# Determinantal modules over preprojective algebras and representations of
Dynkin quivers

Yingjin Bi<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study extension groups of determinantal modules over a
preprojective algebra using the Auslander–Reiten translation of the quiver
associated with it. More precisely, based on the recent work given by Aizenbud
and Lapid, we calculate the extension group of a sort of so-called
determinantal modules, which is an analog of quantum minors in quantum
coordinate rings. In particular, we give an equivalent combinatorial condition
when the product of two quantum minors (up to $q$-power rescaling) belongs to
the dual canonical basis of quantum coordinate rings in the Dynkin case. More
generally, we can check the quasi-commuting condition for any two quantum
cluster monomials with the seeds of quantum minors.

###### Key words and phrases:

Representation theory, Quantum groups, Cluster algebras, Categorification

###### 2020 Mathematics Subject Classification:

17B37; 13F60; 18D20; 81R10

##  1\. Introduction

The present paper attempts to establish a bridge between the representation
theory of a Dynkin quiver $Q$ and the determinantal modules over the
corresponding preprojective algebra $\Lambda_{Q}$ using Auslander–Reiten
theory. The motivation of the paper stems from the cluster structure on
quantum coordinate rings $\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$ for
an element $w\in W$ in the Weyl group (see [10] for more details). In [10]
C.Geiß, B.Leclerc and J.Schröer gave this quantum cluster structure via a
module category $\mathcal{C}_{w}$ on the preprojective algebra $\Lambda_{Q}$.
The key notion in their works is a sort of so-called determinantal modules
over $\Lambda_{Q}$, this notion is an analog of quantum minors in
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$. As quantum minors in
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$ belong to the dual canonical
basis $\mathbf{B}^{*}(w)$ of
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$, our approach can apply to
some problems concerning the dual canonical bases. In particular, this allows
us to describe the quasi-commuting property of quantum cluster monomials with
the seeds of quantum minors.

Before discussing the above statement, let us first recall some well-known
conjectures on dual canonical bases. For two dual canonical bases
$\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ in $\mathbf{B}^{*}(w)$, there are many
conjectures about their product $\mathbf{b}_{1}\mathbf{b}_{2}$: For instance,
Leclerc proposed [16, Conjecture 1 and Conjecture 2] or Berenstein and
Zelevinsky proposed [3, Conjecture 1.7]. In this paper, we focus on the
following problem:

######  Problem 1.

For a Dynkin quiver $Q$, let $w_{0}$ be the unique Weyl group element of
maximal length. Let $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ be two dual
canonical basis elements in $\mathbf{B}^{*}$ of the quantum coordinate ring
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w_{0}))=\mathcal{A}_{q}(\operatorname{\mathfrak{n}})$.
We write $\lambda_{1}$ and $\lambda_{2}$ for the indexing of $\mathbf{b}_{1}$
and $\mathbf{b}_{2}$ respectively. (In this paper, we always consider the
Kostant partitions as the indexing set of dual canonical bases).

Describe the condition $\mathbf{b}_{1}\mathbf{b}_{2}\in
q^{\mathbb{Z}}\mathbf{B}^{*}$ in terms of the combinatorial conditions of
$(\lambda_{1},\lambda_{2})$

In [3], Berenstein and Zelevinsky conjecture that
$\mathbf{b}_{1}\mathbf{b}_{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}$ if and only if
$\mathbf{b}_{1}\mathbf{b}_{2}=q^{n}\mathbf{b}_{2}\mathbf{b}_{1}$ for some
integer $n$. However, Leclerc gave a counterexample to this conjecture and
then proposed the notion of _real_ dual canonical bases, e, t.
$\mathbf{b}^{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}$ and [16, Conjecture 1 and
Conjecture 2].

Classification of the real dual canonical bases has been a difficult open
problem until now. The well-known example is the kind of so-called quantum
minors, which play an important role in the cluster structure on
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$. Recently, in $A_{n}$ type,
Lapid and Minguez [13] classify the real dual canonical bases satisfying a so-
called regular condition. Even if we assume that
$\mathbf{b}_{1},\mathbf{b}_{2}$ are real dual canonical bases, problem 1 is
still difficult to answer (see [16]). In [11] Kang, Kashiwara, Kim and Oh
introduce the $R$-matrix on the category of modules over quiver Hecke algebras
(or over quantum affine algebras) and then describe the condition
$\mathbf{b}_{1}\mathbf{b}_{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}$ in terms of the
denominators of $R$-matrices. However, finding out the meaning of this
denominator is also a difficult problem (see [6]).

There are some results on this problem. In type $A_{n}$, for an integer pair
$(k,j)$, there exists a special Weyl element $w$ such that
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$ is isomorphic to the quantum
coordinate ring $\mathcal{A}_{q}(\mathrm{Mat}_{k\times j})$ of the space of
$k\times j$ matrices (see [10, Section 12.4] for more details). In this case,
if $\mathbf{b}_{1},\mathbf{b}_{2}$ are quantum minors, the condition
$\mathbf{b}_{1}\mathbf{b}_{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}$ is equivalent
to the quasi commuting property, e.t.,
$\mathbf{b}_{1}\mathbf{b}_{2}=q^{n}\mathbf{b}_{2}\mathbf{b}_{1}$ for some
integer $n$. In [15] Leclerc and Zelevinsky introduce a combinatorial notion
of weak separation to describe the quasi-commuting property, and then there is
some progress on quasi-commuting property of any two quantum minors, such as,
Scott’s work [19] and the recent work by Oh, Postnikov and Speyer [17]. Lapid
and Minguez consider this problem using the representations of general linear
groups over non-Archimedean fields,(see [12]). Recently, Li and Bittmann
generalized the notion of weak separation to the more general case of $A_{n}$,
see [2]. However, in the case of any Dynkin quiver with arbitrary orientation,
there are no related results on this problem.

In this paper, we consider the case when $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$
are quantum minors in $\mathcal{A}_{q}(\operatorname{\mathfrak{n}})$ and then
give an equivalent combinatorial condition of the Kostant partitions
$(\lambda_{1},\lambda_{2})$ to describe the condition
$\mathbf{b}_{1}\mathbf{b}_{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}$. More
generally, we can check the quasicommuting condition for any two quantum
cluster monomials with the seeds of quantum minors. We remark that the quantum
minors are dependent on the choice of the reduced expression of $w_{0}$. In
our concerning case, this reduced expression is given by the Coxeter element
$C_{Q}$.

Let us explain this more explicitly. The quantum minors correspond to the so-
called _determinantal_ modules over the preprojective algebra $\Lambda_{Q}$
(see [10]). The Kostant partitions corresponding to determinantal modules are
a sort of special partitions: They can be regarded as a sequence of
Auslander–Reiten translation $\tau$ orbit of an indecomposable representation
of $Q$. Surprisingly, they are similar to the objects in the cluster category
given by the derived category of representations of $Q$ (see [4]).

Based on the work by Geiss, Leclerc and Schröer in [10], the above problem 1
turns out to be a problem when
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})=0$ for
two determinantal modules $\widetilde{{M}}$ and $\widetilde{{N}}$ over
$\Lambda_{Q}$. It seems that it is easy to calculate this group using the
Homology methods of modules over $\Lambda_{Q}$. In fact, the Homology method
is useful only for some special cases of determental modules. (see [10]).

We have to use some new approaches to fix it. Based on Ringel’s work [18] and
Aizenbud and Lapid’s work [1], we calculate the
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})$ for
any two determinantal modules $\widetilde{{M}},\widetilde{{N}}$. More
precisely, in [18], Ringel described a module $\widetilde{{M}}$ over
$\Lambda_{Q}$ as $(M,a)$ where $M$ is the representation of $Q$ and
$a\in\operatorname{Hom}_{Q}({M},\tau{M})$ where $\tau M$ refers to the
Auslander–Reiten translation of $M$. In [1], Aizenbud and Lapid study
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})$ of
representations $\widetilde{{M}}=(M,a)$ and $\widetilde{{N}}=(N,b)$ using the
following map

(1.1)
$\begin{split}r_{a,b};\,\operatorname{Hom}_{Q}({M},{N})&\to\operatorname{Hom}_{Q}({M},\tau{N})\\\
f&\mapsto\tau(f)a-bf\end{split}$

######  Proposition 1.1 ([1, Proposition 9.1]).

For any two modules $\widetilde{{M}}=(M,a)$ and $\widetilde{{N}}=(N,b)$ of
$\Lambda_{Q}$, we have

$0\to\operatorname{Coker}{}r_{a,b}\to\operatorname{Ext}_{\Lambda_{Q}}^{1}(\widetilde{{M}},\widetilde{{N}})\to(\operatorname{Coker}{}r_{b,a})^{*}\to
0$

As a corollary,
$\operatorname{Ext}_{\Lambda_{Q}}^{1}(\widetilde{{M}},\widetilde{{N}})=0$
coincides with the condition that $r_{a,b}$ and $r_{b,a}$ are surjective.
However, it is too tough to calculate the map $r_{a,b}$ for all
$\widetilde{{M}}$ and $\widetilde{{N}}$. In [14, Section 2.4], Lapid and
Minguez give several approaches to calculate extension group
$\operatorname{Ext}_{\Lambda_{Q}}^{1}(\widetilde{{M}},\widetilde{{N}})$, such
as randomized algorithm, and they calculate extension groups of some special
modules.

Unlike their approaches, we interpret $a$ as a nilpotent $r\times r$ matrix
where $r$ is the number of indecomposable direct summands of $M$, Namely,
$M=\oplus_{k=1}^{r}M_{k}$ where each $M_{k}$ is indecomposable. This allows us
to consider the map $r_{a,b}$ as a Lie bracket of $a,b$ on the matrix space
$\mathrm{Mat}_{r\times s}$ where $s$ is the number of indecomposable direct
summands of $N$. This approach makes the map $r_{a,b}$ easier to calculate, as
we only consider the decomposition of representations of $Q$ to calculate the
extension group
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})$.

_Caution : For the composite map $pq:\,M_{1}\stackrel{{\scriptstyle
q}}{{\rightarrow}}M_{2}\stackrel{{\scriptstyle p}}{{\rightarrow}}M_{3}$, it
may be equal to $0$ even if $p,q\neq 0$, For instance, these maps form a
complex with three terms. Fortunately, we can avoid this situation in our
concerning cases._

Since any indecomposable representation $M_{i}$ corresponds to a positive root
$\lambda_{i}\in R^{+}$ by Gabriel’s Theorem. Hence, any representation $M$
gives rise to a root sequence
$({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{r})$ such that
$\lambda_{i}\geq\lambda_{j}$ for any $i<j$, and $M=\oplus_{i}M_{\lambda_{i}}$.
We call this sort root sequences for _Kostant Partitions_ and denote by
$\operatorname{KP}({\alpha})$ the set of Kostant partitions subject to
$\sum_{i}\lambda_{i}=\alpha$ for a dimensional vector $\alpha$. Therefore, the
map $r_{a,b}$ is uniquely determined by the Kostant partitions
$\lambda,\kappa$ associated with modules
$\widetilde{{M}}_{\lambda},\widetilde{{M}}_{\kappa}$, respectively. That means
we should find a combinatorial condition on the pair $(\lambda,\kappa)$ to
describe the condition
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$
and then the condition $\mathbf{b}_{1}\mathbf{b}_{2}\in
q^{\mathbb{Z}}\mathbf{B}^{*}$.

As we pointed out before, The Kostant partitions associated with quantum
minors in our case are of the following form

(1.2) $(\tau^{u}\beta,\tau^{u-1}\beta,\cdots,\tau\beta,\beta)$

where $\tau=C_{Q}$ is the Coxeter element that adapted with the orientation of
$Q$. This implies that the Auslander–Reiten translation of $M_{\beta}$
subjects to $\tau M_{\beta}=M_{\tau\beta}$, where $M_{\beta}$ refers to the
indecomposable representation corresponding to the root $\beta$.

Let $\lambda=(\tau^{u-1}\alpha,\cdots,\alpha)$ and
$\kappa=(\tau^{v-1}\beta,\cdots,\beta)$ be two $\tau$-orbit Kostant
partitions, We abbreviate them by
$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{u})$ and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{v})$. we set

(1.3)
$R(\lambda,\kappa)=\left\\{{i\in[1,v]}\mid\,{\operatorname{Hom}_{Q}({M_{\lambda_{1}}},{M_{\kappa_{i}}})\neq
0}\right\\}$

In what follows, we write $M_{\lambda}$ for the representation associated with
the Kostant partition $\lambda$ and $\widetilde{{M}}_{\lambda}$ for the module
$(M_{\lambda},a)$ such that $a$ is the dual of $M$ in the sense of Definition
2.2. For simplicity, we always write $[M_{\lambda},M_{\kappa}]$ (resp:
$[M_{\lambda},\tau M_{\kappa}]$) for
$\operatorname{dim}\operatorname{Hom}_{Q}({M_{\lambda}},{M_{\kappa}})$ (resp:
$\operatorname{dim}\operatorname{Hom}_{Q}({M_{\lambda}},\tau{M_{\kappa}})$).

We set

(1.4)
$\begin{split}&\operatorname{\mathfrak{m}}(\lambda,\kappa)=\sharp\left\\{{(i,j)}\mid\,{\lambda_{i}-\kappa_{j}\in
R\cup\\{0\\}\text{ and }\lambda_{i}\geq\kappa_{j}}\right\\}\\\
&\operatorname{\mathfrak{p}}(\lambda,\kappa)=\sharp\left\\{{(i,j)}\mid\,{\lambda_{i}+\kappa_{j}\in
R^{+}\text{ and }\lambda_{i}\geq\kappa_{j}}\right\\}\end{split}$

_Assumption : we require that_

(1.5) $\tau^{u+1}\alpha\in R^{+}$

_for any $\tau$-orbit Kostant partition
$\lambda=(\tau^{u}\alpha,\cdots,\tau\alpha,\alpha)$ in the present paper._

Here is one of our main results.

######  Theorem 1.2.

Let $\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{u})$ and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{v})$ subject to the above
assumption for some roots $\alpha,\beta$ and some integers $u,v$. We see that

$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$

if and only if

  1. (1)

If $u\geq v$

$\displaystyle\sharp R(\lambda,\kappa)$
$\displaystyle=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp R(\kappa,\lambda)\cap[u-v+1,u]$
$\displaystyle=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

  2. (2)

If $u<v$

$\displaystyle\sharp
R(\lambda,\kappa)\cap[v-u+1,v]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp
R(\kappa,\lambda)=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

This theorem seems new to most readers. This precisely shows that the map
$r_{a,b}$ is too difficult to calculate in general. Here are some corollary to
reprove some well-known results.

######  Corollary 1.3 ([9]).

For any $\tau$-orbit Kostant partition $\lambda$ we have that
$\widetilde{{M}}_{\lambda}$ is a rigid module over $\Lambda_{Q}$. Namely,
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\lambda}})=0$.

######  Corollary 1.4.

Given two $\tau$-orbit Kostant partition
$\lambda=(\tau^{u}\beta,\cdots,\beta)$ and
$\kappa=(\tau^{u}\alpha,\cdots,\alpha)$ such that $\beta=\tau^{k}\theta_{a}$
and $\alpha=\tau^{k}\theta_{b}$ for some integer $k$ and two different roots
$\theta_{a},\theta_{b}$. We have

(1.6)
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$

Here $\theta_{a}$ and $\theta_{b}$ refers to (2.4).

######  Corollary 1.5 ([9]).

For any $\tau$-orbit Kostant partition $\lambda$, we have that

$\operatorname{dim}\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\tau\lambda}})=1$

Here $\tau\lambda=(\tau^{u+1}\beta,\tau^{u}\beta,\cdots,\tau\beta)$

######  Remark 1.6.

If we consider $\lambda=(\tau^{u}\beta,\cdots,\beta)$ as the Kostant partition
associated with quantum minor $D(a,b)$ for some $a,b$. The first corollary
means $D(a,b)^{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}$; the second corollary means
the formula [10, Lemma 5.2]

(1.7)
$D(\mu\omega_{i_{a}},\nu\omega_{i_{a}})D(\mu\omega_{i_{b}},\nu\omega_{i_{b}})=q^{n}D(\mu\omega_{i_{b}},\nu\omega_{i_{b}})D(\mu\omega_{i_{a}},\nu\omega_{i_{a}})$

for some integer $n$.

The third corollary is an interpretation of the following formula. [10,
Proposition 5.5]

(1.8) $q^{A}D(a,b)D(a^{-},b^{-})=q^{B}D(a,b^{-})D(a^{+},b)+q^{C}\prod_{j\neq
i}D(b^{-}(j),d^{-}(j))$

where $A,B,C$ refer to some integers as in [10, Proposition 5.5]. The third
corollary only shows that $D(a,b)D(a^{-},b^{-})\notin
q^{\mathbb{Z}}\mathbf{B}^{*}$.

For a Dynkin quiver, one defines a Coxeter element
$C_{Q}={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{n}}$ adapted with $Q$. Since Dynkin
quivers are bipartite orientations, e,t. $I=I_{0}\cup I_{1}$, we have that
$w_{0}=c_{Q}^{h/2}$ if the Coxeter element $h$ is even or
$w_{0}=c_{Q}^{\frac{h-1}{2}}c_{1}$ where $c_{1}$ is the $I_{1}$ part of
$c_{Q}$ if $h$ is odd. It gives rise to a word of $w_{0}$ as follows.

(1.9) $\mathbf{i}=[i_{1}i_{2}\cdots i_{n}i_{1}\cdots i_{n}\cdots i_{1}\cdots
i_{n}\cdots i_{k}]$

We remark that $\mathbf{i}$ is adapted to $Q$. We always write $\tau$ for the
Coxeter element $C_{Q}$.

Here is our main theorem.

######  Theorem 1.7.

Let $D(b,d)$ and $D(k,l)$ be two quantum minors induced from the above word
$\mathbf{i}$, let us write
$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{u})$ and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{v})$ for the $\tau$-orbit
Kostant partitions for $D(b,d)$ and $D(k,l)$, respectively.

(1.10) $D(b,d)D(k,l)\in q^{\mathbb{Z}}\mathbf{B}^{*}$

if and only if they satisfy the following condition

  1. (1)

if $u\geq v$

$\displaystyle\sharp R(\lambda,\kappa)$
$\displaystyle=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp R(\kappa,\lambda)\cap[u-v+1,u]$
$\displaystyle=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

  2. (2)

if $u<v$

$\displaystyle\sharp
R(\lambda,\kappa)\cap[v-u+1,v]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp
R(\kappa,\lambda)=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

This phenomenon bears resemblance to the $R$-matrices $R_{D(b,d),D(k,l)}$ as
discussed in [11]. The theorem’s advantage lies in its ability to easily
calculate all quantum minors. In fact, if we interpret the quantum minors in
terms of Kirillov-Reshetikhin modules over quantum affine groups, we can check
condition (1.10) by examining the denominators of the $R$-matrix on Kirillov-
Reshetikhin modules (see [7]). However, it’s important to note that the
quantum minors related to KR modules constitute a specific class of quantum
minors. Our results, on the other hand, are more general as they encompass a
broader range of choices for the orientations of quivers.

More generally, we have the following theorem.

######  Theorem 1.8.

Let $\mathbf{a}=({a}_{1},{a}_{2},\cdots,{a}_{r})$ and
$\mathbf{b}=({b}_{1},{b}_{2},\cdots,{b}_{t})$ be two tuples of nonnegative
integers, and let $Z_{1}=X^{\mathbf{a}}$ and $Z_{2}=Y^{\mathbf{b}}$ be two
quantum cluster monomials in $\mathcal{A}_{q}(\operatorname{\mathfrak{n}})$
with respect to the clusters $({X}_{1},{X}_{2},\cdots,{X}_{r})$ and
$({Y}_{1},{Y}_{2},\cdots,{Y}_{r})$ respectively such that all $X_{i}$ and
$Y_{j}$ are quantum minors and without frozen variables. The following holds

(1.11) $Z_{1}Z_{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}\quad\text{ (equivalently
$Z_{1}Z_{2}=q^{n}Z_{2}Z_{1}$ for some $n\in\mathbb{Z}$)}$

if and only if for any pair $(a_{i},b_{j})$ such that $a_{i}\neq 0$ and
$b_{j}\neq 0$, we have the quantum minors $X_{i},Y_{j}$ satisfies the
condition in Theorem (1.7).

Let’s provide a brief explanation of the structure of this paper. In Section
2, we will define dual representations of $Q$. While this notion may be
familiar to experts in related fields, we will provide a precise definition by
considering the action of automorphism groups on the extension groups. In
Section 3, we will calculate the map $r_{a,b}$ using the results presented in
Section 2. Section 4 will serve as a reminder of the cluster category over
preprojective algebras, and we will use our approach from Section 3 to prove
some well-known results about determinantal modules. Finally, in Section 5, we
will review the cluster structure on the quantum coordinate ring and establish
an equivalent combinatorial condition that holds when the product of any two
quantum minors is a dual canonical base (up to q-power rescaling).

### Notions and conventions

In this paper, a quiver $Q=(I,\Omega)$ is defined as a set of vertices $I$ and
a set of arrows $\Omega$. To avoid confusion, let $n$ denote the number of
vertices. For an arrow $h$, its source and target are denoted by $s(h)$ and
$t(h)$ respectively. For our purposes, we assume that $Q$ is a Dynkin quiver.
The quiver $Q$ gives rise to a symmetric matrix $C_{Q}$, which we will denote
as $C$, by

(1.12) $a_{i,j}=\begin{cases}2&\text{if $i=j$}\\\ -\sharp\\{h:i\to
j\\}-\sharp\\{h:j\to i\\}&\text{otherwise}\end{cases}$

We denote $\alpha_{i}$ as the simple root for the vertex $i\in I$. Let
$Q=\mathbb{Z}[\alpha_{i}]_{i\in I}$ represent its root lattice. The matrix $C$
generates a bilinear form on $Q$, denoted by $(-,-)$. $Q^{+}\subset Q$ refers
to the subset $\mathbb{N}[\alpha_{i}]_{i\in I}$ of $Q$. Define a bilinear form
on $Q^{+}$ by

(1.13) $\langle{\alpha},{\beta}\rangle=\sum_{i\in
I}\alpha_{i}\beta_{i}-\sum_{h\in\Omega}\alpha_{s(h)}\beta_{t(h)}$

It is well known that
$({\alpha,\beta})=\langle{\alpha},{\beta}\rangle+\langle{\beta},{\alpha}\rangle$.
We write $\omega_{i}$ for the fundamental weights and let
$P^{+}=\mathbb{N}[\omega_{i}]_{i\in I}$ be dominant weights. We write $W$ for
its Weyl group generated by simple reflections $s_{i}$ associated with simple
roots $\alpha_{i}$ for any $i\in I$. Denote by $R^{+}$ the set of positive
roots.

Let us fix an orientation of $Q$ such that the indexing set of $I$ satisfying

(1.14) $i_{k}\to i_{j}\text{ then }k<j$

The ground field is assumed to be the complex number field $\mathbb{C}$, we
sometimes write $\operatorname{\mathbf{k}}$ for simplicity. For an element
$\alpha\in Q^{+}$ let $V$ be an $I$-graded space such that

$\underline{\operatorname{dim}}V=\sum_{i_{k}\in
I}\operatorname{dim}V_{i_{k}}\alpha_{i_{k}}=\alpha$

We denote by $G_{\alpha}$ the group $\prod_{i\in I}{\operatorname{GL}}(V_{i})$
and by $E_{\alpha}$ the representation space.

(1.15)
$\bigoplus_{h\in\Omega}\operatorname{Hom}_{\operatorname{\mathbf{k}}}(V_{s(h)},V_{t(h)})$

endowed with a $G_{\alpha}$ action given by $g\cdot
x=(g_{t(h)}x_{h}g_{s(h)}^{-1})_{h\in\Omega}$. We denote by
$\operatorname{Hom}_{I}(V,W)$ the space $\bigoplus_{i\in
I}\operatorname{Hom}_{\operatorname{\mathbf{k}}}(V_{i},W_{i})$ and by
$\operatorname{Hom}_{\Omega}(V,W)$ the space
$\bigoplus_{h\in\Omega}\operatorname{Hom}_{\operatorname{\mathbf{k}}}(V_{s(h)},W_{t(h)})$,
respectively.

### Acknowledge

The author expresses deep gratitude to Professor Cerulli Irelli for his
invaluable help and guidance throughout this paper. The work was completed
during the author’s visit to the University of Sapienza, and the invitation
extended by Professor Cerulli Irelli is highly appreciated. The author also
extends thanks to Professor Qin Fan for their insightful discussions.
Additionally, the author would like to acknowledge Professor Bernard Leclerc
for informing him about the works of Professors Lapid and Minguez and
providing updates on this topic. Special thanks are also owed to Professor
Jian-Rong Li for bringing attention to grammatical errors in this paper and
sharing their recent work. The author acknowledges the financial support
provided by the China Scholarships Council (CSC), Grant No.202006040123.

##  2\. Representation theory of Dynkin quivers

In this paper, we will use representation theory to study Lusztig’s nilpotent
varieties. We will construct some maps related to Lusztig’s nilpotent
varieties. Unlike the viewpoint from the symmetric geometry, we adopt an
approach of representation theory to consider these varieties. In particular,
we try to use the Auslander-Retein translation to analyze Lusztig’s nilpotent
varieties.

Fix a Dynkin quiver $Q$. Let $M=(V,x)$ be a representation of $Q$ where $V$ is
an $I$-graded vector space and $x=(x_{h})_{h\in\Omega}$ is a tuple of matrices
$x_{h}:\,V_{s(h)}\to V_{t(h)}$ for all arrows $h\in\Omega$. The matrices $x$
can be used to represent the representation $M$ without causing confusion. For
a representation $M=(V,x)$, the dimension $\underline{\operatorname{dim}}M$ is
defined as $\sum_{i\in I}\operatorname{dim}(V_{i})\alpha_{i}\in Q^{+}$. Given
two representations $M,N$ of $Q$, $\operatorname{Hom}_{Q}(M,N)$ denotes the
vector space of $Q$-morphisms between $M$ and $N$, and $[M,N]$ (resp:
$[M,N]^{1}$) represents $\operatorname{dim}\operatorname{Hom}_{Q}(M,N)$ (resp:
$\operatorname{dim}\operatorname{Ext}^{1}_{Q}(M,N)$). One has

(2.1)
$\displaystyle\langle\underline{\operatorname{dim}}M,\underline{\operatorname{dim}}N\rangle=[M,N]-[M,N]^{1}$

For any vertex $i\in I$, we associate it with the indecomposable projective
(resp: injective) representation $P_{i}$ (resp: $I_{i}$), and the head (resp:
socle) of $P_{i}$ (resp: $I_{i}$) is the simple module $S_{i}$.

###  2.1. Representations of Dynkin quivers

In this section, we will review the relationship between the root system and
the representations of the Dynkin quiver Q. Let $w\in W$ be an element of the
Weyl group $W$. By considering a reduced expression of $w$ as
$w={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{m}}$, we can obtain a corresponding
word $\mathbf{i}=[{i}_{1}{i}_{2}\cdots{i}_{m}]$. This will then give us a set
of roots in the following manner:

(2.2) $\alpha_{i_{1}};\ s_{i_{1}}(\alpha_{i_{2}});\ \cdots\
{s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{k-1}}(\alpha_{i_{k}});\cdots\
;{s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{m-1}}(\alpha_{i_{m}})$

Let us denote them by
$\beta_{k}={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{k-1}}(\alpha_{i_{k}})$ for
$1\leq k\leq m$ and give them an ordering so that $\beta_{l}<\beta_{k}$ if and
only if $l<k$. we write $w=w_{0}$ for the unique element in $W$ with maximal
length and $C={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{n}}$ for the Coxeter element
of $W$. Denote by $h$ the Coxeter number.

From the given orientation $Q$ described in equation (1.14), we can define a
Coxeter element $C_{Q}={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{n}}$. It is easy to
see that $i_{1}$ is a source and $i_{n}$ is a sink. Since Dynkin quivers are
bipartite orientations, denoted as $I=I_{0}\cup I_{1}$, we can deduce that
$w_{0}=C_{Q}^{h/2}$ if $h$ is even, or $w_{0}=C_{Q}^{\frac{h-1}{2}}C_{1}$
where $C_{1}$ represents the $I_{1}$ portion of $C_{Q}$ if $h$ is odd. It
gives rise to a word of $w_{0}$ as

(2.3) $\mathbf{i}=[i_{1}i_{2}\cdots i_{n}i_{1}\cdots i_{n}\cdots i_{1}\cdots
i_{n}\cdots i_{k}]$

It is easy to see that $\mathbf{i}$ is adapted to $Q$.

Denote by $\theta_{k}$ the root
${s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{k-1}}(\alpha_{i_{k}})$ for any $i_{k}\in
I$. It is easy to see that

(2.4) $\\{\theta_{k}\\}_{k\in[1,n]}=\left\\{{\alpha\in
R^{+}}\mid\,{C^{-1}_{Q}(\alpha)\in R^{-}}\right\\}$

From the above word (2.3) we deduce that the roots $\beta_{l}$ are identified
with $C_{Q}^{r}(\theta_{k})$ for some $r\in\mathbb{N}$.

By Gabriel’s Theorem, the dimension vector of each indecomposable module is
exactly a positive root. We denote by $M_{\beta}$ the module corresponding to
the root $\beta$. If the order of positive roots is given as in (2.2) for the
word (2.3), one has

(2.5)
$\displaystyle\operatorname{Hom}_{Q}(M_{\alpha},M_{\beta})=\operatorname{Ext}^{1}_{Q}(M_{\beta},M_{\alpha})=0\qquad\text{for
$\alpha<\beta$}$

For simplicity, we sometimes write $[\beta_{k},\beta_{l}]$ (resp:
$[\beta_{k},\beta_{l}]^{1}$) for
$\operatorname{dim}\operatorname{Hom}(M_{\beta_{k}},M_{\beta_{l}})$ (resp:
$\operatorname{dim}\operatorname{Ext}_{Q}^{1}(M_{\beta_{k}},M_{\beta_{l}})$).

Since any representation $M$ with $\underline{\operatorname{dim}}M=\alpha$ can
be decomposed as a direct sum of indecomposable representations, we associate
each $M$ with a sequence of roots
$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{r})$ such that
$M\cong\oplus_{k=1}^{r}M_{\lambda_{i}}$ and
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{s}$. Therefore, one can
define the _Kostant partition_ $\lambda$ by

$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{s})$

where $\lambda_{k}\in R^{+}$ such that
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{s}$. Denote by
$\operatorname{KP}({\alpha})$ the set of Kostant partitions that satisfy
$\sum_{i=1}^{s}\lambda_{i}=\alpha$. It is straight to see that the set of
$G_{\alpha}$-orbits in $E_{\alpha}$ (isomorphic classes of representations of
$Q$ with dimensional vector $\alpha$) coincides with
$\operatorname{KP}({\alpha})$ (see (1.15)).

There is an order on representations with dimensional vector $\alpha$ (up to
isomorphism) given by $M\leq N$ if
$\mathbb{O}_{N}\subset\overline{\mathbb{O}_{M}}$ where $M,N\in E_{\alpha}$,
$\mathbb{O}_{N}$ is the $G_{\alpha}$ orbit of $N$, and
$\overline{\mathbb{O}_{M}}$ is the Zariski closure of orbit $\mathbb{O}_{M}$.
Therefore, we define an ordering on $\operatorname{KP}({\alpha})$ so that
$\lambda^{\prime}\leq\lambda$ if their associated representations satisfy
$M_{\lambda^{\prime}}\leq M_{\lambda}$. That is:

(2.6)
$\lambda^{\prime}\leq\lambda\Longleftrightarrow\mathbb{O}_{M_{\lambda}}\subset\overline{\mathbb{O}}_{M_{\lambda^{\prime}}}$

Following [5], we have

(2.7)
$\lambda^{\prime}\leq\lambda\Longrightarrow[M_{\lambda^{\prime}},M_{\lambda^{\prime}}]^{1}\leq[M_{\lambda},M_{\lambda}]^{1}$

###  2.2. An example of $A_{n}$

Let us consider the type $A_{n}$, we denote by $[1,n]$ the set of integers $k$
such that $1\leq k\leq n$. We fix an orientation of $A_{n}$ as follows:

(2.8) $1\longrightarrow 2\longrightarrow\cdots\longrightarrow n$

The Coxeter element $C_{Q}$ is given by
$\tau=C_{Q}:={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{n}}$. Induces a reduced
expression of $w_{0}$ as follows.

$w_{0}=s_{1}s_{2}s_{3}\cdots s_{n}s_{1}s_{2}\cdots s_{n}s_{1}\cdots
s_{n}\cdots s_{k}$

Thus, we obtain a word

(2.9) $\\{1,2,\cdots,n,1,2,\cdots,n,\cdots,1,2,\cdots,n,\cdots,k\\}$

It is well-known that the set $R^{+}$ of positive roots is identified with the
set of segments $[a,b]$ such that $a\leq b$. It is easy to check

(2.10) $\tau([k,l])=C_{Q}([k,l])=[k+1,l+1]\text{ if $l\leq n-1$}$

We write $\beta[k]$ for the root $\tau^{k}\beta$.

The order of $R^{+}$ is given by the following lemma.

######  Lemma 2.1.

The ordering of segments as in (2.2) is given by

$[i,j]<[k,l]\ \text{if $i<k$ or $i=k$ and $j<l$}$

###### Proof.

By Definition (2.4), we get

$\theta_{k}=[1,k]\qquad\text{ for $k\in[1,n]$}$

For any $i$ such that $k+i\leq n$, it follows from (2.10) that
$\tau^{i}\theta_{k}=[i+1,k+i]$. The order of $R^{+}$ in (2.2) and the word
(2.9) implies that $\tau^{i}\theta_{k}<\tau^{j}\theta_{l}$ if and only if
$i<j$ or $i=j$ and $k<l$. In other words, $[i+1,i+k]<[j+1,j+l]$ if and only if
$i<j$ or $i=j$ and $i+k<j+l$. ∎

The Gabriel’s Theorem implies that any indecomposable representations of
$A_{n}$ is of the form $M[a,b]$ such that
$\underline{\operatorname{dim}}{}M[a,b]=\sum\limits_{a\leq i\leq
b}\alpha_{i}$. We list some properties of indecomposable representations as
follows.

(2.11) $\begin{split}[M[i,j],M[k,l]]=1\ &\text{if and only if $k\leq i\leq
l\leq j$}\\\ [M[k,l],M[i,j]]^{1}=1\ &\text{if and only if $k+1\leq i\leq
l+1\leq j$}.\end{split}$

Moreover, for each $[M[k,l],M[i,j]]^{1}=1$, we have the following short exact
sequence

(2.12) $0\to M[i,j]\to M[i,l]\oplus M[k,j]\to M[k,l]\to 0$

where we formally set $M[i,j]=0$ if $i<1$ or $j>n$ or $j<i$.

For each representation $M$ of $A_{n}$, one can decompose every representation

$M\cong M[a_{1},b_{1}]\oplus M[a_{2},b_{2}]\oplus M[a_{m},b_{m}]$

for some $m\geq 1$. We arrange this tuple
$([a_{1},b_{1}],[a_{2},b_{2}],\cdots[a_{m},b_{m}])$ such that
$[a_{k},b_{k}]\geq[a_{k+1},b_{k+1}]$ for any $k\in[1,m-1]$, which is called
_multisegment_. It follows that for any $\alpha\in Q^{+}$, the set of the
Kostant partitions $\operatorname{KP}({\alpha})$ is identified with the set of
multisegments $([a_{1},b_{1}],\cdots,[a_{m},b_{m}])$ such that
$\sum_{i=1}^{m}[a_{k},b_{m}]=\alpha$.

It is straightforward to see any two roots $\alpha=[a,b],\beta=[c,d]$ with the
same length: e,t. $b-a=d-c$, have the relation $\alpha=\beta[k]$ for some
integer $k$. This means that two segments $\alpha=[a,b],\beta=[c,d]$ are in
the same $\tau$-orbit: e,t. $\alpha=\tau^{k}\beta$ for some $k\in\mathbb{Z}$,
if and only if their length are same. It implies that
$\beta_{[j,k]}=\\{\beta_{k},\beta_{k}[-1],\cdots,\beta_{j}[1],\beta_{j}\\}$.

###  2.3. Extension spaces of representations

Given two representations $M=(V,x)$ and $N=(W,y)$, one has an exact sequence

(2.13)
$\begin{split}0\to\operatorname{Hom}_{Q}(M,N)\to\operatorname{Hom}_{I}(V,W)\stackrel{{\scriptstyle
p_{x,y}}}{{\rightarrow}}\operatorname{Hom}_{\Omega}(V,W)\stackrel{{\scriptstyle
e_{x,y}}}{{\rightarrow}}\operatorname{Ext}^{1}_{Q}(M,N)\to 0\end{split}$

where $p_{x,y}$ is $p_{x,y}(f)=(f_{t(h)}x_{h}-y_{h}f_{s(h)})_{h\in\Omega}$.
Recall that for any $\overline{\eta}\in\operatorname{Ext}^{1}_{Q}(M,N)$, there
is a short exact sequence associated with it.

(2.14) $\overline{\eta}:\,0\to N\to E_{\eta}\to M\to 0$

where $E_{\eta}$ refers to the block as $\begin{pmatrix}y&\eta\\\
0&x\end{pmatrix}$ where $\eta$ is an element of
$e_{x,y}^{-1}(\overline{\eta})$. Note that $E_{\eta}$ is independent of the
choice of $\eta\in e_{x,y}^{-1}(\overline{\eta})$ up to isomorphism.

Let us consider the $\operatorname{End}_{Q}(M)$-action on
$\operatorname{Hom}_{\Omega}(V,W)$ by $\eta\cdot f=(\eta_{h}f_{s(h)})$. This
action gives rise to an $\operatorname{End}_{Q}(M)$-action on
$\operatorname{Ext}^{1}_{Q}(M,N)$. To this end, we should check that for each
$\eta\in\operatorname{Im}p_{x,y}$ and $f\in\operatorname{End}_{Q}(M)$, we have
$\eta\cdot f\in\operatorname{Im}p_{x,y}$. Considering
$\eta=(g_{t(h)}x_{h}-y_{h}g_{s(h)})_{h\in\Omega}$ for some
$g\in\operatorname{Hom}_{I}(V,W)$ and the following equation

$\displaystyle\eta\cdot f=$
$\displaystyle(g_{t(h)}x_{h}f_{s(h)}-y_{h}g_{s(h)}f_{s(h)})_{h\in\Omega}$
$\displaystyle=$
$\displaystyle(g_{t(h)}f_{t(h)}x_{h}-y_{h}g_{s(h)}f_{s(h)})_{h\in\Omega}\quad\text{
by $x_{h}f_{s(h)}=f_{t(h)}x_{h}$ for any $f\in\operatorname{End}_{Q}(M)$}$
$\displaystyle=$ $\displaystyle p_{x,y}(gf)$

we obtain our conclusion. That is, $\bar{\eta}\cdot
f=\overline{(\eta_{h}f_{s(h)})_{h\in\Omega}}$ for an element $\eta$ of
$e_{x,y}^{-1}(\bar{\eta})$. It follows that $\operatorname{Ext}^{1}_{Q}(M,N)$
is a module over $\operatorname{End}_{Q}(M)^{op}$. Similarly,
$\operatorname{Ext}^{1}_{Q}(M,N)$ is a module of $\operatorname{End}_{Q}(N)$
via $g\cdot\bar{\eta}=\overline{(g_{t(h)}\eta_{h})_{h\in\Omega}}$ for an
element $\eta$ of $e_{x,y}^{-1}(\bar{\eta})$.

###  2.4. The action of automorphism groups on extension spaces

Let us fix a representation $M=(V,x)$ with dimension vector $\alpha\in Q^{+}$.
Let us consider its self-extension group
$\operatorname{Ext}_{Q}^{1}({M},{M})$. We denote by
$\operatorname{Aut}_{Q}({M})$ the isomorphism group of $M$ in
$\operatorname{End}_{Q}(M)$, which is an open subvariety of
$\operatorname{End}_{Q}(M)$ in the Zariski topology, as it is defined by the
equation that the determinants of elements of $\operatorname{End}_{Q}(M)$ is
not equal to zero. We define an $\operatorname{Aut}_{Q}({M})$ action on
$\operatorname{Ext}^{1}_{Q}(M,M)$ by
$g\star\eta=(g_{t(h)}\eta_{h}g_{s(h)}^{-1})_{h\in\Omega}$. If we consider
$\operatorname{Aut}_{Q}({M})$ as a subset of $G_{\alpha}$, the discussion as
before implies that this action is compatible with the $G_{\alpha}$ on
$E_{\alpha}$. In other words,

(2.15) $g\star\bar{\eta}=\overline{g\cdot\eta}\text{
$\forall\eta\in\operatorname{Hom}_{\Omega}(V,V)$ and $\forall
g\in\operatorname{Aut}_{Q}({M})$}$

We denote by $\mathbf{O}_{\eta}$ the $\operatorname{Aut}_{Q}({M})$-orbit of
$\overline{\eta}\in\operatorname{Ext}^{1}_{Q}(M,M)$. Note that here
$\operatorname{Ext}_{Q}^{1}({M},{M})$ is a quotient space of
$\operatorname{Hom}_{\Omega}(V,V)$, it is not easy to understand these orbits.
Therefore, we consider the dual spaces of
$\operatorname{Hom}_{\Omega}(V,V)^{*}$.

Applying functor $D:=\operatorname{Hom}_{k}(-,k)$ to exact sequence (2.13), we
obtain

(2.16) $\begin{split}0\to
D\operatorname{Ext}^{1}_{Q}(M,N)\stackrel{{\scriptstyle
e^{*}_{x,y}}}{{\rightarrow}}\operatorname{Hom}_{\Omega^{op}}(W,V)\stackrel{{\scriptstyle
p_{x,y}^{*}}}{{\rightarrow}}\operatorname{Hom}_{I}(W,V)\to
D\operatorname{Hom}_{Q}(M,N)\to 0\end{split}$

where $p_{x,y}^{*}$ is the map
$p_{x,y}^{*}(z)=(\sum\limits_{t(h)=i}x_{h}z_{h}-\sum\limits_{s(h)=i}z_{h}y_{h})_{i\in
I}$. Because we can consider $p_{x,y}$ as the operation $fx-yf$ on the space
$\operatorname{Hom}_{k}(V,W)$ after forgetting the $I$-graded structure and
$\operatorname{Hom}_{\Omega}(V,W),\operatorname{Hom}_{I}(V,W)$ can be
considered as subspaces of $\operatorname{Hom}_{k}(V,W)$, the formula of
$p_{x,y}^{*}$ follows from the standard fact.

We denote by $\eta^{t}\in D\operatorname{Ext}_{Q}^{1}({M},{N})$ the dual
object of $\eta\in\operatorname{Ext}^{1}_{Q}(M,N)$. $\eta^{t}$ is the
transpose of matrices of $\eta$ on a given basis. Taking into account the
formula (2.16) and replacing $N$ by $M$, we obtain a representation
$\eta^{t}\in E_{\alpha}^{*}$ for each $\eta\in E_{\alpha}$. The
$G_{\alpha}$-action is compatible with the transpose of matrices. It follows
that $\eta\leq\eta^{\prime}$ if and only if $\eta^{t}\leq\eta^{\prime t}$.

######  Definition 2.2.

Let $M=(V,x)$ be a representation of $Q$. We define the dual representation of
$M$ by

$M^{*}=\left\\{{\eta\in
D\operatorname{Ext}_{Q}^{1}({M},{M})}\mid\,{\eta\leq\eta^{\prime}\quad\forall\eta^{\prime}\in
D\operatorname{Ext}_{Q}^{1}({M},{M})}\right\\}$

Because $D\operatorname{Ext}_{Q}^{1}({M},{M})$ is an irreducible variety and

$D\operatorname{Ext}_{Q}^{1}({M},{M})=\bigcup_{\eta}\overline{\mathbb{O}_{\eta}}\cap
D\operatorname{Ext}_{Q}^{1}({M},{M})$

we get that there is only one unique $\mathbb{O}_{\eta}$ such that
$\overline{\mathbb{O}_{\eta}}\cap
D\operatorname{Ext}_{Q}^{1}({M},{M})=D\operatorname{Ext}_{Q}^{1}({M},{M})$. In
other words, $M^{*}$ is unique up to isomorphism.

##  3\. Auslander–Reiten Theory

In this section, we will recall the Auslander–Reiten theory for Dynkin
quivers. As we discussed before, the action $\operatorname{Aut}_{Q}({M})$ on
the extension space $\operatorname{Ext}_{Q}^{1}({M},{N})$ turns out to be the
action $\operatorname{Aut}_{Q}({M})$ on the space
$D\operatorname{Ext}_{Q}^{1}({M},{N})$. The advantage of the space
$D\operatorname{Ext}_{Q}^{1}({M},{N})$ is that we can consider it as a
subspace of $\operatorname{Hom}_{\Omega}(V,W)$, which is easier than the
quotient space $\operatorname{Ext}_{Q}^{1}({M},{N})$. To describe the space
$D\operatorname{Ext}_{Q}^{1}({M},{N})$, we introduce the notion of
Auslander–Reiten translation for Dynkin quivers.

###  3.1. Auslander–Reiten translation

For any representation $M$, let us consider its minimal projective resolution

$0\to P_{1}\stackrel{{\scriptstyle p_{1}}}{{\longrightarrow}}P_{0}\to M\to 0$

Applying the functor $(-)^{t}:=\operatorname{Hom}_{Q}(-,kQ)$, we obtain

(3.1) $0\to M^{t}\to P_{0}^{t}\stackrel{{\scriptstyle
p_{1}^{t}}}{{\longrightarrow}}P_{1}^{t}\to\operatorname{Coker}p_{1}^{t}\to 0$

then applying $D=\operatorname{Hom}_{k}(-,k)$ to the above exact sequence, we
obtain

$0\to D\operatorname{Coker}p_{1}^{t}\to DP_{1}^{t}\to DP_{0}^{t}\to DM^{t}\to
0$

We call $D\operatorname{Coker}p_{1}^{t}$ the _Auslander–Reiten translation_ of
$M$, which is denoted by $\tau M=(\tau V,\tau x)$. It is easy to see that
$\tau M=D\operatorname{Ext}^{1}_{Q}(M,kQ)$. We list some well-known properties
of Auslander–Reiten translation as follows.

######  Proposition 3.1.

The following propositions are not only true in the Dynkin cases.

  1. (1)

In acyclic cases, we have $\tau M=\Phi_{C_{Q}}M$, where $\Phi_{C_{Q}}$ refers
to the reflection functor corresponding to the Coxeter element $C_{Q}$ (see
Section 2.1). In particular, $\underline{\operatorname{dim}}\tau
V=C_{Q}(\underline{\operatorname{dim}}V)$ as an element in $Q^{+}$ if
$C_{Q}(\underline{\operatorname{dim}}V)\in Q^{+}$.

  2. (2)

In acyclic cases, we have

(3.2) $\begin{split}\operatorname{Hom}_{Q}(M,N)\cong
D\operatorname{Ext}^{1}_{Q}(N,\tau M)\cong
D\operatorname{Ext}^{1}_{Q}(\tau^{-1}N,M)\\\ \operatorname{Hom}_{Q}(M,\tau
N)\cong D\operatorname{Ext}^{1}_{Q}(N,M)\cong D\operatorname{Ext}^{1}_{Q}(\tau
N,\tau M)\end{split}$

  3. (3)

If $M,N$ have no direct summand of projective representations, we have

$\operatorname{Hom}_{Q}(M,N)\cong\operatorname{Hom}_{Q}(\tau M,\tau N);\
\operatorname{Ext}^{1}_{Q}(M,N)\cong\operatorname{Ext}^{1}_{Q}(\tau M,\tau N)$

Suppose $Q$ be a Dynkin quiver, for any two non-projective indecomposable
representations $\beta_{k}$ and $\beta_{l}$, we have
$\beta_{k}<C\beta_{k}\leq\beta_{l}$ if $[\beta_{k},\beta_{l}]^{1}=1$.

  4. (4)

In [18] Ringel inductively constructed the maps $u$ by $u_{h}:(\tau
V)_{t(h)}\to V_{s(h)}$ such that for any ${i\in I}$.

(3.3) $0\to(\tau V)_{i}\stackrel{{\scriptstyle(u_{h},-\tau
x_{h})}}{{\longrightarrow}}\bigoplus_{t(h)=i}V_{s(h)}\oplus\bigoplus_{s(h^{\prime})=i}(\tau
V)_{t(h^{\prime})}\stackrel{{\scriptstyle(x_{h^{\prime}},u_{h^{\prime}})}}{{\longrightarrow}}V_{i}$

and for any element $\eta\in\operatorname{Ker}p_{x,x}^{*}$ as in (2.16), there
exists a unique element $g$ in $\operatorname{Hom}_{Q}(M,\tau M)$ such that
$ug=\eta$. Moreover, for any $f\in\operatorname{Hom}_{Q}(M,N)$, we have the
following.

(3.4) $u_{h}^{N}(\tau f)_{t(h)}=f_{s(h)}u_{h}^{M}\text{, for any arrow }h$

###### Proof.

We only prove $(3)$. Using
$[\beta_{k},\beta_{l}]^{1}=[\beta_{l},C\beta_{k}]=1$ and formula (2.11), we
obtain $\beta_{k}<\beta_{l}$ and $\beta_{k}<C\beta_{k}\leq\beta_{l}$ ∎

Using the identity
$D\operatorname{Ext}_{Q}^{1}({M},{N})=\operatorname{Hom}_{Q}({N},{\tau M})$,
we see that the long exact sequence (2.16) turns out to be the following long
exact sequence.

######  Lemma 3.2.

Let us assume that $M=(V,x)$ and $N=(W,y)$, we have

(3.5) $\begin{split}0\to\operatorname{Hom}_{Q}(N,\tau
M)\stackrel{{\scriptstyle
e^{*}_{x,y}}}{{\rightarrow}}\operatorname{Hom}_{\Omega^{op}}(W,V)\stackrel{{\scriptstyle
p_{x,y}^{*}}}{{\rightarrow}}\operatorname{Hom}_{I}(W,V)\to
D\operatorname{Hom}_{Q}(M,N)\to 0\end{split}$

where $e^{*}_{x,y}$ is given by
$e^{*}_{x,y}(f)=(u_{h}^{M}f_{t(h)})_{h\in\Omega}$.

###### Proof.

For simplicity, we write $u_{h}$ for $u_{h}^{M}$. In order to get the formula,
we first show that
$(u_{h}f_{t(h)})_{h\in\Omega}\in\operatorname{Ker}p_{x,y}^{*}$. Recall that
$f\in\operatorname{Hom}_{Q}({N},\tau{M})$ means that $f_{t(h)}y_{h}=\tau
x_{h}f_{s(h)}$ for any arrow $h:s(h)\to t(h)$. The equation
$p_{x,y}^{*}(z)=(\sum\limits_{t(h)=i}x_{h}z_{h}-\sum\limits_{s(h)=i}z_{h}y_{h})_{i\in
I}$ implies that

$\displaystyle p_{x,y}^{*}((u_{h}f_{t(h)})_{h\in\Omega})$ $\displaystyle=$
$\displaystyle(\sum\limits_{t(h)=i}x_{h}u_{h}f_{i}-\sum\limits_{s(h)=i}u_{h}f_{t(h)}y_{h})_{i\in
I}$ $\displaystyle=$
$\displaystyle(\sum\limits_{t(h)=i}x_{h}u_{h}f_{i}-\sum\limits_{s(h)=i}u_{h}\tau
x_{h}f_{i})_{i\in I}\text{ by $f_{t(h)}y_{h}=\tau x_{h}f_{i}$}$
$\displaystyle=$ $\displaystyle(0\cdot f_{i})_{i\in I}=0\qquad\text{ by
~{}(\ref{eq_uh})}$

Let us show that $e_{x,y}^{*}$ is injective. Let
$f\in\operatorname{Hom}_{Q}({N},\tau{M})$ such that $u_{h}f_{t(h)}=0$ for any
${h\in\Omega}$, we will prove $f_{i}=0$ by induction. If $i$ is a sink,
namely, there exists no arrow $h$ such that $s(h)=i$, then the equation (3.3)
is

(3.6) $0\to(\tau V)_{i}\stackrel{{\scriptstyle
u_{h}}}{{\longrightarrow}}\bigoplus_{t(h)=i}V_{s(h)}\stackrel{{\scriptstyle
x_{h^{\prime}}}}{{\longrightarrow}}V_{i}$

The fact that $u_{h}$ is injective for any $t(h)=i$ implies $f_{i}=0$.

Let us assume that for any $j>i$ we have $f_{j}=0$. Considering the exact
sequence (3.3) and $u_{h}f_{i}=0$, we have $(u_{h}f_{i},-\tau
x_{h}f_{i})=(0,-f_{t(h)}y_{h})=0$, because $h:i\to t(h)$ implies that $t(h)>i$
and then $f_{t(h)}=0$. Since $(u_{h},-\tau x_{h})$ is injective as in (3.3),
then $f_{i}=0$ for any vertex $i\in I$.

The fact $\operatorname{Im}e_{x,y}^{*}\subset\ker p_{x,y}^{*}$ and
$\operatorname{dim}\operatorname{Hom}_{Q}({N},{\tau
M})=\operatorname{dim}\operatorname{Ext}_{Q}^{1}({M},{N})$ by Proposition 3.1
implies that $\operatorname{Hom}_{Q}({M},\tau{N})$ is the kernel of
$p_{x,y}^{*}$ via the map $e_{x,y}^{*}$. ∎

Let us consider the representations of the opposite quiver $Q^{op}$. For any
two $I$-graded vector spaces $V,W$, one can consider
$\operatorname{Hom}_{\Omega^{op}}(V,W)$ as the dual space of
$\operatorname{Hom}_{\Omega}(W,V)$. It follows that one can consider
$E_{\alpha}^{*}$ as the representation space $E_{\alpha}(Q^{op})$ for any
$\alpha\in Q^{+}$. Let $M^{*}=(V,a)$ and $N^{*}=(W,b)$ be two representations
of $Q^{op}$ such that $a\in\operatorname{Hom}_{Q}(M,\tau
M)\subset\operatorname{Hom}_{\Omega^{op}}(V,V)$ and
$b\in\operatorname{Hom}_{Q}(N,\tau
N)\subset\operatorname{Hom}_{\Omega^{op}}(W,W)$ via $e_{x,x}^{*}$ and
$e_{y,y}^{*}$ (3.5). By (2.13), we have

(3.7)
$\begin{split}0\to\operatorname{Hom}_{Q^{\operatorname{op}}}(M^{*},N^{*})\to\operatorname{Hom}_{I}(V,W)\stackrel{{\scriptstyle
p_{a,b}}}{{\longrightarrow}}\operatorname{Hom}_{\Omega^{op}}(V,W)\to\operatorname{Ext}_{Q^{\operatorname{op}}}^{1}(M^{*},N^{*})\to
0\end{split}$

Here $p_{a,b}(f)=(f_{s(h)}a_{\bar{h}}-b_{\bar{h}}f_{t(h)})_{h\in\Omega}$.

Denote by $i_{x,y}$ the injective map
$\operatorname{Hom}_{Q}(M,N)\stackrel{{\scriptstyle
i_{x,y}}}{{\longrightarrow}}\operatorname{Hom}_{I}(V,W)$.

######  Lemma 3.3.

The restriction of $p_{a,b}$ to $\operatorname{Hom}_{Q}(M,N)$ becomes the
following map.

$\displaystyle r_{a,b}:\,\operatorname{Hom}_{Q}(M,N)$
$\displaystyle\to\operatorname{Hom}_{Q}(M,\tau N)$ $\displaystyle f$
$\displaystyle\mapsto\tau(f)a-bf$

###### Proof.

We only show the following commutative diagram.

(3.8)
$\textstyle{\operatorname{Hom}_{Q}(M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{x,y}}$$\scriptstyle{r_{a,b}}$$\textstyle{\operatorname{Hom}_{Q}(M,\tau
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{x,y}^{*}}$$\textstyle{\operatorname{Hom}_{I}(V,W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{a,b}}$$\textstyle{\operatorname{Hom}_{\Omega^{op}}(V,W)}$

Consider

$\displaystyle e_{x,y}^{*}r_{a,b}(f)=$ $\displaystyle
e_{x,y}^{*}(\tau(f)a-bf)$ $\displaystyle=$
$\displaystyle(u_{h}^{N}\tau(f)a-u_{h}^{N}bf)_{h\in\Omega}\qquad\text{ by
Lemma \ref{lem:dualpxy}}$ $\displaystyle=$
$\displaystyle(fu_{h}^{M}a-e_{y,y}^{*}(b)f)_{h\in\Omega}\qquad\text{ by
~{}(\ref{eq_uhf})}$ $\displaystyle=$
$\displaystyle(fe_{x,x}^{*}(a)-e_{y,y}^{*}(b)f)_{h\in\Omega}$ $\displaystyle=$
$\displaystyle p_{a,b}i_{x,y}(f)\qquad\qquad\qquad\quad\text{ by Lemma
\ref{lem:dualpxy}}$

∎

###  3.2. Representations and root system

In this section, we will give an interpretation of the condition
$[M_{\beta},M_{\alpha}]=1$ in terms of the relationship between roots $\beta$
and $\alpha$. This seems like an interesting link. Due to Proposition 3.1, in
what follows, we denote by $\tau$ the Coxeter element $C_{Q}$ in Section 2.1.

######  Lemma 3.4.

Suppose that $\alpha<\beta\in R^{+}$. If
$[M_{\beta},M_{\alpha}]=[\beta,\alpha]=0$, then we have ${\alpha+\beta}\in
R^{+}$ or $({\alpha,\beta})=0$. Moreover, we have $0\to M_{\beta}\to
M_{{\alpha+\beta}}\to M_{\alpha}\to 0$ when ${\alpha+\beta}\in R^{+}$.

###### Proof.

If $[\beta,\alpha]=0$, it implies from formula (2.5) that
$\langle{\beta},{\alpha}\rangle=[\beta,\alpha]-[\beta,\alpha]^{1}=0$. On the
other hand, we deduce
$\langle{\alpha},{\beta}\rangle=[\alpha,\beta]-[\alpha,\beta]^{1}=-[{\alpha,\beta}]^{1}$
by formula (2.5). If $-[{\alpha,\beta}]^{1}=0$, it implies that
$({\alpha,\beta})=0$. If $-[{\alpha,\beta}]^{1}\neq 0$, then the fact that
$[{\alpha,\beta}]^{1}\leq 1$ for any roots ${\alpha,\beta}$ implies that
$-[{\alpha,\beta}]^{1}=-1$. Hence, we obtain $({\alpha,\beta})=-1$. It follows
that $s_{\beta}(\alpha)=\alpha-({\alpha,\beta})\beta={\alpha+\beta}\in R^{+}$.

Take a nonzero extension
$E\in\operatorname{Ext}_{Q}^{1}({M_{\alpha}},{M_{\beta}})$, The [5, Lemma 4.1]
implies that $E<M_{\alpha}\oplus M_{\beta}$. It is easy to see by (2.7) and
$[M_{\alpha},M_{\alpha}]^{1}=[M_{\beta},M_{\beta}]^{1}=0$ that

$[E,E]^{1}<[M_{\alpha}\oplus M_{\beta},M_{\alpha}\oplus M_{\beta}]^{1}=1$

This implies that $E$ is a rigid representation with dimension vector
${\alpha+\beta}$; e,t.$\operatorname{Ext}_{Q}^{1}({E},{E})=0$. We can see that
$E\cong
M_{{\alpha+\beta}}\in\operatorname{Ext}_{Q}^{1}({M_{\alpha}},{M_{\beta}})$
because the indecomposable representation $M_{{\alpha+\beta}}$ associated with
root ${\alpha+\beta}$ is also a rigid representation and all rigid
representations with same dimension vector are isomorphic. In other words, we
get the following short exact sequence:

$0\to M_{\beta}\to M_{{\alpha+\beta}}\to M_{\alpha}\to 0$

∎

######  Proposition 3.5.

Let $\alpha<\beta\in R^{+}$. There are four cases as follows:

  1. (1)

$[\beta,\alpha]=[\beta,\tau\alpha]=1$;

  2. (2)

${\alpha+\beta}\in R^{+}$ if and only if $[\beta,\alpha]=0$ and
$[\beta,\tau\alpha]=1$

  3. (3)

$\beta-\alpha\in R$ if and only if $[\beta,\alpha]=1$ and
$[\beta,\tau\alpha]=0$

  4. (4)

$[\beta,\alpha]=[\beta,\tau\alpha]=0$.

###### Proof.

It is straight to see cases (1), (4). Let us focus on cases $(2),(3)$. The
relation (2.5) implies that

$[\alpha,\beta]=[\beta,\alpha]^{1}=0$

It follows from (2.1) and Proposition 3.1 that

$\langle{\alpha},{\beta}\rangle=-[\alpha,\beta]^{1}=-[\beta,\tau\alpha]$

and

$\langle{\beta},{\alpha}\rangle=[\beta,\alpha]$

Therefore, we obtain

$({\alpha,\beta})=\langle{\alpha},{\beta}\rangle+\langle{\beta},{\alpha}\rangle=[\beta,\alpha]-[\beta,\tau\alpha]$

In the case $(2)$, the condition $({\alpha,\beta})=-1$ is equivalent to
$s_{\alpha}(\beta)=\beta-({\alpha,\beta})\alpha=\beta+\alpha$ in $R^{+}$. This
condition coincides with $[\beta,\alpha]=0$ and $[\beta,\tau\alpha]=1$, as
$[\beta,\alpha]\leq 1$ and $[\beta,\tau\alpha]\leq 1$.

Similarly, in the case $(3)$. $({\alpha,\beta})=1$ coincides with the
condition $[\beta,\alpha]=1$ and $[\beta,\tau\alpha]=0$, it is equivalent to
$s_{\alpha}(\beta)=\beta-({\alpha,\beta})\alpha=\beta-\alpha\in R$. ∎

###  3.3. AR-orbits Kostant partitions

In this section, we will consider a sort of special Kostant partition: _$\tau$
-orbit Kostant partition_. This notion is inspired by the quantum minors
$D(p,q)$ in the quantum nilpotent coordinate ring
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}})$, where $p,q$ are positive
integers such that $p<q$.(see [10] or Section 5). In our cases, the Kostant
partitions for these $D(p,q)$ are the sequences of the following form

(3.9) $(\tau^{r}\beta,\tau^{r-1}\beta_{p},\cdots,\beta)$

for some $r\in\mathbb{N}$ and $\beta\in R^{+}$ such that $\tau^{r+1}\beta\in
R^{+}$. On the other hand, in [4] Buan, Marsh, Reineke, Reiten and Todorov
constructed the cluster category using the $\tau$-quotient of the derived
category of representations of $Q$. Therefore, it is important to consider
$\tau$-orbit Kostant partitions.

######  Example 3.6.

In the case of Example 2.2, the $\tau$-orbit Kostant partitions are of the
form multisegments

$\lambda=([a+r,b+r],\cdots,[a+1,b+1],[a,b])$

###  3.4. The map $r_{a,b}$

Consider the map $r_{a,b}$ in Lemma 3.3. This map plays an important role in
the following sections. This map is used to calculate the
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})$ for
modules $\widetilde{{M}},\widetilde{{N}}$ over $\Lambda_{Q}$ (see [1,
Proposition 9.1]). We will only focus on the case of $\tau$-orbit Kostant
partitions in this section. In fact, it is impossible to describe the map
$r_{a,b}$ in all cases, even in the case of $A_{n}$ with usual orientation
(see [14, Section 2.4]).

Fix a representation $M=(V,x)$ of $Q$ such that
$\underline{\operatorname{dim}}V=\alpha\in Q^{+}$, and let
$a\in\operatorname{Hom}_{Q}({M},\tau{M})$. Consider the Kostant partition
$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{r})\in\operatorname{KP}({\alpha})$
associated with $M$. As $\lambda_{i}>\lambda_{j}$ if $i<j$, we can deduce from
the equation (2.5) that

(3.10) $\begin{split}\Pi:\operatorname{Hom}_{Q}({M},{M})&\cong\bigoplus_{i\leq
j}\operatorname{Hom}_{Q}({M_{\lambda_{i}}},{M_{\lambda_{j}}})\\\
f&\mapsto(f_{i,j})\end{split}$

where $f_{i,j}$ is the composition of the embedding $M_{\lambda_{i}}\to M$,
$f:M\to M$, and the projection $M\to M_{\lambda_{j}}$. Similarly, we have

(3.11)
$\begin{split}\operatorname{Hom}_{Q}({M},\tau{M})&\cong\bigoplus_{i<j}\operatorname{Hom}_{Q}({M_{\lambda_{i}}},\tau{M_{\lambda_{j}}})\\\
a&\mapsto(a_{i,j})\end{split}$

Note that
$\operatorname{Hom}_{Q}({M_{\lambda_{i}}},\tau{M_{\lambda_{i}}})=D\operatorname{Ext}_{Q}^{1}({M_{\lambda_{i}}},{M_{\lambda_{i}}})=0$
by Proposition 3.1. As a result, any element
$a\in\operatorname{Hom}_{Q}({M},\tau{M})$ turns out to be a nilpotent matrix.

$a\in\mathrm{Mat}_{r\times r}$

Let $N,K$ be two presentations of $Q$ and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{t})$ and
$\mu=({\mu}_{1},{\mu}_{2},\cdots,{\mu}_{k})$ be their corresponding Kostant
partitions, respectively. It is easy to see from the definition that the
composition of the following map

$\displaystyle\operatorname{Hom}_{Q}({M},{N})\times\operatorname{Hom}_{Q}({N},{K})$
$\displaystyle\to\operatorname{Hom}_{Q}({M},{K})$ $\displaystyle(f,g)$
$\displaystyle\mapsto gf$

is identified with the composition of matrices as follows.

$\displaystyle\operatorname{Hom}_{Q}({M},{N})\times\operatorname{Hom}_{Q}({N},{K})$
$\displaystyle\to\operatorname{Hom}_{Q}({M},{K})$
$\displaystyle((f_{i,j}),(g_{i,j}))$
$\displaystyle\mapsto(\sum_{p=1}^{t}f_{i,p}g_{p,j})$

This mean by the isomorphism $\Pi$ as in (3.11), we obtain

(3.12) $\Pi(gf)=\Pi(f)\Pi(g)$

If we write $N=(W,y)$ and take an element of
$b\in\operatorname{Hom}_{Q}({N},\tau{N})$, the map $r_{a,b}$ becomes the
following map under this identity.

(3.13) $\begin{split}\mathrm{Mat}_{r\times t}&\to\mathrm{Mat}_{r\times t}\\\
f&\mapsto af-fb\end{split}$

_Assumption: Throughout this paper, we assume that the $\tau$-orbit Kostant
partitions $\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{u})$ have
no projective roots, e.t., $\tau\lambda_{1}\in R^{+}$. As a result of this
,$\tau(f)=f$._

######  Remark 3.7.

One of the reasons is that we only focus determinantal modules over the
preprojective algebra $\Lambda_{Q}$. If there is a projective indecomposable
representation $P_{i}$ in the direct summand of a module $M$, then its
corresponding determinantal module $\widetilde{{M}}$ is exactly an injective
(and projective ) module over $\Lambda_{Q}$. Therefore, we obtain
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})=\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{N}}},{\widetilde{{M}}})=0$
for all modules $\widetilde{{N}}$ of $\Lambda_{Q}$; Another reason is that
those determinantal modules with projective indecomposable representations in
their direct summands correspond to the frozen variable in quantum cluster
algebras.

######  Lemma 3.8.

Let $a=(a_{i,j})_{i,j\in[1,r]}$ and $b=(b_{i,j})_{i,j\in[1,t]}$ be two upper-
triangular nilpotent matrices such that $a_{i,i+1}=b_{j,j+1}=1$ for all
$i\in[1,r]$ and all $j\in[1,t]$. Consider the subspace $Z(a,b)$ of
$Mat_{r\times t}$ consisting of elements $f=(f_{i,j})_{i\in[1,r];j\in[1,t]}$
satisfying the equation

$af-fb=0$

We have $\operatorname{dim}Z(a,b)=\operatorname{min}\\{r,t\\}$

###### Proof.

Let us first define the map $r_{a,b}:Mat_{r\times s}\to Mat_{r\times s}$ by
sending $f$ to $af-fb$. Consider the following condition

$r_{a,b}(f)=af-fb=0$

We consider the composition of maps into the composition of matrices. There
are two kinds of paths from $i$ to $j$:

$i\stackrel{{\scriptstyle a_{i,d}}}{{\rightarrow}}{d}\stackrel{{\scriptstyle
f_{d,j}}}{{\rightarrow}}j$

and

$i\stackrel{{\scriptstyle f_{i,s}}}{{\rightarrow}}s\stackrel{{\scriptstyle
b_{s,j}}}{{\rightarrow}}j$

Hence, for a pair $(i,j)$ of vertices, we have

$\displaystyle r_{a,b}(f)_{i,j}=$
$\displaystyle\sum_{d=1}^{r}a_{i,d}f_{d,j}-\sum_{s=1}^{t}f_{i,s}b_{s,j}$
$\displaystyle=$
$\displaystyle\sum_{d=i+1}^{r}a_{i,d}f_{d,j}-\sum_{s=1}^{j-1}f_{i,s}b_{s,j}\quad\text{
by $a_{i,d}=b_{s,j}=0$ for $i\geq d$ and $s\geq j$}$

We remark that here $f_{d,j}$ satisfies $d\geq i+1$ and $f_{i,s}$ satisfies
$s<j$.

Let us first show $f_{r,1}=0$. Consider

$\displaystyle r_{a,b}(f)_{r-1,1}=$ $\displaystyle
a_{r-1,r}f_{r,1}-\sum_{s=1}^{t}f_{r-1,s}b_{s,1}$ $\displaystyle=$
$\displaystyle a_{r-1,r}f_{r,1}=f_{r,1}=0\quad\text{ by $b_{s,1}=0$ for all
$s\geq 1$}$

Next, we will show that

(3.14) $f_{r,j}=0\qquad\text{except $f_{r,t}$}$

by induction on $j$. Let us assume that $f_{r,k}=0$ for any $k<j$ and consider
the pair $(r,j)$. The above equation becomes

$\displaystyle r_{a,b}(f)_{r,j+1}=$ $\displaystyle
a_{r,r}f_{r,j+1}-\sum_{s=1}^{j}f_{r,s}b_{s,j+1}$ $\displaystyle=$
$\displaystyle-\sum_{s=1}^{<j}f_{r,s}b_{s,j+1}-f_{r,j}b_{j,j+1}$
$\displaystyle=$ $\displaystyle-f_{r,j}b_{j,j+1}\text{ by induction
$f_{r,s}=0$ for all $s<j$}$ $\displaystyle=$ $\displaystyle-f_{r,j}=0.$

Similarly, we will show that $f_{i,1}=0$ except $f_{1,1}$ using the induction
on $i$. Assume that $f_{d,1}=0$ for all $d>i$ and consider

$\displaystyle r_{a,b}(f)_{i-1,1}=$
$\displaystyle\sum_{d=i}^{r}a_{i-1,d}f_{d,1}-0$ $\displaystyle=$
$\displaystyle\sum_{d=i+1}^{r}a_{i-1,d}f_{d,1}+a_{i-1,i}f_{i,1}$
$\displaystyle=$ $\displaystyle f_{i,1}=0\text{ by induction $f_{d,1}=0$ for
all $d>i$}$

Next, we will deduce that $f_{i,j}=0$ if $i>j$ except $(r,t)$ by induction on
$d=i-j$. As we discussed before, for $d=r-1$ we have $f_{r,1}=0$. Let us
assume that $f_{i^{\prime},j^{\prime}}=0$ for all $i^{\prime}>j^{\prime}$ and
$i^{\prime}-j^{\prime}>i-j$, and then consider

$\displaystyle r_{a,b}(f)_{i-1,j}=$
$\displaystyle\sum_{d=i}^{r}a_{i-1,d}f_{d,j}-\sum_{s=1}^{j-1}f_{i-1,s}b_{s,j}$
$\displaystyle=$ $\displaystyle
a_{i-1,i}f_{i,j}+\sum_{d=i+1}^{r}a_{i-1,d}f_{d,j}-\sum_{s=1}^{<j-1}f_{i-1,s}b_{s,j}-f_{i-1,j-1}b_{j-1,j}$

The conditions $d-j\geq i+1-j>i-j$ and $i-s\geq i-(j-1)>i-j$ lead to
$f_{d,j}=0$ and $f_{i,s}=0$ by our assumption. We deduce
$r_{a,b}(f)_{i-1,j}=a_{i-1,i}f_{i,j}-f_{i-1,j-1}b_{j-1,j}=f_{i,j}-f_{i-1,j-1}=0$
from $a_{i-1,i}=b_{i-1,j-1}=1$. Hence, we obtain $f_{i,j}=f_{i-j+1,1}$ by
repeating the relation $f_{i,j}=f_{i-1,j-1}$. However, the relation $i>j$
leads to $i-j+1>1$, and then implies from $f_{i,1}=0$ for all $i>1$ that
$f_{i,j}=f_{i-j+1,1}=0$

Therefore, we get that

$\displaystyle
r_{a,b}(f)_{i,j}=\sum_{d=i+1}^{j}a_{i,d}f_{d,j}-\sum_{s=i}^{j-1}f_{i,s}b_{s,j}$

Next, we will show that $f_{i,i}=f_{j,j}$ for any
$i,j\in\operatorname{min}\\{s,t\\}$. It is enough to check
$f_{i,i}=f_{i+1,i+1}$ for all $i<\operatorname{min}\\{s,t\\}$. This follows
from

$\displaystyle r_{a,b}(f)_{i,i+1}=$ $\displaystyle
a_{i,i+1}f_{i+1,i+1}-f_{i,i}b_{i,i+1}$ $\displaystyle=$ $\displaystyle
f_{i+1,i+1}-f_{i,i}=0$

Let us give a tuple of elements

$(f_{1,i})_{i\in[1,t]}$

We will show that this tuple of elements yields a unique solution of the
equation $r_{a,b}(f)=0$. As above, we see $f_{i,i}=f_{1,1}$ for all
$i\leq\operatorname{min}\\{r,t\\}$. Let $k\leq t-1$, we will show that
$f_{i,i+k}$ is determined by $f_{1,j}$ for $j\leq k$. In other words,

$f_{i,i+k}\in\langle f_{1,1},f_{1,2},\cdots f_{1,k+1}\rangle$

where $\langle f_{1,1},f_{1,2},\cdots f_{1,k}\rangle$ is the subspace
generated by $f_{1,1},\cdots,f_{1,k}$. We show this by induction on $k$, we
have seen that $f_{1,1}=f_{i,i}$ for all $i\leq\operatorname{min}\\{s,t\\}$.
Let us assume that

$f_{i,i+l}\in\langle f_{1,1},\cdots,f_{1,l+1}\rangle\qquad\text{for any $l<k$
and all }i$

and then consider

$\displaystyle r_{a,b}(f)_{i,i+k+1}=$
$\displaystyle\sum_{d=i+1}^{i+k+1}a_{i,d}f_{d,i+k+1}-\sum_{s=i}^{i+k}f_{i,s}b_{s,i+k+1}$
$\displaystyle=$
$\displaystyle\sum_{d>i+1}^{i+k+1}a_{i,d}f_{d,i+k+1}-\sum_{s=i}^{i+k-1}f_{i,s}b_{s,i+k+1}$
$\displaystyle\qquad\qquad\qquad\qquad+(a_{i,i+1}f_{i+1,i+k+1}-f_{i,i+k}b_{i+k,i+k+1})$

Due to $i+k+1-d<k$ and $s-i<k$ and our assumption, we get

$f_{d,i+k+1},f_{i,s}\in\langle f_{1,1},\cdots,f_{1,k}\rangle\qquad\text{for
any $d,s$ in the above equation}$

Thus, $r_{a,b}(f)_{i,i+k+1}=0$ implies

$f_{i+1,i+1+k}=f_{i,i+k}+b_{i}$

where $b_{i}\in\langle f_{1,1},\cdots,f_{1,k}\rangle$. It follows
$f_{i,i+k}=f_{1,k+1}+c_{i}$ where $c_{i}\in\langle
f_{1,1},\cdots,f_{1,k}\rangle$. That means

$f_{i,i+k}\in\langle f_{1,1},f_{1,2},\cdots f_{1,k+1}\rangle\text{ for any $i$
such that $i+k\leq t$}$

Let us calculate the dimension of $Z(a,b)$. Combining our claims, we have

  1. (1)

$f_{r,j}=0$ except $f_{r,t}$;

  2. (2)

$f_{i,j}=0$ for all $i>j$;

  3. (3)

$f_{i,i+k}\in\langle f_{1,1},\cdots f_{1,t}\rangle$

If $r\geq t$, The condition $(1)$ follows from the condition $(2)$. This means
that the tuple $(f_{1,i})_{i\in[1,t]}$ is a basis of the equation
$r_{a,b}(f)=0$.

If $r<t$, the condition $(1)$ implies that $f_{r,k}=0$ for $k\in[r,t-1]$.
Thus, $f_{r,r}=0$ leads to $f_{1,1}=0$. Based on our proof of the condition
$(3)$, we see that $f_{1,j}=0$ for $j\in[1,t-r]$. Thus,
$(f_{1,i})_{i\in[t-r+1,t]}$ is a basis of the equation $r_{a,b}(f)=0$. ∎

######  Remark 3.9.

We should remark that the composite map $af-fb$ can be considered as the
matrix composition only in some special cases, in particular for the _$\tau$
-orbit Kostant partitions_. Because the composite map
$ba:M_{i}\stackrel{{\scriptstyle
a}}{{\rightarrow}}M_{j}\stackrel{{\scriptstyle b}}{{\rightarrow}}M_{k}$ maybe
equal to $0$ even if $a,b\neq 0$. For instance, The $AR$-relations in the
$AR$-quiver. In the case of AR-orbit Kostant partitions, there are no
$AR$-relations in the composite map $af-fb$. More importantly, we only use the
coefficients $a_{i,i+1}$ and $b_{i,i+1}$ in our proof of Lemma 3.8. Because
$a_{i,i+1}=\operatorname{Id}\in\operatorname{Hom}_{Q}({M_{i}},{M_{i}})$ and
$b_{i,i+1}=\operatorname{Id}\in\operatorname{Hom}_{Q}({N_{i}},{N_{i}})$.
Hence, this doesn’t happen in our case.

We should mention that the other $a_{i,j}$ for $i+1<j$ does not affect the
dimension of the solutions of the equation

$af-fb=0$

It is easy to see that from our proof. In addition, the dimension of $Z(a,b)$
will increase if $a_{k,k+1}=0$ for some $k$.

######  Lemma 3.10.

Let $\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{m})$ be an
$\tau$-orbit Kostant partition without projective roots, the dual
representation
$M_{\lambda}^{*}=(a_{i,j})\in\operatorname{Hom}_{Q}({M_{\lambda}},{M_{\lambda}})$
is a nilpotent matrix subject to $a_{i,i+1}=1$ for all $i\in[1,m]$.

###### Proof.

In fact, $\widetilde{{M_{\lambda}}}$ is a rigid module over the preprojective
algebra $\Lambda_{Q}$. That means, the space $\operatorname{Hom}_{Q}(M,\tau
M)$ has an open $\operatorname{Aut}_{Q}({M_{\lambda}})$-orbit
$\mathbf{O}_{a}$, (see Theorem 4.4). The dimension of $\mathbf{O}_{a}$ is
equal to

$[M_{\lambda},M_{\lambda}]-\operatorname{dim}Z(a,a)$

Because
$\operatorname{Hom}_{Q}({M_{\lambda}},{M_{\lambda}})=\operatorname{Lie}(\operatorname{Aut}_{Q}({M_{\lambda}}))$
and
$\operatorname{Lie}(Z_{\operatorname{Aut}_{Q}({M_{\lambda}})}(\eta))=Z(a,a)$.
Using Remark 3.9, we see that $M_{\lambda}^{*}=(a_{i,j})$ subject to
$a_{i,i+1}=1$ for all $i\in[1,m]$. ∎

######  Theorem 3.11.

Let
$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{r})\in\operatorname{KP}({\alpha})$
and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{t})\in\operatorname{KP}({\beta})$
be two $\tau$-orbit Kostant partitions with condition 3.4. Let
$a=M_{\lambda}^{*}\in\operatorname{Hom}_{Q}({M_{\lambda}},\tau{M_{\lambda}})$
and
$b=M_{\kappa}^{*}\in\operatorname{Hom}_{Q}({M_{\kappa}},\tau{M_{\kappa}})$.
Set

$R(\lambda,\kappa)=\\{k\mid\,[\lambda_{1}.\kappa_{k}]=1\\}$

Let us consider the map

$r_{a,b}:\operatorname{Hom}_{Q}({M_{\lambda}},{M_{\kappa}})\to\operatorname{Hom}_{Q}({M_{\lambda}},\tau{M_{\kappa}})$

We have

(3.15) $\operatorname{dim}\operatorname{Ker}{}r_{a,b}=\begin{cases}\sharp
R(\lambda,\kappa)\text{ if $r\geq t$}\\\ \sharp[t-r+1,t]\cap
R(\lambda,\kappa)\text{ if $r<t$}\end{cases}$

In particular, $r_{a,b}$ is surjective if and only if

(3.16) $[M_{\lambda},M_{\kappa}]-[M_{\lambda},\tau
M_{\kappa}]=\begin{cases}\sharp R(\lambda,\kappa)\text{ if $r\geq t$}\\\
\sharp[t-r+1,t]\cap R(\lambda,\kappa)\text{ if $r<t$}\end{cases}$

###### Proof.

It is easy to see by the fact $\lambda_{i}=\tau\lambda_{i+1}$ and
$\kappa_{k}=\tau\kappa_{k+1}$ that
$[\lambda_{i},\tau\lambda_{i+1}]=[\kappa_{k},\tau\kappa_{k+1}]=1$ for any
$i,k$. Hence, $a,b$ satisfies the assumption in Lemma 3.8.

In the case $r\geq t$, by the above Lemma, we have that
$(f_{1,k})_{k\in[1,t]}$ is a basis for the equation $r_{a,b}=0$. On the other
hand, $f_{1,k}\neq 0$ if and only if $k\in R(\lambda,\kappa)$. Hence, we
obtain $\operatorname{dim}\operatorname{Ker}r_{a,b}=\sharp[1,t]\cap
R(\lambda,\kappa)$. Similarly, in the case $r<t$, we have that
$(f_{1,k})_{k\in[t-r+1,t]}$ is a basis for the equation $r_{a,b}=0$, then
$\operatorname{dim}\operatorname{Ker}r_{a,b}=\sharp[t-r+1,t]\cap
R(\lambda,\kappa)$. ∎

The following is one of our main results.

######  Theorem 3.12.

Let $\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{u})$ and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{v})$ be two $\tau$-orbit
Konstant partitions with condition 3.4, and put

(3.17)
$\begin{split}&\operatorname{\mathfrak{m}}(\lambda,\kappa)=\sharp\left\\{{(i,j)}\mid\,{\lambda_{i}-\kappa_{j}\in
R\cup\\{0\\}\text{ and }\lambda_{i}\geq\kappa_{j}}\right\\}\\\
&\operatorname{\mathfrak{p}}(\lambda,\kappa)=\sharp\left\\{{(i,j)}\mid\,{\lambda_{i}+\kappa_{j}\in
R^{+}\text{ and }\lambda_{i}\geq\kappa_{j}}\right\\}\end{split}$

we have

$\displaystyle[M_{\lambda},M_{\kappa}]-[M_{\lambda},\tau
M_{\kappa}]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$

In particular, $r_{a,b}$ is surjective if and only if

  1. (1)

If $u\geq v$

$\displaystyle\sharp
R(\lambda,\kappa)=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$

  2. (2)

If $u<v$

$\displaystyle\sharp
R(\lambda,\kappa)\cap[v-u+1,v]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$

###### Proof.

Let $\Gamma_{\lambda,\kappa}$ be the set

$\left\\{{(i,j)}\mid\,{[\lambda_{i},\kappa_{j}]=1}\right\\}$

and $\Gamma_{\lambda,\tau\kappa}$ be the set

$\left\\{{(i,j)}\mid\,{[\lambda_{i},\kappa_{j-1}]=1}\right\\}$

We have

(3.18)
$\Gamma_{\lambda,\tau\kappa}\cap\Gamma_{\lambda,\kappa}=\left\\{{(i,j)}\mid\,{[\lambda_{i},\kappa_{j}]=[\lambda_{i},\kappa_{j-1}]=1}\right\\}$

and

(3.19)
$\Gamma_{\lambda,\kappa}\cap\Gamma_{\lambda,\tau\kappa}^{c}=\left\\{{(i,j)}\mid\,{[\lambda_{i},\kappa_{j}]=1;[\lambda_{i},\kappa_{j-1}]=0}\right\\}$

Proposition 3.5 implies

(3.20)
$\Gamma_{\lambda,\kappa}\cap\Gamma_{\lambda,\tau\kappa}^{c}=\left\\{{(i,j)}\mid\,{\lambda_{i}-\kappa_{j}\in
R\cup\\{0\\}\text{ and }\lambda_{i}\geq\kappa_{j}}\right\\}$

On the other hand,

(3.21)
$\Gamma_{\lambda,\tau\kappa}\cap\Gamma_{\lambda,\kappa}^{c}=\left\\{{(i,j)}\mid\,{[\lambda_{i},\kappa_{j}]=0;[\lambda_{i},\kappa_{j-1}]=1}\right\\}$

Using Proposition 3.5, we get

(3.22)
$\Gamma_{\lambda,\tau\kappa}\cap\Gamma_{\lambda,\kappa}^{c}=\left\\{{(i,j)}\mid\,{\lambda_{i}+\kappa_{j}\in
R^{+}\text{ and }\lambda_{i}\geq\kappa_{j}}\right\\}$

Then we have the following equation

(3.23) $\begin{split}&[M_{\lambda},M_{\kappa}]-[M_{\lambda},\tau
M_{\kappa}]\\\
&=\Gamma_{\lambda,\kappa}\cap\Gamma_{\lambda,\tau\kappa}^{c}-\Gamma_{\lambda,\tau\kappa}\cap\Gamma_{\lambda,\kappa}^{c}\\\
&=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)\end{split}$

Therefore, $r_{a,b}$ is surjective if and only if $r_{a,b}$ is surjective if
and only if

  1. (1)

If $u\geq v$

$\displaystyle\sharp
R(\lambda,\kappa)=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$

  2. (2)

If $u<v$

$\displaystyle\sharp
R(\lambda,\kappa)\cap[v-u+1,v]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$

∎

##  4\. Representation theory of preprojective algebras

In this section, we briefly recall some basic notions of preprojective
algebras. The goal of this section is to interpret the properties of modules
of preprojective algebras using the results we obtained before. Thus, the
conclusions in this section may be well known in other papers. We only try to
give a new point of view to consider these problems.

Let us consider a quiver $Q$ and its opposite quiver $Q^{op}$. One defines an
algebra $\Lambda_{Q}$ as a quotient algebra of $k(Q\cup Q^{op})$ by the ideal
$I$ which is generated by

$\sum\limits_{s(h)=i,t(h^{\prime})=i}\bar{h}h-h^{\prime}\bar{h^{\prime}}$

for any ${i\in I}$. We simply denote it by $\Lambda$

The modules $\widetilde{{M}}$ over $\Lambda$ are of the form $(V,x,a)$ where
$x_{h}:V_{s(h)}\to V_{t(h)}$ and $a_{\bar{h}}:V_{t(h)}\to V_{s(h}$ subject to
the relation
$\sum\limits_{s(h)=i,t(h^{\prime})=i}a_{\bar{h}}x_{h}-x_{h^{\prime}}a_{\bar{h^{\prime}}}=0$
for any ${i\in I}$. For any two modules $\widetilde{{M}},\widetilde{{N}}$ of
$\Lambda$. one has

(4.1)
$\begin{split}D\operatorname{Ext}_{\Lambda}^{1}(\widetilde{{M}},\widetilde{{N}})&=\operatorname{Ext}_{\Lambda}^{1}(\widetilde{{N}},\widetilde{{M}})\\\
\operatorname{dim}\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{\widetilde{{N}}})=\operatorname{dim}\operatorname{Hom}_{\Lambda}({\widetilde{{M}}},{\widetilde{{N}}})+&\operatorname{dim}\operatorname{Hom}_{\Lambda}({\widetilde{{N}}},{\widetilde{{M}}})-(\underline{\operatorname{dim}}\widetilde{{M}},\underline{\operatorname{dim}}\widetilde{{N}})\end{split}$

For a module $\widetilde{{M}}$ and a vertex $i_{k}$, let us consider
$\operatorname{soc}(\widetilde{{M}})=\bigoplus_{i_{p}\in I}S_{i_{p}}^{n_{p}}$,
where $S_{i_{k}}$ is the simple module for the vertex $i_{k}$. We denote by
$\operatorname{soc}_{i_{k}}(\widetilde{{M}})$ the submodule
$S_{i_{k}}^{n_{k}}$ of $\operatorname{soc}(\widetilde{{M}})$. Denote by
$\widetilde{{I}}_{i_{k}}$ the projective (injective) module of $\Lambda$ for
the vertex $i_{k}$.

For any $k\in[1,r]$ where $r$ is the length of $w_{0}$, let us consider
$\beta_{k}$ and set the following subset of $R^{+}$:

(4.2)
$\begin{split}\tau^{\mathbf{i}}\beta_{k}=\\{\beta_{j}\mid\,i_{j}=i_{k}\\}\quad\,&\tau^{\mathbf{i}}(k)=\\{i_{j}\mid\,i_{j}=i_{k}\\}\\\
k^{+}=\operatorname{min}\\{j>k\mid\,i_{j}\in\tau^{\mathbf{i}}(k)\\};\quad&k^{-}=\operatorname{max}\\{j<k\mid\,i_{j}\in\tau^{\mathbf{i}}(k)\\}\\\
C_{Q}\beta_{k}=\beta_{k^{+}}&\text{ if $k^{+}\neq\emptyset$}\\\
\beta_{[j,k]}=(\beta_{j},C_{Q}\beta_{j},\cdots,C^{-1}_{Q}\beta_{k},&\beta_{k})\text{
for $i_{j}=i_{k}$ and $j\leq k$}\\\
l(\beta_{[j,k]})=\sharp\beta_{[j,k]}\end{split}$

###  4.1. Cluster category of preprojective algebras

In this section, we briefly review the cluster category of preprojective
algebras (see [9] and [10] for more detail). Give an $w\in W$ as in Section
2.1, and fix a reduced expression $w={s}_{i_{1}}{s}_{i_{2}}\cdots{s}_{i_{r}}$
and $\mathbf{i}=[{i}_{1}{i}_{2}\cdots{i}_{r}]$. Denote by $R(w)$ the set of
roots associated with it, (see (2.2)). For any $\beta_{k}\in R(w)$, we define
a module $\widetilde{{V}}_{k}$ associated with it by

$0=M_{0}\subset
M_{1}\subset\cdots\subset\widetilde{{V}}_{k}\subset\widetilde{{I}}_{i_{k}}$

such that $\operatorname{soc}_{i_{k-p+1}}(I_{i_{k}}/M_{p-1})=M_{p}/M_{p-1}$.
Set

(4.3) $\widetilde{{V}}_{\mathbf{i}}=\oplus_{k=1}^{r}\widetilde{{V}}_{k}$

Define $\mathcal{C}_{\mathbf{i}}$ (or $\mathcal{C}_{w}$) as the full
subcategory of $\Lambda-\operatorname{mod}$ consisting of objects as quotient
modules of some direct sum of a finite number of
$\widetilde{{V}}_{\mathbf{i}}$. For $j\in I$, let
$k_{j}:=\operatorname{max}\\{k\in[1,r]\mid\,i_{k}=j\\}$. Define
$I_{j}=\widetilde{{V}}_{k_{j}}$ and set $I_{\mathbf{i}}=\oplus_{j\in I}I_{j}$,
which is an injective module in $\mathcal{C}_{w}$.

######  Definition 4.1.

For any $k\leq l$, one defines

(4.4) $\widetilde{{V}}[k,l]=\widetilde{{V}}_{l}/\widetilde{{V}}_{k^{-}}$

We call $\widetilde{{V}}[k,l]$ the _determinantal modules_.

###  4.2. Ringel’s interpretation of the cluster category

In what follows, we assume that $w=w_{0}$ and $\mathbf{i}$ is (2.3). In the
Dynkin case, it is well-known that

$\mathcal{C}_{w_{0}}\cong\Lambda_{Q}-\operatorname{mod}$

In [18] Ringel gave an interpretation of the category
$\Lambda_{Q}-\operatorname{mod}$ via Auslander–Reiten theory of $Q$. More
precisely, We define a category $\mathcal{D}_{Q}$ by: its objects consist of a
pair $(M,a)$ where $M$ is a representation of $Q$ and
$a\in\operatorname{Hom}_{Q}(M,\tau M)$; the morphisms from $(M,a)$ to $(N,b)$
are the morphisms $f$ in $\operatorname{Hom}_{Q}(M,N)$ such that
$bf=\tau(f)a$. By Lemma 3.3, we have
$\operatorname{Hom}_{\mathcal{D}_{Q}}((M,a),(N,b))=\ker
r_{a,b}=\operatorname{Hom}_{Q}(M,N)\cap\operatorname{Hom}_{Q^{op}}(a,b)$.

Recall that a module $\widetilde{{M}}=(V,x,a)$ of $\Lambda$ where
$x\in\operatorname{Hom}_{\Omega}(V,V)$ and
$a\in\operatorname{Hom}_{\Omega^{op}}(V,V)$ such that $p_{x,x}^{*}(a)=0$, (see
(2.16)). The Lemma 3.2 implies that there exists a unique element
$f\in\operatorname{Hom}_{Q}({M},\tau{M})$ such that $a=e_{x,x}^{*}(f)$.
Therefore, we can consider $a$ as an element of
$\operatorname{Hom}_{Q}({M},\tau{M})$ via $e_{x,x}^{*}$. This proves that
$\mathcal{D}_{Q}\cong\Lambda_{Q}-\operatorname{mod}$.

There is a canonical functor $\Psi:\mathcal{D}_{Q}\to kQ-\operatorname{mod}{}$
by sending $(M,a)$ to $M$.

######  Definition 4.2.

Let $M$ be a representation of $Q$. We define
$M_{\beta}^{\tau^{+}}=\oplus_{k\in\mathbb{N}}\tau^{k}M_{\beta}$ for an
indecomposable representation $M_{\beta}$ of $Q$. Similarly, we define
$M_{\beta}^{\tau^{-}}=\oplus_{k\in\mathbb{N}}\tau^{-k}M_{\beta}$.

Let $w_{0}$ be the Weyl element with maximal length and
$\mathbf{i}=[i_{1}i_{2}\cdots i_{r}]$ is the word given by (2.3). For any
$k\in[1,r-n]$, we define $V_{k}=M_{\beta_{k}}^{\tau^{-}}$. Namely,

$V_{k}=\bigoplus_{i_{l}\in\tau^{\mathbf{i}}(k);l\leq k}M_{\beta_{l}}$

If $k^{+}=\emptyset$, we set $V_{k}=V_{k^{-}}\oplus P_{k}$, where $P_{k}$ is
the projective representation associated with $i_{k}$. We define

(4.5) $V[k,l]=\bigoplus_{\beta_{j}\in\beta_{[k,l]}}M_{\beta_{j}}\text{for
$i_{k}=i_{l}$}$

and set

(4.6) $V_{\mathbf{i}}=\bigoplus_{k\in[1,m]}V_{k}$

Next, Let us define the following category:

######  Definition 4.3.

Let us define $\mathcal{D}_{w_{0}}$ by the category consisting of
representations $M$ of $Q$ such that

$0\to T^{\prime}\to T\to M\to 0$

where $T^{\prime},T\in\operatorname{add}({V_{\mathbf{i}}})$. It is easy to see
$\mathcal{D}_{w_{0}}=kQ-\operatorname{mod}$.

Following [18], we have the following theorem:

######  Theorem 4.4.

There is a functor $\Psi_{w}:\Lambda_{Q}-\operatorname{mod}\to
kQ-\operatorname{mod}$ by sending $\widetilde{{M}}=(V,x,a)$ to $M=(V,x)$.
Moreover, the following conditions hold:

  1. (1)

For any $k\leq l$, we have $\Psi_{w}(\widetilde{{V}}[k,l])=V[k,l]$, and
$\widetilde{{V}}[k,l]=(V[k,l],\eta)$ for $\eta=V[k,l]^{*}$ (see Definition
2.2);

  2. (2)

$\Psi_{w}$ is an exact functor;

  3. (3)

We have that $\widetilde{{T}}=(T,T^{*})$ is a rigid module over $\Lambda_{Q}$
if and only if $\operatorname{Ext}_{Q}^{1}({T},{T})$ admits an open
$\operatorname{Aut}_{Q}({T})$-orbit.

###### Proof.

Following [18, Theorem A], we obtain $(1),(2)$. Let us consider the condition
$(3)$.

Denote by $\underline{\operatorname{dim}}\widetilde{{T}}=\alpha$, and by
$\Lambda_{\alpha}$ the representation variety of $\Lambda_{Q}$ with dimension
vector $\alpha$. It is well known that

(4.7) $\operatorname{dim}\Lambda_{\alpha}=\operatorname{dim}E_{\alpha}$

$\widetilde{{T}}$ is a rigid module over $\Lambda_{Q}$, e.t.,
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{T}}},{\widetilde{{T}}})=0$. The
equation (4.1) implies that

(4.8)
$\operatorname{dim}\operatorname{Hom}_{\Lambda}({\widetilde{{T}}},{\widetilde{{T}}})=\langle\alpha,\alpha\rangle$

Since $\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})$ is an open dense
subvariety of
$\operatorname{Hom}_{\Lambda}({\widetilde{{T}}},{\widetilde{{T}}})$, then

(4.9)
$\operatorname{dim}\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})=\operatorname{dim}\operatorname{Hom}_{\Lambda}({\widetilde{{T}}},{\widetilde{{T}}})=\langle\alpha,\alpha\rangle$

Let us consider the $G_{\alpha}$-orbit $\mathbb{O}_{\widetilde{{T}}}$ of
$\widetilde{{T}}$, it is well known that

(4.10)
$\operatorname{dim}\mathbb{O}_{\widetilde{{T}}}=\operatorname{dim}G_{\alpha}-\operatorname{dim}Z_{G_{\alpha}}(\widetilde{{T}})$

where $Z_{G_{\alpha}}(\widetilde{{T}})$ refers to the center of
$\widetilde{{T}}$, e.t., $Z_{G_{\alpha}}(\widetilde{{T}})=\left\\{{g\in
G_{\alpha}}\mid\,{g\cdot\widetilde{{T}}=\widetilde{{T}}}\right\\}$. Recall
that for an element $g\in G_{\alpha}$ the action $g\cdot\widetilde{{T}}$ is
given by

$g\cdot\widetilde{{T}}=(g_{t(h)}\widetilde{{T}}_{h}g_{s(h)}^{-1})_{h\in\Omega\cup\Omega^{op}}$

Hence, the condition $g\cdot\widetilde{{T}}=\widetilde{{T}}$ is equivalent to
$g_{t(h)}\widetilde{{T}}_{h}=\widetilde{{T}}_{h}g_{s(h)}$ for all arrows
$h\in\Omega\cup\Omega^{op}$. In other words,
$Z_{G_{\alpha}}(\widetilde{{T}})=\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})$.
The equation (4.10) turns out to be

$\displaystyle\operatorname{dim}\mathbb{O}_{\widetilde{{T}}}=$
$\displaystyle\operatorname{dim}G_{\alpha}-\operatorname{dim}\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})$
$\displaystyle=$
$\displaystyle\operatorname{dim}G_{\alpha}-\langle{\alpha},{\alpha}\rangle$ by
(4.9) $\displaystyle=$
$\displaystyle\operatorname{dim}G_{\alpha}-(\operatorname{dim}G_{\alpha}-\operatorname{dim}E_{\alpha})\quad$
by (1.13) $\displaystyle=$
$\displaystyle\operatorname{dim}E_{\alpha}=\operatorname{dim}\Lambda_{\alpha}$
by (4.7)

This implies that $\mathbb{O}_{\widetilde{{T}}}$ is an open subvariety of
$\Lambda_{\alpha}$.

Let us consider $T$ in $\widetilde{{T}}=(T,T^{*})$ as a representation of $Q$.
The $G_{\alpha}$-action on $\widetilde{{T}}$ is given by

(4.11) $g\cdot(T,T^{*})=(g\cdot T,g\cdot T^{*})\text{ for all $g\in
G_{\alpha}$}$

Therefore, it is easy to see that

$\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})=\operatorname{Aut}_{Q}({T})\cap
Z_{G_{\alpha}}(T^{*})$

Since $T^{*}$ is an element of $\operatorname{Hom}_{Q}({T},\tau{T})$ and
$\operatorname{Aut}_{Q}({T})$ is a subgroup of $G_{\alpha}$, the
$\operatorname{Aut}_{Q}({T})$-action is given by

(4.12) $g\cdot(T,T^{*})=(T,g\cdot T^{*})\text{ for all
$g\in\operatorname{Aut}_{Q}({T})$}$

Therefore, we see that the center of $T^{*}$ in $\operatorname{Aut}_{Q}({T})$
is identified with

(4.13) $\operatorname{Aut}_{Q}({T})\cap
Z_{G_{\alpha}}(T^{*})=\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})$

This implies that the dimension of $\operatorname{Aut}_{Q}({T})$-orbit of
$T^{*}$ is

$\displaystyle\operatorname{dim}\mathbf{O}_{T^{*}}=$
$\displaystyle\operatorname{dim}\operatorname{Aut}_{Q}({T})-\operatorname{dim}\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})$
$\displaystyle=$
$\displaystyle\operatorname{dim}\operatorname{Hom}_{Q}({T},{T})-\langle{\alpha},{\alpha}\rangle$
by (4.9) $\displaystyle=$
$\displaystyle\operatorname{dim}\operatorname{Ext}_{Q}^{1}({T},{T})$ by 2.1
$\displaystyle=$
$\displaystyle\operatorname{dim}\operatorname{Hom}_{Q}({T},\tau{T})$ by
Proposition 3.1

Therefore, we obtain that $\mathbf{O}_{T^{*}}$ is an open orbit in
$\operatorname{Hom}_{Q}({T},\tau{T})$. Applying the functor
$D=\operatorname{Hom}_{k}(-,k)$, we obtain
$\operatorname{Ext}_{Q}^{1}({T},{T})$ admits an open
$\operatorname{Aut}_{Q}({T})$-orbit.

On the other hand, if $\operatorname{Ext}_{Q}^{1}({T},{T})$ admits an open
$\operatorname{Aut}_{Q}({T})$-orbit, then we will show that
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{T}}},{\widetilde{{T}}})=0$
where $\widetilde{{T}}=(T,T^{*})$ and $T^{*}$ is the dual representation of
$T$. The dimension of this open $\operatorname{Aut}_{Q}({T})$-orbit
$\mathbf{O}_{T^{*}}$ implies that the dimension of the center of $T^{*}$ is
equal to

$\operatorname{dim}Z_{\operatorname{Aut}_{Q}({T})}(T^{*})=\operatorname{dim}\operatorname{Aut}_{Q}({T})-\operatorname{dim}\operatorname{Ext}_{Q}^{1}({T},{T})=[T,T]-[T,T]^{1}$

The relation 2.1 implies that
$\operatorname{dim}Z_{\operatorname{Aut}_{Q}({T})}(T^{*})=\langle{\alpha},{\alpha}\rangle$.
The equation (4.13) leads to

$Z_{\operatorname{Aut}_{Q}({T})}(T^{*})=\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})$

Thus, we obtain by (1.13) and (4.7) that

$\operatorname{dim}\operatorname{Aut}_{\Lambda}({\widetilde{{T}}})=\operatorname{dim}\operatorname{Hom}_{\Lambda}({\widetilde{{T}}},{\widetilde{{T}}})=\langle{\alpha},{\alpha}\rangle$

The equation (4.1) implies that

$\operatorname{dim}\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{T}}},{\widetilde{{T}}})=2\operatorname{dim}\operatorname{Hom}_{\Lambda}({\widetilde{{T}}},{\widetilde{{T}}})-(\alpha,\alpha)=0$

We prove that $\widetilde{{T}}$ is a rigid module over $\Lambda_{Q}$. ∎

Following [1], we have the following key proposition.

######  Proposition 4.5 ([1, Corollary 9.3]).

For any two representations $\widetilde{{M}}=(V,x,a)$ and
$\widetilde{{N}}=(W,y,b)$ of $\Lambda_{Q}$, we have

$0\to\operatorname{Coker}{}r_{a,b}\to\operatorname{Ext}_{\Lambda_{Q}}^{1}(\widetilde{{M}},\widetilde{{N}})\to(\operatorname{Coker}r_{b,a})^{*}\to
0$

###### Proof.

Following [1, equation 9.5], we have the following long exact sequence

(4.14)
$\begin{split}0&\to\operatorname{Hom}_{\Lambda_{Q}}(\widetilde{{M}},\widetilde{{N}})\to\operatorname{Hom}_{Q}({M},{N})\stackrel{{\scriptstyle
r_{a,b}}}{{\rightarrow}}\operatorname{Hom}_{Q}({M},\tau{N})\\\
&\stackrel{{\scriptstyle
e_{a,b}}}{{\rightarrow}}\operatorname{Ext}_{\Lambda_{Q}}^{1}(\widetilde{{M}},\widetilde{{N}})\to\operatorname{Ext}_{Q}^{1}({M},{N})\stackrel{{\scriptstyle
r_{b,a}^{*}}}{{\rightarrow}}\operatorname{Ext}_{Q}^{1}({M},{\tau
N})\to(\operatorname{Ker}r_{b,a})^{*}\to 0\end{split}$

where $e_{a,b}(\eta)=\left\\{M\oplus N,\begin{pmatrix}a&\eta^{t}\\\
0&b\end{pmatrix}\right\\}$. ∎

###  4.3. Calculate the extension groups of determinantal modules

In what follows, we will assume that the word $\mathbf{i}$ is (2.3). Following
Section 4.1 and Section 4.2, we will calculate the extension groups of the
determinantal modules $\widetilde{{V}}[a,b]$ and $\widetilde{{V}}[c,d]$. In
Section 4.2, we see that the Kostant partitions corresponding to determinantal
modules are of the form $\tau$-orbit Kostant partitions.

For an $\tau$-orbit Kostant partition $\lambda=(\tau^{u}\beta,\cdots,\beta)$,
Lemma 3.10 implies that $M_{\lambda}^{*}=(a_{i,j})$ for
$\widetilde{{M}}_{\lambda}$ subject to $a_{i,i+1}=1$ for each $i$.

######  Theorem 4.6.

Let $\lambda=(\tau^{u}\alpha,\cdots,\tau\alpha,\alpha)$ and
$\kappa=(\tau^{v}\beta,\cdots,\tau\beta,\beta)$ subject to the assumption 3.4
for some roots $\alpha,\beta$ and some integers $u,v$. We see that

$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$

if and only if

  1. (1)

If $u\geq v$

$\displaystyle\sharp R(\lambda,\kappa)$
$\displaystyle=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp R(\kappa,\lambda)\cap[u-v+1,u]$
$\displaystyle=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

  2. (2)

If $u<v$

$\displaystyle\sharp
R(\lambda,\kappa)\cap[v-u+1,v]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp
R(\kappa,\lambda)=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

###### Proof.

By Proposition 4.5, we see that
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$
if and only if $r_{a,b}$ and $r_{b,a}$ are surjective. By Theorem 3.12, we get
our conclusion. ∎

Next, we give some important examples in [9]. To this end, we first give a
useful lemma.

######  Lemma 4.7.

For two $\tau$-orbit Kostant partitions
$\lambda=(\tau^{u}\alpha,\cdots,\alpha)$ and
$\kappa=(\tau^{u}\beta,\cdots,\beta)$. We always write
$\lambda=({\lambda}_{1},{\lambda}_{2},\cdots,{\lambda}_{u+1})$ and
$\kappa=({\kappa}_{1},{\kappa}_{2},\cdots,{\kappa}_{u+1})$ for
$\lambda,\kappa$. Suppose

$T(\lambda,\kappa)=\left\\{{j\in\mathbb{Z}}\mid\,{[\lambda_{1},\kappa_{j}]=1}\right\\}=\bigsqcup_{p\in[1,m]}[u_{p},v_{p}]\subset[0,+\infty)$

and

$[\lambda_{i},\kappa_{j}]=0\text{ if }j-i<-1$

Here, we assume that $[\tau^{u}\alpha,\kappa_{j}]=0$ when $\kappa_{j}\in
R^{-}$. We have that

$\displaystyle R(\lambda,\kappa)$ $\displaystyle=T(\lambda,\kappa)\cap[1,u+1]$
$\displaystyle\operatorname{\mathfrak{M}}(\lambda,\kappa)$
$\displaystyle=\bigsqcup_{i=1}^{u+1}\left\\{{p\in[1,m]}\mid\,{2-u_{p}\leq
i\leq u-u_{p}+2}\right\\}$
$\displaystyle\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle=\sum_{p=1}^{m^{\prime}}p(v_{p+1}-v_{p})$

where $m^{\prime}=\operatorname{max}\left\\{{p\in[1,m]}\mid\,{v_{p}\leq
u+1}\right\\}$.

###### Proof.

It is easy to see that $R(\lambda,\kappa)=T(\lambda,\kappa)\cap[1,u+1]$. Let
us consider $\operatorname{\mathfrak{M}}(\lambda,\kappa)$. Recall that

$\operatorname{\mathfrak{M}}(\lambda,\kappa)=\left\\{{(i,j)\in[u+1\times
u+1]}\mid\,{\lambda_{i}-\kappa_{j}\in R\cup\\{0\\}\text{ and
}\lambda_{i}\geq\kappa_{j}}\right\\}$

By Proposition 3.5, the condition $\lambda_{i}-\kappa_{j}\in R\cup\\{0\\}$ is
equivalent to

$[\lambda_{i},\kappa_{j}]=1\text{ and }[\lambda_{i},\kappa_{j-1}]=0$

Applying the functor $\tau^{i-1}$, we get

$[\lambda_{1},\kappa_{j-i+1}]=1\text{ and
}[\lambda_{1},\kappa_{j-i}]=0\quad\text{ if }\kappa_{j-i-1}\in R^{+}$

By our assumption, $-1\leq j-i<u$, the condition
$\kappa_{j-i+1}=\tau^{u-(j-i)}\beta\in R^{+}$ always holds, as $\tau^{u}\beta$
is not a root of a projective module, e.t., $\tau^{u+1}\beta\notin R^{-}$.

If we write $p-1=j-i$, then the equation turns out to be

$[\lambda_{1},\kappa_{p}]=1\text{ and }[\lambda_{1},\kappa_{p-1}]=0$

we get $p=u_{p}$, e.t., $j=u_{p}+i-1$ for some $p$. If we fix a number
$i\in[1,u+1]$ and set

$\Gamma_{i}=\left\\{{j\in[1,u+1]}\mid\,{(i,j)\in\operatorname{\mathfrak{M}}(\lambda,\kappa)}\right\\}$

As we discussed before, we obtain

$\Gamma_{i}=\\{u_{p}+i-1\\}_{p\in[1,m]}\cap[1,u+1]$

The number of $\Gamma_{i}$ is the same with the number of $p\in[1,m]$ such
that $1\leq u_{p}+i-1\leq u+1$, e.t.,

$2-u_{p}\leq i\leq u-u_{p}+2$

hence,

$\Gamma_{i}=\left\\{{p\in[1,m]}\mid\,{2-u_{p}\leq i\leq u-u_{p}+2}\right\\}$

and then

$\operatorname{\mathfrak{M}}(\lambda,\kappa)=\bigsqcup_{i=1}^{u+1}\left\\{{p\in[1,m]}\mid\,{2-u_{p}\leq
i\leq u-u_{p}+2}\right\\}$

Let us consider the set $\operatorname{\mathfrak{P}}(\lambda,\kappa)$. Recall
that

$\operatorname{\mathfrak{P}}(\lambda,\kappa)=\left\\{{(i,j)\in[u+1\times
u+1]}\mid\,{\lambda_{i}+\kappa_{j}\in R^{+}\text{ and
}\lambda_{i}\geq\kappa_{j}}\right\\}$

By Proposition 3.5, the condition $\lambda_{i}+\kappa_{j}\in R^{+}$ is
equivalent to

$[\lambda_{i},\kappa_{j}]=0\text{ and }[\lambda_{i},\kappa_{j-1}]=1$

Applying the functor $\tau^{i-1}$, we get

$[\lambda_{1},\kappa_{j-i+1}]=0\text{ and
}[\lambda_{1},\kappa_{j-i}]=1\quad\text{ if }\tau^{u-j+i+1}\beta\in R^{+}$

If we write $p=j-i$, then the equation turns out to be

$[\lambda_{1},\kappa_{p+1}]=0\text{ and }[\lambda_{1},\kappa_{p}]=1$

we get $p=v_{p}$, e.t., $j=v_{p}+i$ for some $p$. If we fix a number
$i\in[1,u+1]$ and set

$\Upsilon_{i}=\left\\{{j\in[1,u+1]}\mid\,{(i,j)\in\operatorname{\mathfrak{P}}(\lambda,\kappa)}\right\\}$

As we discussed before, we obtain

$\Upsilon_{i}=\\{v_{p}+i\\}_{p\in[1,m]}\cap[1,u+1]$

The number of $\Upsilon_{i}$ is the same with the number of $p\in[1,m]$ such
that $1\leq v_{p}+i\leq u+1$, e.t.,

$1-v_{p}\leq i\leq u-v_{p}+1$

hence,

$\Upsilon_{i}=\left\\{{p\in[1,m]}\mid\,{1\leq i\leq u-v_{p}+1}\right\\}\text{
by }v_{p}\geq 0$

Set

(4.15) $\displaystyle
m^{\prime}=\operatorname{max}\left\\{{p\in[1,m]}\mid\,{v_{p}\leq
u+1}\right\\}$ (4.16) $\displaystyle
v^{\prime}_{0}=0=u+1-v_{m^{\prime}+1},v^{\prime}_{1}=$ $\displaystyle
u-v_{m^{\prime}}+1,v^{\prime}_{2}=u-v_{m^{\prime}-1}+1,\cdots,v^{\prime}_{m^{\prime}}=u-v_{1}+1$

we have

$\sharp\Upsilon_{i}=m^{\prime}-k+1\text{ if
}k\in(v^{\prime}_{k-1},v^{\prime}_{k}]$

and then

$\displaystyle\operatorname{\mathfrak{p}}(\lambda,\kappa)=\sum_{i=1}^{u+1}\sharp\Upsilon_{i}=\sum_{k=1}^{m^{\prime}}(v^{\prime}_{k}-v^{\prime}_{k-1})(m^{\prime}-k+1)=\sum_{p=1}^{m^{\prime}}p(v_{p+1}-v_{p})$

∎

Here we will give another proof of some well-known results concerning
determinantal modules:

######  Corollary 4.8 ([9]).

Let $\lambda=(\tau^{u}\alpha,\cdots,\alpha)$ and
$\kappa=(\tau^{u}\beta,\cdots,\beta)$ be two $\tau$-orbit Kostant partitions
subject to Assumption 3.4. Suppose that

$\displaystyle T(\lambda,\kappa)\subset[1,\infty)\text{ and
}T(\kappa,\lambda)\subset[1,\infty)$

and

$\displaystyle[$ $\displaystyle\lambda_{i},\kappa_{j}]=0\text{ if }j-i<-1$
$\displaystyle[$ $\displaystyle\lambda_{i},\kappa_{j}]=0\text{ if }j-i<-1$

We have

$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$

In particular, $\widetilde{{M}}_{\lambda}$ is a rigid module over
$\Lambda_{Q}$.

###### Proof.

Using Lemma 4.7, we get

$R(\lambda,\kappa)=\bigsqcup_{p,v_{p}\leq u+1}[u_{p},v_{p}]$

Since $u_{p}\geq 1$, then $2-u_{p}\leq i\leq u-u_{p}+2$ becomes

$i\leq u-u_{p}+2$

Set

(4.17)
$u^{\prime}_{0}=0=u+2-u_{m^{\prime}+1},u^{\prime}_{1}=u-u_{m^{\prime}}+2,u^{\prime}_{2}=u-u_{m^{\prime}-1}+2,\cdots,u^{\prime}_{m^{\prime}}=u-u_{1}+2$

we have

$\sharp\Gamma_{i}=m^{\prime}-p+1\text{ if
}i\in(u^{\prime}_{p-1},u^{\prime}_{p}]$

it is easy to see that

$\displaystyle\operatorname{\mathfrak{m}}(\lambda,\kappa)$
$\displaystyle=\sum_{i=1}^{u+1}\Gamma_{i}$
$\displaystyle=\sum_{p=1}^{m^{\prime}}(u^{\prime}_{p}-u^{\prime}_{p-1})(m^{\prime}-p+1)$
$\displaystyle=\sum_{p=1}^{m^{\prime}}(u_{m^{\prime}-p+2}-u_{m^{\prime}-p+1})(m^{\prime}-p+1)$
$\displaystyle=\sum_{k=1}^{m^{\prime}}k(u_{k+1}-u_{k})$

Following 4.7 we obtain

$\operatorname{\mathfrak{p}}(\lambda,\kappa)=\sum_{k=1,v_{k}\leq
u+1}k(v_{k+1}-v_{k})$

Hence, we get

$\displaystyle\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle=\sum_{k=1,v_{k}\leq u+1}k[(v_{k}-u_{k})-(v_{k+1}-u_{k+1})]$
$\displaystyle=\sum_{k=1}^{m^{\prime}}(k-(k-1))(v_{k}-u_{k})-m^{\prime}(v_{m^{\prime}+1}-u_{m^{\prime}+1})$
$\displaystyle=\sum_{k=1}^{m^{\prime}}(k-(k-1))(v_{k}-u_{k})-m^{\prime}(u+1-(u+2))\text{
by ~{}(\ref{eq_vprime}) and ~{}(\ref{eq_uprime})}$
$\displaystyle=\sum_{k=1}^{m^{\prime}}(v_{k}-u_{k})+m^{\prime}=\sharp
R(\lambda,\kappa)$

here $m^{\prime}$ is the number such that $v_{m^{\prime}}\leq u+1$ but
$v_{m^{\prime}+1}>u+1$ if it exists. Similarly, we have
$\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)=\sharp
R(\kappa,\lambda)$. Our claim follows from Theorem 4.6. ∎

######  Corollary 4.9 ([9, Theorem 13.1]).

Let $\lambda=(\tau^{u}\beta,\tau^{u-1}\beta,\cdots,\beta)$ be an $\tau$-orbit
Kostant partition, we have that

$\operatorname{dim}\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\tau\lambda}})=1$

Here $\tau\lambda=(\tau^{u+1}\beta,\tau^{u}\beta,\cdots,\tau\beta)$.

###### Proof.

Let us consider the map $r_{\lambda,\tau\lambda}$. Recall

$T(\lambda,\tau\lambda)=\left\\{{i\in\mathbb{Z}}\mid\,{[\tau^{u}\beta,\tau^{u+2-i}\beta]=1}\right\\}\subset[2,\infty)$

and

$[\tau^{u-i+1}\beta,\tau^{u-j+2}\beta]=0\text{ if }j-i<-1$

Therefore, we can us the proof of Corollary 4.8 to show that
$r_{\lambda,\tau\lambda}$ is surjective.

Next, we consider the map $r_{\tau\lambda,\lambda}$. We see that

$T(\tau\lambda,\lambda)=\left\\{{i\in\mathbb{Z}}\mid\,{[\tau^{u+1}\beta,\tau^{u+1-i}\beta]=1}\right\\}=[0,v_{1}]\bigcup\sqcup_{i=1}^{m}[u_{i},v_{i}]\subset[0,\infty)$

In other words,

$R(\tau\lambda,\lambda)=T(\tau\lambda,\lambda)\cap[1,u+1]=[1,v_{1}]\bigcup\sqcup_{i=1}^{m^{\prime}}[u_{i},v_{i}]$

We denote by $m^{\prime}$ the maximal integer $p$ such that $v_{p}\leq u+1$.

Let us consider $\operatorname{\mathfrak{M}}(\tau\lambda,\lambda)$. It is easy
to see that $(\tau\lambda,\lambda)$ satisfies the assumption of Lemma 4.7. For
the condition $2-u_{p}\leq i\leq u-u_{p}+2$, we have $2\leq i\leq u+2$ if
$p=1$. That means $i=1$ is not in this segment.

$\Gamma_{1}=\\{u_{2},u_{3},\cdots,u_{m^{\prime}}\\}$

For any $i\geq 2$, we can only consider $i\leq u-u_{p}+2$. Set

$u^{\prime}_{0}=0=u+2-u_{m^{\prime}+1},u^{\prime}_{1}=u-u_{m^{\prime}}+2,\cdots,u^{\prime}_{m^{\prime}-1}=u-u_{2}+2,u^{\prime}_{m^{\prime}}=u+1$

we have

$\sharp\Gamma_{i}=m-p+1\text{ if }i\in(u^{\prime}_{p-1},u^{\prime}_{p}]\text{
and }i\neq 1$

and then

$\displaystyle\operatorname{\mathfrak{m}}(\tau\lambda,\lambda)=$
$\displaystyle\sharp\Gamma_{1}+\sum_{i\geq 2}\sharp\Gamma_{i}$
$\displaystyle=$ $\displaystyle
m^{\prime}-1+m^{\prime}(u^{\prime}_{1}-1)+\sum_{p\geq
2}^{m^{\prime}}(m^{\prime}-p+1)(u^{\prime}_{p}-u^{\prime}_{p-1})$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{m^{\prime}}(m^{\prime}-p+1)(u^{\prime}_{p}-u^{\prime}_{p-1})-1$
$\displaystyle=$ $\displaystyle\sum_{p=1}^{m^{\prime}}p(u_{p+1}-u_{p})-1$

Hence, we have

$\operatorname{\mathfrak{m}}(\tau\lambda,\lambda)-\operatorname{\mathfrak{p}}(\tau\lambda,\lambda)=\sum_{p=1}^{m^{\prime}}p(u_{p+1}-u_{p})-\sum_{p=1}^{m^{\prime}}p(v_{p+1}-v_{p})-1=\sharp
R(\tau\lambda,\lambda)-1$

It implies that
$\operatorname{dim}\operatorname{Coker}r_{\tau\lambda,\lambda}=1$. By
Proposition 4.5, we get

$\operatorname{dim}\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\tau\lambda}})=1$

∎

######  Remark 4.10.

The above three corollaries are well-known results in [10]. We will show that
the determinantal modules $\widetilde{{M}}_{\lambda}$ correspond to the
quantum minors under the isomorphism 5.1. The condition
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})=0$
means that the corresponding quantum minors
$\mathbf{b}_{\lambda}^{*},\mathbf{b}_{\kappa}^{*}$ are quasicommuting, e.t.,
$\mathbf{b}_{\lambda}^{*}\mathbf{b}_{\kappa}^{*}=q^{n}\mathbf{b}_{\kappa}^{*}\mathbf{b}_{\lambda}^{*}$
for some integer $n$.

######  Example 4.11.

In this section, we focus on $A_{5}$ as in Section 2.2. The set of the
$\tau$-orbit Kostant partitions is given by

$\displaystyle([1],[2],[3],[4])\,$ $\displaystyle\&\,([k],[k+1],\cdots,[l])$
$\displaystyle([1,2],[2,3],[3,4])\,$
$\displaystyle\&\,([k,k+1],\cdots,[l,l+1])$ $\displaystyle([1,3],[2,4])\,$
$\displaystyle\&\,([1,4])$

We list the multsegments $(\lambda,\kappa)$ such that
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})\neq
0$. First, the pair $(\lambda,\tau\lambda)$ subject to this condition. For any
pair such that $\operatorname{supp}\lambda\cup\operatorname{supp}\kappa$ is a
segment and $b(\lambda)=e(\kappa)+1$ or $b(\kappa)=e(\lambda)+1$. To prove it,
we only note that $R(\lambda,\kappa)=R(\kappa,\lambda)=0$,
$[\widetilde{{M}}_{\lambda},\widetilde{{M}}_{\kappa}]=[\widetilde{{M}}_{\kappa},\widetilde{{M}}_{\lambda}]=0$,
and

$[\widetilde{{M}}_{\lambda},\widetilde{{M}}_{\kappa}]^{1}=1\text{ using the
equation ~{}(\ref{eq:extindecom})}$

In $\operatorname{supp}\lambda\cup\operatorname{supp}\kappa$ is a segment and
$\operatorname{supp}\lambda\cap\operatorname{supp}\kappa=\\{k\\}\neq\operatorname{supp}\lambda,\operatorname{supp}\kappa$
for some $k\in I$. we can check that
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})\neq
0$ case by case. More generally, we can check
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{\lambda}},{\widetilde{{M}}_{\kappa}})\neq
0$ if and only if $\operatorname{supp}\lambda$ and $\operatorname{supp}\kappa$
are linked. In particular, in this case, we have that our condition for
$\mathbf{b}_{\lambda}^{*}\mathbf{b}_{\kappa}^{*}\in
q^{\mathbb{Z}}\mathbf{B}^{*}$ coincides with the conditions as in [12,
Corollary 5.10]. We expect our conditions are equivalent to the Condition in
[12, Corollary 5.10] for the case $A_{n}$ with orientation as in Section
(2.2), but we can’t know how to prove it.

##  5\. Quantum minors and determinatal modules

In this section, we briefly recall the notion of quantum minors. For a Weyl
element $w\in W$, one has a quantum unipotnet group
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$. Please see [10] for more
details. For the word $\mathbf{i}$ of an reduced expression of $w$, C.Geiß,
B.Leclerc and J.Schröer define the quantum minors $D(k,l)$ in [10, Section
5.4]. In this section, we assume that $w=w_{0}$ and $\mathbf{i}$ as in 2.3.

Recall Section 4.1, we have a cluster category $\mathcal{C}_{w}$. C.Geiß,
B.Leclerc and J.Schröer construct a quantum cluster algebra
$\mathcal{A}_{\mathbb{Q}(q)}(\mathcal{C}_{w})$ on $\mathcal{C}_{w}$, (see [10,
Section 10]). The following is the main result in their paper [10].

######  Theorem 5.1.

[10, Theorem 12.3] There is an isomorphism
$\Upsilon:\mathcal{A}_{\mathbb{Q}(q)}(\mathcal{C}_{w})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$
by sending $\widetilde{{V}}_{k}$ to $D(0,k)$ for any $k\in[1,r]$. Moreover,
$\Upsilon(\widetilde{{V}}[k^{+},l])=D(k,l)$.

Let us assume that $\mathbf{i}$ is the word as in (2.3) and recall the
notations in (4.2). We set $l(b,d)=l(\lambda(b,d))$. In other words,
$l(b,d)=u+1$ if $\lambda(b,d)=(\tau^{u}\beta=\beta_{d},\cdots,\beta)$.
Following Theorem 3.12, we have

######  Theorem 5.2.

Let $D(b,d)$ and $D(k,l)$ be two quantum minors induced from the word (2.3).
Let us write $\lambda(b,d)=(\tau^{u}\alpha,\cdots,\alpha)$ and
$\lambda(k,l)=(\tau^{v}\beta,\cdots,\beta)$. we have that $D(b,d)D(k,l)$ is a
dual canonical base in $\mathcal{A}_{q}(\operatorname{\mathfrak{n}}(w))$ (up
to $q$-power scaling) if and only if they satisfy the following condition

  1. (1)

if $u\geq v$

$\displaystyle\sharp R(\lambda,\kappa)$
$\displaystyle=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp R(\kappa,\lambda)\cap[v-u+1,v]$
$\displaystyle=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

  2. (2)

If $u<v$

$\displaystyle\sharp
R(\lambda,\kappa)\cap[v-u+1,v]=\operatorname{\mathfrak{m}}(\lambda,\kappa)-\operatorname{\mathfrak{p}}(\lambda,\kappa)$
$\displaystyle\sharp
R(\kappa,\lambda)=\operatorname{\mathfrak{m}}(\kappa,\lambda)-\operatorname{\mathfrak{p}}(\kappa,\lambda)$

###### Proof.

By [9], we have all quantum minors $D(b,d)$ are dual semicanonical bases
$\delta_{\widetilde{{V}}[a^{+},b]}$. The multiplication formula in [8] implies
that
$\delta_{\widetilde{{V}}[b^{+},d]}\delta_{\widetilde{{V}}[k^{+},l]}=q^{n}\delta_{\widetilde{{V}}[b^{+},d]\oplus\widetilde{{V}}[k^{+},l]}$
for some integer $q\in\mathbb{Z}$. ∎

More generally, let $Z_{1}$ and $Z_{2}$ be two quantum cluster monomials in
$\mathcal{A}_{q}(\operatorname{\mathfrak{n}})$ with respect to a cluster
$({X}_{1},{X}_{2},\cdots,{X}_{r})$ and $({Y}_{1},{Y}_{2},\cdots,{Y}_{r})$
respectively such that each $X_{i}$ and $Y_{j}$ are quantum minors. Using this
theorem, one can check the following condition.

(5.1) $Z_{1}Z_{2}=q^{n}Z_{2}Z_{1}\qquad\text{for some integer $n$}$

Because the module $\widetilde{{M}}_{Y}$ (resp:$\widetilde{{M}}_{Z}$) over
$\Lambda_{Q}$ corresponding to $Y$ (resp:$Z$) are the direct sum of finite
determinantal modules $\widetilde{{V}}[a,b]$. The condition
$Z_{1}Z_{2}=q^{n}Z_{2}Z_{1}$ for some $n$ is identified with the following
condition via Theorem 5.1

$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{Z_{1}}},{\widetilde{{M}}_{Z_{2}}})=0$

Based on Theorem 5.2 and
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}\oplus\widetilde{{N}}},{-})=\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}},{-})\oplus\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{N}}},{-})$,
we can check the condition
$\operatorname{Ext}_{\Lambda}^{1}({\widetilde{{M}}_{Z_{1}}},{\widetilde{{M}}_{Z_{2}}})=0$
for any two quantum cluster monomials $Y,Z$ and then check the condition
$Z_{1}Z_{2}=q^{n}Z_{2}Z_{1}$ for some $n$. Therefore, we obtain the following
theorem.

######  Theorem 5.3.

Let $\mathbf{a}=({a}_{1},{a}_{2},\cdots,{a}_{r})$ and
$\mathbf{b}=({b}_{1},{b}_{2},\cdots,{b}_{t})$ be two tuples of positive
integers, and let $Z_{1}=X^{\mathbf{a}}$ and $Z_{2}=Y^{\mathbf{b}}$ be two
quantum cluster monomials in $\mathcal{A}_{q}(\operatorname{\mathfrak{n}})$
with respect to the clusters $({X}_{1},{X}_{2},\cdots,{X}_{r})$ and
$({Y}_{1},{Y}_{2},\cdots,{Y}_{r})$ such that all $X_{i}$ and $Y_{j}$ are
quantum minors. The following holds

(5.2) $Z_{1}Z_{2}\in q^{\mathbb{Z}}\mathbf{B}^{*}\quad\text{ (equivalently
$Z_{1}Z_{2}=q^{n}Z_{2}Z_{1}$ for some $n\in\mathbb{Z}$)}$

if and only if for any pair $(a_{i},b_{j})$ such that $a_{i}\neq 0$ and
$b_{j}\neq 0$, we have the quantum minors $X_{i},Y_{j}$ satisfies the
condition in Theorem 5.2.

## References

  * [1] A. Aizenbud and E. Lapid. _A binary operation on irreducible components of Lusztig’s nilpotent varieties I: definition and properties._ Pure and Applied Mathematics Quarterly, (2022)
  * [2] L. Bittmann and J-R. Li _On the simplicity of the tensor product of two simple modules of quantum affine algebras._. arXiv preprint arXiv:2203.17268 (2022).
  * [3] A. Berenstein, A. Zelevinsky, _String Bases for Quantum Groups of Type $A_{r}$_. Kyoto University. Research Institute for Mathematical Sciences [RIMS], 1992.
  * [4] A.Buan, R. Marsh, M. Reineke, I. Reiten, and G. Todorov. _Tilting theory and cluster combinatorics._ Advances in mathematics 204, no. 2 (2006): 572-618.
  * [5] B. Crawley-Boevey, _Geometry of representations of algebras_ , Lecture notes, (1993)
  * [6] R. Fujita, Graded quiver varieties and singularities of normalized R-matrices for fundamental modules. Selecta Mathematica 28.1 (2022): 1-45.
  * [7] R. Fujita and K. Murakami. _Deformed Cartan matrices and generalized preprojective algebras I: Finite type._ International Mathematics Research Notices 2023.8 (2023): 6924-6975.
  * [8] C. Geiss, B. Leclerc, and J. Schroer, _Semicanonical bases and preprojective algebras II: A multiplication formula._ Compositio Mathematica 143.5 (2007): 1313-1334.
  * [9] C. Geiss, B. Leclerc, and J. Schroer, _Kac–Moody groups and cluster algebras._ Advances in Mathematics 228 (2011): 329-433.
  * [10] C. Geiss,, Leclerc, B., and Schröer, J._Cluster structures on quantum coordinate rings_ Selecta Mathematica, 19 (2013),337-397.
  * [11] S. Kang and M. Kashiwara and M. Kim and S.J. Oh, _Monoidal categorification of cluster algebras_ , Journal of the American Mathematical Society, 31(2), 2018, 349-426.
  * [12] E. Lapid and A. Minguez _On parabolic induction on inner forms of the general linear group over a non-archimedean local field._ Selecta Mathematica 22.4 (2016): 2347-2400.
  * [13] E. Lapid and A. Minguez _Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field_, Advances in Mathematics. (2018) 339 (113–190)
  * [14] E. Lapid and A. Minguez _A binary operation on irreducible components of Lusztig’s nilpotent varieties $\\{$II$\\}$: applications and conjectures for representations of $GL\\_n$ over a non-archimedean local field_. Pure and Applied Mathematics Quarterly, (2022)
  * [15] B. Leclerc and A. Zelevinsky, _Quasicommuting families of quantum Plücker coordinates._ American Mathematical Society Translations (1998): 85-108.
  * [16] B. Leclerc, _Imaginary vectors in the dual canonical basis of $U_{q}(\operatorname{\mathfrak{n}})$_ Transformation Groups 1.8 (2003): 95-104.
  * [17] S.J. Oh, A. Postnikov and D. E. Speyer, _Weak separation and plabic graphs._ Proceedings of the London Mathematical Society 110.3 (2015): 721-754.
  * [18] C,M. Ringel, _The preprojective algebra of a quiver_ Algebras and modules, II (Geiranger, 1996). (1998) 24; 467–480.
  * [19] J. Scott, _Quasi-commuting families of quantum minors._ Journal of Algebra 290.1 (2005): 204-220.