcal E}(U \times V)$.

Using the previous proposition, and some well-known results
on the representation theory of the algebra $\Lambda$,
see \cite{GSc} and \cite{Ri},
one can show that
\[
{\mathcal V}(C_\alpha,C_\alpha) = \{ C_{\alpha+\alpha},
C_\beta \}.
\]
Note that ${\rm ext}_{\Lambda}^1(C_\alpha,C_\alpha) = 0$,
since ${\rm Ext}_{\Lambda}^1(M_\lambda,M_\mu) = 0$ for
all $\lambda \not= \mu$. 
But one can show that 
${\rm dim}\, {\rm Ext}_{\Lambda}^1(M_\lambda,M_\lambda) = 2$, 
see \cite[Section 6]{GSc}.
For any $M_\lambda$ there is a short exact sequence
\[
0 \to M_\lambda \to M_\lambda(2) \to M_\lambda \to 0,
\]
where $M_\lambda(2)$ is the module of quasi-length two in the same 
Auslander-Reiten component as $M_\lambda$ (it is known that
$M_\lambda$ lies in a homogeneous tube).
Additionally to this `natural' self-extension,  
there exists a short exact sequence
\[
0 \to M_\lambda \to P_2 \oplus P_4 \to M_\lambda \to 0.
\]
Motivated by our above analysis, 
one might ask whether the following is true:


\vspace{0.2cm}\begin{Qu}
Are the following two statements equivalent?
\begin{itemize}

\item[(1)]
$\lambda_{C,D}^E \not= 0$;

\item[(2)] 
$E \in {\mathcal V}(C,D) \cup {\mathcal V}(D,C)$.

\end{itemize}
\end{Qu}\vspace{0.2cm}

A positive answer to the above question would imply the following:
%
\begin{itemize}

\item 
If an irreducible component $C$ contains an open orbit, then
\[
b^*(C)^2 = v^m b^*(\overline{C \oplus C})
\]
for some $m \in \Z$.
Here we use that ${\rm Ext}^1_{\Lambda}(M,M) = 0$ if and only if
${\mathcal O}(M)$ is an open orbit.
This is a special feature of preprojective algebras.

\item
If irreducible components $C$ and $D$ contain non-empty stable open
subsets $U \subseteq C$ and $V \subseteq D$ such that
${\rm Ext}_{\Lambda}^1(M,N) = 0$ for all $M \in U, N \in V$, then
$b^*(C)$ and $b^*(D)$ are multiplicative.
Here we use \cite[Theorem 1.3]{CBSc}.

\item
If an irreducible component $C$ contains an open orbit, and
if $D$ is an irreducible component such that
${\rm ext}_{\Lambda}^1(C,D) = 0$, then
$b^*(C)$ and $b^*(D)$ are multiplicative.
Again this uses \cite[Theorem 1.3]{CBSc}.

\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{CH}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{BZ}
{\it A. Berenstein, A. Zelevinsky}, 
String bases for quantum groups of type $A\sb r$. 
I.M. Gelfand Seminar, 51--89, 
Adv. Soviet Math. \textbf{16}, Part 1, 
Amer. Math. Soc., Providence, RI (1993). 

\bibitem{CBSc}
{\it W. Crawley-Boevey, J. Schr\"oer},
Irreducible components of varieties of modules.
Preprint arXiv:math.AG/0103100 (2001), 1--20.
To appear in J. Reine Angew. Math.

\bibitem{GSc}
{\it C. Gei{\ss}, J. Schr\"oer},
Varieties of modules over tubular algebras.
Preprint (2002), 1--20. To appear in Colloq. Math. 

\bibitem{KS}
{\it M. Kashiwara, Y. Saito}, 
Geometric construction of crystal bases.
Duke Math. J. \textbf{89} (1997), 9--36.

\bibitem{Le} 
{\it B. Leclerc},
Imaginary vectors in the dual canonical basis of $U_q(n)$. 
Preprint math.QA/0202148 (2002), 1--9. 

\bibitem{LNT}
{\it B. Leclerc, M. Nazarov, J.-Y. Thibon}, 
Induced representations of affine Hecke algebras and canonical bases 
of quantum groups.
Preprint arXiv:math.QA/0011074 (2000), 1--33. 

\bibitem{L1}
{\it G. Lusztig},
Canonical bases arising from quantized enveloping algebras. II. 
Common trends in mathematics and quantum field theories
(Kyoto, 1990). Progr. Theoret. Phys. Suppl. No. 102 (1990), 
175--201 (1991).

\bibitem{L2}
{\it G. Lusztig},
Quivers, perverse sheaves, and quantized enveloping algebras. 
J. Amer. Math. Soc. \textbf{4} (1991), no. 2, 365--421.

\bibitem{MR}
{\it R. Marsh, M. Reineke},
Private communication.
Bielefeld, February 2002.

\bibitem{R} 
{\it M. Reineke},
Multiplicative properties of dual canonical bases of quantum groups. 
J. Algebra \textbf{211} (1999), 134--149.
 
\bibitem{Ri}
{\it C.M. Ringel},
Tame algebras and integral quadratic forms. 
Lecture Notes in Mathematics \textbf{1099},
Springer-Verlag, Berlin (1984), xiii+376pp.

\bibitem{Ri2}
{\it C.M. Ringel},
The preprojective algebra of a quiver. 
Algebras and modules II (Geiranger, 1996), 467--480, CMS Conf. Proc.
\textbf{24}, Amer. Math. Soc., Providence, RI (1998). 


\end{thebibliography}


\end{document}

math0208209.extracted_bib
\begin{thebibliography}{CH}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{BZ}
{\it A. Berenstein, A. Zelevinsky}, 
String bases for quantum groups of type $A\sb r$. 
I.M. Gelfand Seminar, 51--89, 
Adv. Soviet Math. \textbf{16}, Part 1, 
Amer. Math. Soc., Providence, RI (1993). 

\bibitem{CBSc}
{\it W. Crawley-Boevey, J. Schr\"oer},
Irreducible components of varieties of modules.
Preprint arXiv:math.AG/0103100 (2001), 1--20.
To appear in J. Reine Angew. Math.

\bibitem{GSc}
{\it C. Gei{\ss}, J. Schr\"oer},
Varieties of modules over tubular algebras.
Preprint (2002), 1--20. To appear in Colloq. Math. 

\bibitem{KS}
{\it M. Kashiwara, Y. Saito}, 
Geometric construction of crystal bases.
Duke Math. J. \textbf{89} (1997), 9--36.

\bibitem{Le} 
{\it B. Leclerc},
Imaginary vectors in the dual canonical basis of $U_q(n)$. 
Preprint math.QA/0202148 (2002), 1--9. 

\bibitem{LNT}
{\it B. Leclerc, M. Nazarov, J.-Y. Thibon}, 
Induced representations of affine Hecke algebras and canonical bases 
of quantum groups.
Preprint arXiv:math.QA/0011074 (2000), 1--33. 

\bibitem{L1}
{\it G. Lusztig},
Canonical bases arising from quantized enveloping algebras. II. 
Common trends in mathematics and quantum field theories
(Kyoto, 1990). Progr. Theoret. Phys. Suppl. No. 102 (1990), 
175--201 (1991).

\bibitem{L2}
{\it G. Lusztig},
Quivers, perverse sheaves, and quantized enveloping algebras. 
J. Amer. Math. Soc. \textbf{4} (1991), no. 2, 365--421.

\bibitem{MR}
{\it R. Marsh, M. Reineke},
Private communication.
Bielefeld, February 2002.

\bibitem{R} 
{\it M. Reineke},
Multiplicative properties of dual canonical bases of quantum groups. 
J. Algebra \textbf{211} (1999), 134--149.
 
\bibitem{Ri}
{\it C.M. Ringel},
Tame algebras and integral quadratic forms. 
Lecture Notes in Mathematics \textbf{1099},
Springer-Verlag, Berlin (1984), xiii+376pp.

\bibitem{Ri2}
{\it C.M. Ringel},
The preprojective algebra of a quiver. 
Algebras and modules II (Geiranger, 1996), 467--480, CMS Conf. Proc.
\textbf{24}, Amer. Math. Soc., Providence, RI (1998). 


\end{thebibliography}