Quivers with relations for symmetrizable Cartan matrices V. Caldero-Chapoton formula

TL;DR

This paper extends the Caldero-Chapoton formula to skew-symmetrizable cluster algebras by using categories of locally free modules over specific Gorenstein algebras, linking representation theory and Lie algebra structures.

Contribution

It generalizes the Caldero-Chapoton formula to a broader class of cluster algebras using new algebraic and categorical frameworks.

Findings

Generalization of Caldero-Chapoton formula to skew-symmetrizable cases

Introduction of categories of locally free modules over Gorenstein algebras

New result on PBW basis of convolution algebra

Abstract

We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in Part I. The proof relies on the realization of the positive part of the enveloping algebra of a simple Lie algebra of the same finite type as a convolution algebra of constructible functions on representation varieties of $H$, given in Part III. Along the way, we obtain a new result on the PBW basis of this convolution algebra.

Equations

$$b_{ij}=\left\{\begin{matrix}c_{ij}&\mbox{if $(j,i)\in\Omega$,}\\ -c_{ij}&\mbox{if $(i,j)\in\Omega$,}\\ 0&\mbox{otherwise.}\end{matrix}\right.$$

$$X_M=\sum _{\mathbf{r}\in {\mathbb{N}}^n}\chi ({\mathrm{Gr}}_{l.f.}(\mathbf{r},M))\prod _{i=1}^n{v}_i^{-⟨\mathbf{r},\underline{\mathrm{rank}}(E_i){⟩}_H-⟨\underline{\mathrm{rank}}(E_i),\underline{\mathrm{rank}}(M)-\mathbf{r}{⟩}_H},$$

$$X_{M(\beta )}=x(\beta ).$$

$${\theta }_{\mathbf{a}}:={\theta }_{{\beta }_1}^{(a_1)}\ast \cdots \ast {\theta }_{{\beta }_r}^{(a_r)},(\mathbf{a}\in {\mathbb{N}}^r)$$

$$M_{\mathbf{a}}:=M({\beta }_1{)}^{a_1}\oplus \cdots \oplus M({\beta }_r{)}^{a_r},(\mathbf{a}\in {\mathbb{N}}^r),$$

$${\theta }_{\mathbf{a}}(M_{\mathbf{b}})={\delta }_{\mathbf{a},\mathbf{b}}.$$

$$F_M(t_1,\dots ,t_n):=\sum _{\mathbf{r}\in {\mathbb{N}}^n}\chi ({\mathrm{Gr}}_{l.f.}(\mathbf{r},M))t_1^{r_1}\cdots t_n^{r_n},$$

$$0\to M\to \underset{1\le k\le n}{⨁}{I}_k^{a_k}\to \underset{1\le k\le n}{⨁}{I}_k^{b_k}\to 0,$$

$$g_M:=(b_k-a_k{)}_{1\le k\le n}\in {\mathbb{Z}}^n.$$

$$Q_1:=\{{\alpha }_{ij}:j\to i\mid (i,j)\in \Omega \}\cup \{{\epsilon }_i:i\to i\mid 1\le i\le n\}.$$

$$s_i(\Omega ):=\{(r,s)\in \Omega \mid i\notin \{r,s\}\}\cup \{(s,r)\in {\Omega }^{\ast }\mid i\in \{r,s\}\},$$

$$H:=H(C,D,\Omega ):=KQ/I$$

$${\epsilon }_i^{c_i}=0;$$

$${\epsilon }_i^{|c_{ji}|}{\alpha }_{ij}={\alpha }_{ij}{\epsilon }_j^{|c_{ij}|}.$$

$$H_i:=K[{\epsilon }_i]/({\epsilon }_i^{c_i})$$

$$⟨-,-{⟩}_H:{\mathbb{Z}}^n\times {\mathbb{Z}}^n\to \mathbb{Z}$$

$$\langle\alpha_{i},\alpha_{j}\rangle_{H}:=\left\{\begin{matrix}c_{i}c_{ij}&\mbox{if $(j,i)\in\Omega$,}\\ c_{i}&\mbox{if $i=j$,}\\ 0&\mbox{otherwise.}\end{matrix}\right.$$

$$⟨\underline{\mathrm{rank}}(M),\underline{\mathrm{rank}}(N){⟩}_H=\dim {\mathrm{Hom}}_H(M,N)-\dim {\mathrm{Ext}}_H^1(M,N),$$

$${\dim }_{\mathbb{C}}{\mathrm{Hom}}_H(M(\beta ),M(\gamma ))$$

$${\dim }_{\mathbb{C}}{\mathrm{Ext}}_H^1(M(\beta ),M(\gamma ))$$

$$\mathcal{F}(H):=\underset{\mathbf{d}\in {\mathbb{N}}^n}{⨁}\mathcal{F}(H{)}_{\mathbf{d}},$$

$$\mathbf{r}:=\mathbf{d}/D:=(d_1/c_1,\dots ,d_n/c_n)$$

$$(f\ast g)(M):=\sum _{m\in \mathbb{C}}m\cdot \chi (\{U\in \mathrm{rep}(H,\mathbf{d})\mid U\subseteq M\text{mbox}andf(U)g(M/U)=m\}),$$

$$1_M(N):=\{\begin{array}{ll}1& \text{if M≅N},\\ 0& \text{otherwise}.\end{array}$$

$$\mathcal{M}(H)=\underset{\mathbf{d}\in {\mathbb{N}}^n}{⨁}\mathcal{M}(H{)}_{\mathbf{d}}$$

$$\mathcal{M}(H{)}_{\mathbf{d}}:=\mathcal{F}(H{)}_{\mathbf{d}}\cap \mathcal{M}(H).$$

$$\mathrm{supp}(f):=\{M\in \mathrm{rep}(H,\mathbf{d})\mid f(M)\ne 0\}$$

$$\mathcal{P}(\mathcal{M})=\underset{\beta \in {\Delta }^{+}(C)}{⨁}\mathcal{P}(\mathcal{M}{)}_{\beta }$$

$$\mathfrak{n}=\underset{\beta \in {\Delta }^{+}(C)}{⨁}{\mathfrak{n}}_{\beta }.$$

$$\dim \mathcal{P}(\mathcal{M}{)}_{\beta }=\dim {\mathfrak{n}}_{\beta }=1$$

$$\mathcal{M}(H{)}_{\mathrm{gr}}^{\ast }:=\underset{\mathbf{d}\in {\mathbb{N}}^n}{⨁}\mathcal{M}(H{)}_{\mathbf{d}}^{\ast }$$

Full text

The first named author acknowledges partial support from CoNaCyT grant no. 239255.

1. 1 Introduction and main results
2. 2 Quivers with relations associated with Cartan matrices
3. 3 Multiplicativity of $X_{M}$
4. 4 Dependence on the symmetrizer
5. 5 Filtrations of preprojective modules
6. 6 PBW bases for convolution algebras
7. 7 $F$-polynomials and $g$-vectors
8. 8 $g$-vectors and injective coresolutions
9. 9 Another expression of $F_{M}$
10. 10 $\varphi_{M(\beta)}$ as a generalized minor
11. 11 Comparison with [YZ]
12. 12 Proof of Theorem 1.2 (d)
13. 13 Examples

1. Introduction and main results

Fomin and Zelevinsky [FZ2] have shown that the classification of cluster algebras of finite type is identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems. For cluster algebras of type $A$, $D$, $E$, Caldero and Chapoton [CC] proved a geometric formula for the cluster expansions of cluster variables with respect to an acyclic initial seed. If $Q$ is the quiver (of the same type $A$, $D$, $E$) attached to this initial seed, the coefficients of the cluster expansion are equal to the Euler characteristics of the Grassmannians of sub-representations of the indecomposable representations of $Q$. In this paper, we generalize the Caldero-Chapoton formula to all finite types by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in [GLS3].

Let us state our result more precisely. Let $H=H(C,D,\Omega)$ be the Iwanaga-Gorenstein algebra introduced in [GLS3, Section 1.4]. (The definition of $H$ is recalled in Section 2.) Here $C=(c_{ij})$ is an $n\times n$ Cartan matrix, $D={\rm diag}(c_{i})$ a symmetrizer for $C$ (that is, $DC$ is a symmetric positive definite matrix), and $\Omega$ an acyclic orientation. We take as base field the field $\mathbb{C}$ of complex numbers.

Let $\operatorname{rep}_{\mathrm{l.f.}}(H)\subseteq\operatorname{rep}(H)$ be the exact subcategory of locally free $H$-modules [GLS3, Section 1.5]. We denote by $\langle\,\cdot,\cdot\,\rangle_{H}$ its homological bilinear form, defined in [GLS3, Section 4]. For $M,N\in\operatorname{rep}_{\mathrm{l.f.}}(H)$ the integer $\langle\,M,N\,\rangle_{H}$ depends only on the rank vectors $\underline{\operatorname{rank}}(M)$ and $\underline{\operatorname{rank}}(N)$. We denote by $E_{i}$ the indecomposable locally free $H$-module with rank vector $\underline{\operatorname{rank}}(E_{i})=(\delta_{ij}\mid 1\leq j\leq n)$.

By [GLS3, Theorem 1.3], the map $M\mapsto\underline{\operatorname{rank}}(M)$ induces a bijection between the set of isomorphism classes of indecomposable locally free rigid $H$-modules and the set $\Delta^{+}(C)$ of positive roots for $C$. We denote by $M(\beta)\in\operatorname{rep}_{\mathrm{l.f.}}(H)$ the indecomposable rigid module with rank vector the positive root $\beta$.

Given $M\in\operatorname{rep}_{\mathrm{l.f.}}(H)$ and ${\mathbf{r}}\in\mathbb{N}^{n}$, let ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)$ be the quasi-projective variety of locally free submodules $N$ of $M$ with rank vector $\underline{\operatorname{rank}}(N)={\mathbf{r}}$. We denote by $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))$ its Euler characteristic.

Let $B=(b_{ij})$ be the $n\times n$ skew-symmetrizable matrix defined by

[TABLE]

Let $R=\mathbb{Z}[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$ be the Laurent polynomial ring in $n$ indeterminates $u_{1},\ldots,u_{n}$. Let $\mathcal{A}\subset R$ be the coefficient-free cluster algebra with initial seed $\Sigma=(B,(u_{1},\ldots,u_{n}))$. (Note that $\Sigma$ is acyclic.) By [FZ2] this is a skew-symmetrizable cluster algebra of the same Cartan type as $C$, whose cluster variables $x(\beta)$ (other than $u_{1},\ldots,u_{n}$) are naturally labelled by the positive roots $\beta\in\Delta^{+}(C)$.

Inspired by [CC, Section 3.1] we make the following definition.

Definition 1.1**.**

For $M\in\operatorname{rep}_{\mathrm{l.f.}}(H)$ set

[TABLE]

where $v_{i}:=u_{i}^{1/c_{i}}$.

It is easy to check that $X_{M}$ is a Laurent polynomial in the variables $u_{i}$ (see below Lemma 7.1). The aim of this paper is to prove the following theorem.

Theorem 1.2**.**

• (a)

For every locally free module $M$ we have $X_{M}\in\mathcal{A}$.

• (b)

For every locally free modules $M$ and $N$, we have $X_{M}\cdot X_{N}=X_{M\oplus N}$.

• (c)

The map $M\mapsto X_{M}$ induces a bijection between isomorphism classes of locally free indecomposable rigid $H$-modules and cluster variables of $\mathcal{A}$ other than $u_{1},\ldots,u_{n}$. More precisely, for every $\beta\in\Delta^{+}(C)$ there holds:

[TABLE]

• (d)

The map $M\mapsto X_{M}$ induces a bijection between isomorphism classes of locally free rigid $H$-modules and cluster monomials of $\mathcal{A}$ which do not contain the cluster variables $u_{1},\ldots,u_{n}$.

Note that when $C$ is symmetric and $D=I_{n}$ the identity matrix, the algebra $H$ is isomorphic to the path algebra over $\mathbb{C}$ of the Dynkin quiver given by $C$ and the orientation $\Omega$. In this case all $H$-modules are locally free, and we have $v_{i}=u_{i}$. Moreover, indecomposable modules over this path algebra are always rigid. Thus, one sees that Theorem 1.2(c) then reduces to the classical Caldero-Chapoton formula [CC, Theorem 3.4].

However, it does not seem possible to extend the proof of [CC] to our setup. We will instead rely on the results of [GLS5] and [YZ]. Here is a sketch of our proof. By [FZ3], cluster expansions of cluster variables are completely determined by their $g$-vectors and their $F$-polynomials. In [YZ] a description of the $g$-vectors and $F$-polynomials of the cluster variables of $\mathcal{A}$ (with respect to its initial cluster $(u_{1},\ldots,u_{n})$) is given in terms of the simply-connected complex algebraic group $G$ with the same Cartan type as $\mathcal{A}$. Let $N$ be a maximal unipotent subgroup of $G$, with Lie algebra $\mathfrak{n}$. In [GLS5] we have given a geometric construction of the enveloping algebra $U(\mathfrak{n})$ as a convolution algebra ${\mathcal{M}}(H)$ of constructible functions on representation varieties of locally free $H$-modules. In particular for every positive root $\beta$ we obtained a primitive element $\theta_{\beta}$ of ${\mathcal{M}}(H)$ such that $\theta_{\beta}(M(\beta))=1$ [GLS5, Theorem 6.1]. Write $\Delta^{+}(C)=\{\beta_{1},\ldots,\beta_{r}\}$. The normalized ordered products

[TABLE]

then form a Poincaré-Birkhoff-Witt basis of ${\mathcal{M}}(H)$. Putting

[TABLE]

we then prove:

Theorem 1.3**.**

Let the elements of $\Delta^{+}(C)$ be numbered so that $\operatorname{Hom}_{H}(M(\beta_{i}),M(\beta_{j}))=0$ if $i<j$. (This is possible in view of [GLS5, Corollary 5.8].) For every $\mathbf{a},\mathbf{b}\in\mathbb{N}^{r}$, we have

[TABLE]

Note that in contrast to the classical quiver theory, the support of the constructible function $\theta_{\mathbf{a}}$ is not in general reduced to the orbit of $M_{\mathbf{a}}$ in its representation variety. So Theorem 1.3 is not at all obvious. We prove it by establishing the existence of certain filtrations of $M(\beta)$ (Theorem 5.7).

Let ${\mathcal{M}}(H)^{*}_{\rm gr}$ denote the graded dual of ${\mathcal{M}}(H)$, and for a locally free $H$-module $M$, denote by $\delta_{M}\in{\mathcal{M}}(H)^{*}_{\rm gr}$ the linear form given by evaluation at $M$. By Theorem 1.3 the basis of ${\mathcal{M}}(H)^{*}_{\rm gr}$ dual to the PBW-basis consists of all (commutative) monomials in the linear forms $\delta_{M(\beta)}$. To prove Theorem 1.2, we introduce for every locally free $H$-module $M$ its $F$-polynomial

[TABLE]

and we give another expression of $F_{M}$ in terms of $\delta_{M}$ (Proposition 9.2). We then show using Theorem 1.3 that $\delta_{M(\beta)}$ is equal to the restriction to $N$ of a generalized minor of $G$, and hence obtain an expression of $F_{M(\beta)}$ as the evaluation of this minor on a certain product of one-parameter subgroups of $N$ (Corollary 10.3). This expression of $F_{M(\beta)}$ is very similar to the expression of the $F$-polynomial of $x(\beta)$ given in [YZ]. Using some commutation relations in $G$ together with classical properties of the generalized minors, we can show that $F_{M(\beta)}$ is in fact equal to the $F$-polynomial of $x(\beta)$ (Theorem 11.3). This is the main step in the proof of Theorem 1.2.

We note that our proof also provides a representation-theoretic interpretation of $g$-vectors. Indeed, for a locally free $H$-module $M$ with injective coresolution

[TABLE]

where $I_{k}$ denotes the injective hull of $E_{k}$, define

[TABLE]

We show (Proposition 8.2) that the $g$-vector of $x(\beta)$ coincides with $g_{M(\beta)}$. To summarize, we have:

Theorem 1.4**.**

For $\beta\in\Delta^{+}(C)$, the $F$-polynomial and $g$-vector of the cluster variable $x(\beta)$ with respect to the initial seed $(u_{1},\ldots,u_{n})$ are $F_{M(\beta)}$ and $g_{M(\beta)}$, as defined by equations (1.1) and (1.2).

We conclude this introduction by mentioning related work in the literature. Several methods have already been explored to obtain generalizations of the Caldero-Chapoton formula to the skew-symmetrizable case. Demonet [D] has used $\Gamma$-equivariant categories of modules over preprojective algebras (where $\Gamma$ is the cyclic group generated by a diagram automorphism) to obtain cluster characters for acyclic symmetrizable cluster algebras in the spirit of [GLS1], see [D, Theorem C]. In [R1, R2], Rupel has used categories of representations of valued quivers over finite fields to obtain a quantum analogue of the Caldero-Chapoton formula for acyclic symmetrizable cluster algebras. Theorem 1.2 provides yet another approach based on categories of locally free $H$-modules.

The paper is organized as follows. Section 2 recalls the definition of the algebras $H$ and of the convolution algebras ${\mathcal{M}}(H)$, as well as the main results from [GLS3, GLS5] which we will need. In Section 3 we prove Theorem 1.2 (b). Sections 5 and 6 contain the proof of Theorem 1.3. In Section 7 we introduce the $g$-vector $g_{M}$ and the $F$-polynomial $F_{M}$ of a locally free $H$-module $M$, and we express $X_{M}$ in terms of $g_{M}$ and $F_{M}$. Section 8 shows that $g_{M(\beta)}$ is equal to the $g$-vector of the cluster variable $x(\beta)$. Sections 9, 10, and 11 contain the proof that $F_{M(\beta)}$ is equal to the $F$-polynomial of $x(\beta)$. Section 12 concludes with the proof of Theorem 1.2 (d). Finally Section 13 illustrates Theorem 1.2 with examples.

2. Quivers with relations associated with Cartan matrices

In this section, we recall some definitions and results from [GLS3].

2.1. The algebras $H(C,D,\Omega)$

Let $C=(c_{ij})\in M_{n}(\mathbb{Z})$ be a Cartan matrix. This means that

• (C1)

$c_{ii}=2$ for all $i$;

• (C2)

$c_{ij}\leq 0$ for all $i\not=j$;

• (C3)

There is a diagonal integer matrix $D={\rm diag}(c_{1},\ldots,c_{n})$ with $c_{i}\geq 1$ for all $i$ such that $DC$ is a symmetric positive definite matrix.

The matrix $D$ appearing in (C3) is called a symmetrizer of $C$. The symmetrizer $D$ is minimal if $c_{1}+\cdots+c_{n}$ is minimal.

An orientation of $C$ is a subset $\Omega\subset\{1,\ldots,n\}\times\{1,\ldots,n\}$ such that the following hold:

• (i)

$\{(i,j),(j,i)\}\cap\Omega\not=\varnothing$ if and only if $c_{ij}<0$;

• (ii)

If $(i,j)\in\Omega$ then $(j,i)\not\in\Omega$.

For an orientation $\Omega$ of $C$ let $Q:=Q(C,\Omega):=(Q_{0},Q_{1},s,t)$ be the quiver with the set of vertices $Q_{0}:=\{1,\ldots,n\}$ and with the set of arrows

[TABLE]

(Thus we have $s(\alpha_{ij})=j$ and $t(\alpha_{ij})=i$ and $s(\varepsilon_{i})=t(\varepsilon_{i})=i$, where $s(a)$ and $t(a)$ denote the starting and terminal vertex of an arrow $a$, respectively.) Let $Q^{\circ}:=Q^{\circ}(C,\Omega)$ be the quiver obtained from $Q$ by deleting all loops $\varepsilon_{i}$. Clearly, $Q^{\circ}$ is an acyclic quiver.

For an orientation $\Omega$ of $C$ and some $1\leq i\leq n$ let

[TABLE]

where $\Omega^{*}:=\{(s,r)\mid(r,s)\in\Omega\}$ denotes the orientation opposite to $\Omega$.

For a quiver $Q=Q(C,\Omega)$ and a symmetrizer $D={\rm diag}(c_{1},\ldots,c_{n})$ of $C$, let

[TABLE]

where $KQ$ is the path algebra of $Q$ over a field $K$, and $I$ is the ideal of $KQ$ defined by the following relations:

• (H1)

For each $i$ we have

[TABLE]

• (H2)

For each $(i,j)\in\Omega$ we have

[TABLE]

Let $\operatorname{rep}(H)$ be the category of finite-dimensional $H$-modules. Such a module $M$ is given by a $K$-vector space $M_{i}$ at each vertex $i$ of $Q$, and linear maps $M(a):M_{s(a)}\to M_{t(a)}$ for each arrow $a$ of $Q$ satisfying the defining relations of $H$. In particular the endomorphism $M(\varepsilon_{i})$ endows $M_{i}$ with the structure of an $H_{i}$-module, where

[TABLE]

is a truncated polynomial ring. An $H$-module $M$ is locally free if $M_{i}$ is a free $H_{i}$-module for all $i$. The rank of a free $H_{i}$-module $M_{i}$ is denoted by $\operatorname{rank}(M_{i})$. For a locally free $H$-module $M$ let $\underline{\operatorname{rank}}(M):=(\operatorname{rank}(M_{1}),\ldots,\operatorname{rank}(M_{n}))$ be the rank vector of $M$. Let $\operatorname{rep}_{\rm l.f.}(H)\subseteq\operatorname{rep}(H)$ be the subcategory of locally free $H$-modules. In particular, let $E_{i}:=H_{i}$ seen as an $H$-module.

We refer to [GLS3] for further details.

2.2. Recollection of results

Theorem 2.1** ([GLS3, Theorem 1.2]).**

The algebra $H$ is a $1$-Iwanaga-Gorenstein algebra. For $M\in\operatorname{rep}(H)$ the following are equivalent:

• (i)

$\operatorname{proj.dim}(M)\leq 1$;

• (ii)

$\operatorname{inj.dim}(M)\leq 1$;

• (iii)

$\operatorname{proj.dim}(M)<\infty$;

• (iv)

$\operatorname{inj.dim}(M)<\infty$;

• (v)

$M$* is locally free.*

Define a bilinear form

[TABLE]

by

[TABLE]

(Here $\alpha_{1},\cdots,\alpha_{n}$ denotes the standard basis of $\mathbb{Z}^{n}$.) For $M,N\in\operatorname{rep}_{\rm l.f.}(H)$ we have

[TABLE]

see [GLS3, Section 4]. An $H$-module $M$ is said to be rigid if $\operatorname{Ext}_{H}^{1}(M,M)=0$.

Let $\tau_{H}$ be the Auslander-Reiten translation for the algebra $H$, and let $\tau_{H}^{-}$ be the inverse Auslander-Reiten translation. An indecomposable $H$-module $M$ is preprojective (resp. preinjective) if there exists some $k\geq 0$ such that $M\cong\tau_{H}^{-k}(P)$ (resp. $M\cong\tau_{H}^{k}(I)$) for some indecomposable projective $H$-module $P$ (resp. indecomposable injective $H$-module $I$). (The usual definition of a preprojective or preinjective module $M$ requires some additional conditions on the Auslander-Reiten component containing $M$.) An indecomposable $H$-module $M$ is called $\tau$-locally free, if $\tau_{H}^{k}(M)$ is locally free for all $k\in\mathbb{Z}$.

Since we assume that $C$ is a Cartan matrix, we can state:

Theorem 2.2** ([GLS3, Theorem 1.3]).**

There are only finitely many isomorphism classes of $\tau$-locally free $H$-modules, and the following hold:

• (i)

The map $M\mapsto\underline{\operatorname{rank}}(M)$ yields a bijection between the set of isomorphism classes of $\tau$-locally free $H$-modules and the set $\Delta^{+}(C)$ of positive roots of the semisimple complex Lie algebra associated with $C$.

• (ii)

For an indecomposable $H$-module $M$ the following are equivalent:

• (a)

$M$* is preprojective;*

• (b)

$M$* is preinjective;*

• (c)

$M$* is $\tau$-locally free;*

• (d)

$M$* is locally free and rigid.*

For $\beta\in\Delta^{+}(C)$, let $M(\beta)$ be the indecomposable locally free rigid $H$-module with $\underline{\operatorname{rank}}(M(\beta))=\beta$. Note that for $1\leq i\leq n$ we have $M(\alpha_{i})=E_{i}$.

2.3. Representations of modulated graphs

With the datum of $(C,D,\Omega)$ and of a field $F$ having field extensions $F_{i}$ of degree $c_{i}$ for every $1\leq i\leq n$, we can associate as in [GLS5, Section 2] a modulated graph in the sense of Dlab and Ringel [DR1]. Let $T$ denote, as in [GLS5, Section 2], the tensor $F$-algebra attached to this modulated graph. This is a finite-dimensional hereditary $F$-algebra of finite representation type given by the Cartan type of $C$. For each positive root $\beta\in\Delta^{+}(C)$ we write $X(\beta)$ for the unique (up to isomorphism) indecomposable $T$-module with dimension vector $\beta$.

In the sequel we will use the following comparison result proved in [GLS5].

Proposition 2.3** ([GLS5, Proposition 5.5]).**

For every $\beta,\gamma\in\Delta^{+}(C)$, we have

[TABLE]

2.4. Convolution algebras

In this section, assume that $K=\mathbb{C}$. For a dimension vector $\mathbf{d}=(d_{1},\ldots,d_{n})\in\mathbb{N}^{n}$, let $\operatorname{rep}(H,\mathbf{d})$ denotes the affine variety of $H$-modules with dimension vector $\mathbf{d}$. Let $G(\mathbf{d})=\prod_{i}\operatorname{GL}(d_{i},\mathbb{C})$, which acts on $\operatorname{rep}(H,\mathbf{d})$ by conjugation. Let

[TABLE]

where ${\mathcal{F}}(H)_{\mathbf{d}}$ is the $\mathbb{C}$-vector space of constructible functions $\operatorname{rep}(H,\mathbf{d})\to\mathbb{C}$ taking constant values on $G(\mathbf{d})$-orbits. For $\mathbf{d}$ such that $\operatorname{rep}_{\rm l.f.}(H,\mathbf{d})\not=\varnothing$ let

[TABLE]

be the associated rank vector. Let $\operatorname{rep}_{\rm l.f.}(H,{\mathbf{r}})$ be the subvariety of $\operatorname{rep}(H,\mathbf{d})$ consisting of the locally free $H$-modules with rank vector ${\mathbf{r}}$.

The space ${\mathcal{F}}(H)$ is endowed with an associative convolution product defined by

[TABLE]

where $f\in{\mathcal{F}}(H)_{\mathbf{d}}$, $g\in{\mathcal{F}}(H)_{\mathbf{e}}$, $M\in\operatorname{rep}(H,\mathbf{d}+\mathbf{e})$, and $\chi$ denotes the topological Euler characteristic. (Note that the sum is finite because $f$ and $g$ are constructible.)

For $M\in\operatorname{rep}(H)$ define $1_{M}\in{\mathcal{F}}(H)$ by

[TABLE]

For $1\leq i\leq n$ let $\theta_{i}:=1_{E_{i}}$. Let

[TABLE]

be the subalgebra of ${\mathcal{F}}(H)$ generated by $\{\theta_{1},\ldots,\theta_{n}\}$, where

[TABLE]

For $f\in{\mathcal{M}}(H)_{\mathbf{d}}$ let

[TABLE]

be the support of $f$. We have $\operatorname{supp}(f)\subseteq\operatorname{rep}_{\rm l.f.}(H,\mathbf{d}/D)$, [GLS5, Lemma 4.2].

By [GLS5, Theorem 1.1], there is a Hopf algebra isomorphism $U(\mathfrak{n})\to{\mathcal{M}}(H)$, where $U(\mathfrak{n})$ is the enveloping algebra of the positive part $\mathfrak{n}$ of the semisimple Lie algebra $\mathfrak{g}$ with Cartan matrix $C$. This map sends the Chevalley generator $e_{i}$ of $U(\mathfrak{n})$ to the constructible function $\theta_{i}$.

Recall that the comultiplication in ${\mathcal{M}}(H)$ is given by $\theta_{i}\mapsto\theta_{i}\otimes 1+1\otimes\theta_{i}$. A constructible function $f\in{\mathcal{M}}(H)_{\mathbf{d}}$ is primitive if and only if $\operatorname{supp}(f)$ consists just of indecomposable modules, [GLS5, Lemma 4.6].

The algebra ${\mathcal{M}}(H)$ is isomorphic to the enveloping algebra $U({\mathcal{P}}({\mathcal{M}}))$ of the Lie algebra ${\mathcal{P}}({\mathcal{M}})\subset{\mathcal{M}}(H)$ of primitive elements. We have a root space decomposition

[TABLE]

which (under the isomorphism $U(\mathfrak{n})\to{\mathcal{M}}(H)$) corresponds to the root space decomposition

[TABLE]

Since we are in the Dynkin case, we have

[TABLE]

for all $\beta\in\Delta^{+}(C)$. Using this and [GLS5, Theorem 6.1] we have

Theorem 2.4**.**

For each $\beta\in\Delta^{+}(C)$ there exists a unique primitive element $\theta_{\beta}\in{\mathcal{P}}({\mathcal{M}})_{\beta}$ such that $\theta_{\beta}(M(\beta))=1$.

For a finite-dimensional $\mathbb{C}$-vector space $V$, let $V^{*}$ be its dual. Let

[TABLE]

be the graded dual of the Hopf algebra ${\mathcal{M}}(H)$. For each locally free $H$-module $M$, let $\delta_{M}\in{\mathcal{M}}(H)_{\rm gr}^{*}$ be the evaluation function defined by

[TABLE]

It follows from the description of the comultiplication of ${\mathcal{M}}(H)$ in [GLS5, Section 4.2] that these delta functions are multiplicative, in the sense that for any locally free $H$-modules $M$ and $N$ one has

[TABLE]

3. Multiplicativity of $X_{M}$

In this section, we give a direct proof that the Laurent polynomials $X_{M}$ have a similar multiplicative property, that is, of Theorem 1.2 (b). This could also be deduced from (2.1), in view of Lemma 7.1 and Proposition 9.2.

Lemma 3.1**.**

Let $M$ and $N$ be locally free $H$-modules. For any rank vector ${\mathbf{r}}\in\mathbb{N}^{n}$ we have

[TABLE]

Proof.

Let $d=\sum_{i}c_{i}r_{i}$ be the corresponding dimension, and consider the ordinary Grassmannian parametrizing all dimension $d$ subspaces of the vector space $M\oplus N$. For $\lambda\in\mathbb{C}^{*}$, consider the linear automorphism $\phi_{\lambda}$ of $M\oplus N$ defined by

[TABLE]

This yields a $\mathbb{C}^{*}$-action on the ordinary Grassmannian defined by

[TABLE]

whose fixed points are the subspaces $L$ satisfying $L=(L\cap M)\oplus(L\cap N)$.

Since $\phi_{\lambda}$ is in fact an $H$-module automorphism, if $L$ is a locally free submodule of $M\oplus N$ then $\phi_{\lambda}(L)$ is also locally free, so this action restricts to a $\mathbb{C}^{*}$-action on ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M\oplus N)$. Suppose that $L$ is a locally free fixed point for this action. Then $L=(L\cap M)\oplus(L\cap N)$. In particular, $L_{i}=(L\cap M)_{i}\oplus(L\cap N)_{i}$ for every $i$. This is a free $H_{i}$-module, so by Krull-Schmidt $(L\cap M)_{i}$ and $(L\cap N)_{i}$ are also free over $H_{i}$. Hence $L\cap M$ and $L\cap N$ are both locally free $H$-submodules of $M$ and $N$. Now the claimed identity follows from the fact that $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M\oplus N))$ is equal to the Euler characteristic of its fixed point subset under the $\mathbb{C}^{*}$-action. Indeed, by the above discussion this fixed point subset is isomorphic to

[TABLE]

∎

Corollary 3.2**.**

For locally free $H$-modules $M$ and $N$ we have

[TABLE]

Proof.

From the definition of $X_{M}$ and the bilinearity of $\langle\,\cdot,\cdot\,\rangle$ it follows that the monomial

[TABLE]

occurs in $X_{M}\cdot X_{N}$ with coefficient

[TABLE]

By Lemma 3.1, this is $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M\oplus N))$, hence the result. ∎

4. Dependence on the symmetrizer

The algebra $H=H(C,D,\Omega)$ and its category of locally free modules depend a lot on the choice of the symmetrizer $D$. But as shown by Theorem 2.2, the classification of the indecomposable rigid locally free modules does not depend on $D$. Moreover, we also have the following result.

Proposition 4.1**.**

For every $\beta\in\Delta^{+}(C)$, the Laurent polynomial $X_{M(\beta)}$ is independent of $D$.

Proof.

By [GLS4, Corollary 1.3], the Euler characteristic $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M(\beta)))$ is independent of $D$. The result then follows immediately from the definition of $X_{M(\beta)}$. ∎

Corollary 4.2**.**

If $C$ is symmetric, then Theorem 1.2 (c) holds.

Proof.

If $C$ is symmetric and $D=I_{n}$, Theorem 1.2 (c) holds by [CC]. Hence, by Proposition 4.1, it also holds for an arbitrary symmetrizer $D=kI_{n}\ (k\geq 1)$. ∎

5. Filtrations of preprojective modules

The results in this section can be found in Omlor’s Master Thesis [O]. Using the theory developed in [GLS3], these are straightforward generalizations of similar results by Dlab and Ringel [DR2], who study the representation theory of species of Dynkin type. For convenience, we include proofs.

As before, let $H=H(C,D,\Omega)$ with $C$ of Dynkin type. We label the positive roots $\beta_{1},\ldots,\beta_{r}$ in $\Delta^{+}(C)$ such that $\operatorname{Hom}_{H}(M(\beta_{i}),M(\beta_{j}))=0$ for all $i<j$. (This is always possible by [GLS5, Corollary 5.8].)

Lemma 5.1**.**

We have $\operatorname{Ext}_{H}^{1}(M(\beta_{j}),M(\beta_{i}))=0$ for all $j\geq i$.

Proof.

We use the Auslander-Reiten formula

[TABLE]

(Here we used that the modules $M(\beta_{j})$ have projective dimension at most one.) We have $\tau_{H}(M(\beta_{j}))\cong M(\beta_{s})$ for some $s>j$. If $\operatorname{Hom}_{A}(M(\beta_{i}),\tau_{H}(M(\beta_{j})))\not=0$ for some $j\geq i$, then $i\geq s>j$, a contradiction. ∎

Lemma 5.2**.**

Let $M\in\operatorname{rep}(H)$ such that there is a filtration

[TABLE]

such that for each $1\leq i\leq t$ we have $M_{i}/M_{i-1}\cong M(\beta_{j})$ for some $1\leq j\leq r$. Let $a_{j}$ be the number of indices $i$ such that $M_{i}/M_{i-1}\cong M(\beta_{j})$. Then there is a filtration

[TABLE]

such that for each $1\leq j\leq r$ we have $N_{j-1}/N_{j}\cong M(\beta_{j})^{a_{j}}$.

Proof.

This follows by induction and by our observation that $\operatorname{Ext}_{H}^{1}(M(\beta_{j}),M(\beta_{i}))=0$ for all $j\geq i$. ∎

Lemma 5.3**.**

For $\gamma\in\Delta^{+}(C)$, let

[TABLE]

be a filtration such that for each $1\leq j\leq r$ we have $N_{j-1}/N_{j}\cong M(\beta_{j})^{a_{j}}$ for some $a_{j}\geq 0$. Suppose that at least two multiplicities $a_{j}$ are non-zero. Then there is no filtration of the form

[TABLE]

such that for each $1\leq j\leq r$ the module $M_{j}/M_{j-1}$ is locally free of rank $a_{j}\beta_{j}$.

Proof.

Let $s$ be maximal with $a_{s}\not=0$. Thus the first filtration yields a non-zero homomorphism $M(\beta_{s})\to M(\gamma)$. Suppose for a contradiction that the filtration

[TABLE]

exits. Then, we have an epimorphism $f\colon M(\gamma)\to X$ with $X:=M_{s}/M_{s-1}$. Since we are in the Dynkin case, we have

[TABLE]

This implies that

[TABLE]

Thus there is a non-zero homomorphism $g\colon X\to M(\beta_{s})$. It follows that $g\circ f\colon M(\gamma)\to M(\beta_{s})$ is non-zero, a contradiction to [GLS5, Corollary 5.8]. ∎

Note that in the situation of Lemma 5.3, if exactly one $a_{j}$ is non-zero, then it has to be equal to one, since a non-trivial multiple of a root is never a root.

Lemma 5.4**.**

Let $k$ be a source in $Q(C,\Omega)^{\circ}$, and let $\alpha+\alpha_{k}=\gamma$ with $\alpha,\gamma\in\Delta^{+}(C)$. Then there exists a short exact sequence

[TABLE]

Proof.

For any rank vector ${\mathbf{r}}$, $Z_{\mathbf{r}}:=\operatorname{rep}_{\rm l.f.}(H,{\mathbf{r}})$ is a maximal irreducible component of the generalized nilpotent variety $\Pi({\mathbf{r}})$ introduced and studied in [GLS6, Section 4].

The modules $M(\gamma)$ and $M(\alpha)$ are both rigid, and their orbit closures are $Z_{\gamma}$ and $Z_{\alpha}$, respectively.

We now use the notation from [GLS6, Section 5]. Since $k$ is a source in $Q(C,\Omega)^{\circ}$ we have $Z_{\gamma}\in\operatorname{Irr}(\Pi({\mathbf{r}})^{k,p})^{\rm max}$ with $p=\gamma_{k}>0$. We get

[TABLE]

(Like $Z_{\gamma}$, the irreducible component $\tilde{f}_{k}^{*}(Z_{\gamma})$ just consists of $H$-modules. Thus it has to be equal to $Z_{\alpha}$.) It follows that there is a short exact sequence

[TABLE]

This finishes the proof. ∎

There is an obvious dual of Lemma 5.4 where one assumes that $k$ is a sink in $Q(C,\Omega)^{\circ}$.

Lemma 5.5**.**

Let $\alpha+\beta=\gamma$ with $\alpha,\beta,\gamma\in\Delta^{+}(C)$. Then there exists a short exact sequence

[TABLE]

or a short exact sequence

[TABLE]

Proof.

In [GLS3, Section 9] we developed reflection functors for generalized preprojective algebras. These restrict to reflection functors

[TABLE]

for $k$ a sink in $Q(C,\Omega)^{\circ}$, and

[TABLE]

for $k$ a source in $Q(C,\Omega)^{\circ}$.

Now we can proceed as in [DR2]. Namely, there is a sequence $(i_{1},\ldots,i_{t})$ such that $M(\alpha^{\prime}):=F_{i_{1}}^{-}\cdots F_{i_{t}}^{-}(M(\alpha))\not=0$ and $M(\beta^{\prime}):=F_{i_{1}}^{-}\cdots F_{i_{t}}^{-}(M(\beta))\not=0$, and one if these modules is locally free injective of rank $\alpha_{k}$ for some $k$. Thus, without loss of generality we can assume that $M(\beta^{\prime})\cong E_{k}$ with $E_{k}$ injective. Set $M(\gamma^{\prime}):=F_{i_{1}}^{-}\cdots F_{i_{t}}^{-}(M(\gamma))$. (Note that the modules $M(\alpha^{\prime})$, $M(\beta^{\prime})$ and $M(\gamma^{\prime})$ are modules over $H(C,D,s_{i_{1}}\cdots s_{i_{t}}(\Omega))$.) Applying Lemma 5.4 we get a short exact sequence

[TABLE]

Since the functors $F_{i}^{+}$ are exact on short exact sequences of $\tau$-locally free modules which do not have non-zero projective direct summands isomorphic to $E_{i}$, we can apply $F_{i_{t}}^{+}\cdots F_{i_{1}}^{+}$ to the sequence above and get a short exact sequence

[TABLE]

Note that in the entire proof we strongly used [GLS3, Proposition 9.6] and [GLS3, Theorem 11.10], compare also the discussion in [GLS5, Section 5]. ∎

A proof of the following lemma can be found in [B, Ch. VI, §1, Proposition 19].

Lemma 5.6**.**

Let $(\gamma,\gamma_{1},\ldots,\gamma_{t})$ be a sequence of positive roots in $\Delta^{+}(C)$ such that

[TABLE]

Then there is a permutation $\pi$ of $\{1,\ldots,t\}$ such that

[TABLE]

is a positive root for each $1\leq s\leq t$.

Theorem 5.7**.**

Let $(\gamma,\gamma_{1},\ldots,\gamma_{t})$ be a sequence of positive roots in $\Delta^{+}(C)$ such that

[TABLE]

Then there is a permutation $\pi$ of $\{1,\ldots,t\}$ and a filtration

[TABLE]

such that $M_{i}/M_{i-1}\cong M(\gamma_{\pi(i)})$ for all $1\leq i\leq t$.

Proof.

The proof is a straightforward induction using Lemmas 5.5 and 5.6. ∎

6. PBW bases for convolution algebras

We keep our numbering convention for the positive roots $\beta_{1},\ldots,\beta_{r}$. Recall from Theorem 2.4 the constructible functions $\theta_{\beta_{i}}$, and for $a_{i}\geq 0$ define

[TABLE]

For $\mathbf{a}=(a_{1},\ldots,a_{r})\in\mathbb{N}^{r}$ let

[TABLE]

Then

[TABLE]

is a PBW basis of ${\mathcal{M}}(H)$. Set $\delta_{\mathbf{a}}:=\delta_{M_{\mathbf{a}}}$. We can now prove the next theorem, which is a reformulation of Theorem 1.3.

Theorem 6.1**.**

For $\mathbf{a},\mathbf{b}\in\mathbb{N}^{r}$ we have

[TABLE]

Proof.

Note that for degree reasons, we have $\delta_{\mathbf{a}}(\theta_{\mathbf{b}})=0$ unless $\sum_{i}a_{i}\beta_{i}=\sum_{i}b_{i}\beta_{i}$. Let us consider first the case when $\mathbf{a}=e_{k}$ is the $k$th standard basis vector of $\mathbb{Z}^{r}$. In other words, we have $M_{\mathbf{a}}=M(\beta_{k})$. By Theorem 2.4,

[TABLE]

Next, let $\mathbf{b}=(b_{1},\ldots,b_{r})\in\mathbb{N}^{r}$ with $\mathbf{b}\not=e_{k}$. Then

[TABLE]

We can assume that

[TABLE]

Now Theorem 5.7 combined with Lemma 5.2 yields a filtration

[TABLE]

such that for each $1\leq j\leq r$ we have $N_{j-1}/N_{j}\cong M(\beta_{j})^{b_{j}}$. Using Lemma 5.3 we see that there is no filtration of the form

[TABLE]

such that for each $1\leq j\leq r$ the module $M_{j}/M_{j-1}$ is locally free of rank $b_{j}\beta_{j}$. But this implies that

[TABLE]

Thus $\delta_{M(\beta_{k})}$ is a dual PBW generator, and the theorem is proved in the case $\mathbf{a}=e_{k}$.

The general case now follows from general Hopf algebra considerations. Indeed, by (2.1), for $\mathbf{a}\in\mathbb{N}^{r}$ we have

[TABLE]

Now, if $\varphi_{1},\ldots,\varphi_{k}\in{\mathcal{M}}(H)^{*}_{\rm gr}$ and $f\in{\mathcal{M}}(H)$, we have

[TABLE]

where $\Delta_{k}:{\mathcal{M}}(H)\to{\mathcal{M}}(H)^{\otimes k}$ denotes the iterated comultiplication. Hence

[TABLE]

Since $\theta_{\beta_{k}}$ is a primitive element of ${\mathcal{M}}(H)$,

[TABLE]

Hence, using (6.1) and the fact that $\Delta_{a_{1}+\cdots+a_{r}}$ is an algebra homomorphism, we see that $\delta_{\mathbf{a}}(\theta_{\mathbf{b}})$ can be nonzero only if $\mathbf{b}=\mathbf{a}$. Moreover in that case, since the coefficient of

[TABLE]

in

[TABLE]

is equal to 1, we get $\delta_{\mathbf{a}}(\theta_{\mathbf{a}})=1$. This finishes the proof. ∎

Corollary 6.2**.**

The set

[TABLE]

is the dual of the PBW basis

[TABLE]

7. $F$-polynomials and $g$-vectors

Fomin and Zelevinsky [FZ3] have shown how to express cluster variables in terms of $F$-polynomials and $g$-vectors. Let us recall this formula for the coefficient-free cluster algebra $\mathcal{A}$.

For $j=1,\ldots,n$, define:

[TABLE]

To every cluster variable $x(\beta)$, Fomin and Zelevinsky attach a $g$-vector $g_{\beta}\in\mathbb{Z}^{n}$ [FZ3, Section 6], and an $F$-polynomial $F_{\beta}\in\mathbb{Z}[t_{1},\ldots,t_{n}]$ [FZ3, Definition 3.3]. One then has [FZ3, Corollary 6.3]:

[TABLE]

where $u^{g_{\beta}}=u_{1}^{g_{\beta,1}}\cdots u_{n}^{g_{\beta,n}}$ and $z=(z_{1},\ldots,z_{n})$. Note that the denominator of [FZ3, Corollary 6.3] does not occur in (7.1) because $\mathcal{A}$ is coefficient-free.

Lemma 7.1**.**

Let $M$ be a locally free $H$-module with rank vector $\underline{\operatorname{rank}}(M)=(m_{i})$. Define $g_{M}\in\mathbb{Z}^{n}$ by:

[TABLE]

and $F_{M}\in\mathbb{Z}[t_{1},\ldots,t_{n}]$ by:

[TABLE]

Then

[TABLE]

Proof.

We have

[TABLE]

hence, for ${\mathbf{r}}=(r_{i})\in\mathbb{N}^{n}$,

[TABLE]

On the other hand, using the definition of $\langle\,\cdot,\cdot\,\rangle$ we have

[TABLE]

Therefore using the definition of $g_{M}$, we get

[TABLE]

which yields

[TABLE]

as required. ∎

Therefore, the proof of Theorem 1.2 (c) is reduced to proving that, for every positive root $\beta$ of $C$, we have

[TABLE]

8. $g$-vectors and injective coresolutions

Let $M$ be a locally free $H$-module. By Theorem 2.1, $M$ has an injective coresolution of length 1:

[TABLE]

Let $I^{0}=\bigoplus_{1\leq k\leq n}I_{k}^{a_{k}}$ and $I^{1}=\bigoplus_{1\leq k\leq n}I_{k}^{b_{k}}$ be the decompositions of $I^{0}$ and $I^{1}$ into indecomposable summands.

Proposition 8.1**.**

We have $g_{M}=(b_{k}-a_{k})_{1\leq k\leq n}$.

Proof.

We have $\langle\,\underline{\operatorname{rank}}(E_{i}),\underline{\operatorname{rank}}(I_{j})\,\rangle_{H}=\delta_{ij}c_{i}$, by definition of the homological bilinear form $\langle\,\cdot,\cdot\,\rangle_{H}$. Comparing this equation with (7.2) and taking into account the explicit expression of $\langle\,\cdot,\cdot\,\rangle_{H}$ given in Section 2.2, we get that

[TABLE]

For an arbitrary locally free $H$-module, we have $\underline{\operatorname{rank}}(M)=\underline{\operatorname{rank}}(I^{0})-\underline{\operatorname{rank}}(I^{1})$, and the result follows from the fact that, by (7.2), the map $\underline{\operatorname{rank}}(M)\mapsto g_{M}$ is $\mathbb{Z}$-linear. ∎

From now on, we fix a $(+)$-admissible sequence ${\mathbf{i}}=(i_{1},\ldots,i_{n})$ [GLS3, Section 2.5]. This means that $\{i_{1},\ldots,i_{n}\}=\{1,\ldots,n\}$, $i_{1}$ is a sink of the acyclic quiver $Q^{\circ}(C,\Omega)$, and $i_{k}$ is a sink of $Q^{\circ}(C,s_{i_{k-1}}\cdots s_{i_{1}}(\Omega))$ for $2\leq k\leq n$. Let $c=s_{i_{1}}\cdots s_{i_{n}}$. Then $c$ is a Coxeter element of $W$. Moreover, $c$ is such that the initial seed of the cluster algebra ${\mathcal{A}}$ which we have defined from the Cartan matrix $C$ and the orientation $\Omega$, coincides with the initial seed of the cluster algebra of [YZ] defined from $C$ and $c$.

Proposition 8.2**.**

For $\beta\in\Delta^{+}(C)$, we have $g_{M(\beta)}=g_{\beta}$.

Proof.

By [YZ, Theorem 1.8], the linear map sending $g_{\beta}$ to $\beta$ can be described in the following way. Let $\gamma_{\beta}$ be the weight of $\mathfrak{g}$ whose coordinates on the basis $\{\varpi_{k}\}$ of fundamental weights are the components of $g_{\beta}$. Then, $c^{-1}(\gamma_{\beta})-\gamma_{\beta}=\beta$.

We know [GLS3, Lemma 3.3] that the indecomposable injective $H$-module $I_{i_{k}}$ has rank vector

[TABLE]

Also, by the previous proposition $g_{I_{i_{k}}}$ expressed as above as a weight is equal to $-\varpi_{i_{k}}$. Now

[TABLE]

This proves that $g_{I_{k}}=g_{\underline{\operatorname{rank}}(I_{k})}$ for $1\leq k\leq n$. The result follows since $(\underline{\operatorname{rank}}(I_{k}))_{1\leq k\leq n}$ is a basis of $\mathbb{Z}^{n}$. ∎

9. Another expression of $F_{M}$

From now on, we fix a $(+)$-admissible sequence ${\mathbf{i}}=(i_{1},\ldots,i_{n})$.

For a locally free $H$-module $M$ and an arbitrary sequence ${\mathbf{j}}=(j_{1},\ldots,j_{s})$ of elements of $\{1,\ldots,n\}$, let ${\mathcal{F}}_{{\mathbf{j}}}(M)$ denote the quasi-projective variety of filtrations

[TABLE]

where $M_{k}/M_{k-1}\cong E_{j_{k}}$ for $1\leq k\leq s$.

Proposition 9.1**.**

Let $M$ be a locally free $H$-module with rank vector ${\mathbf{m}}\in\mathbb{N}^{n}$. Let ${\mathbf{r}}\in\mathbb{N}^{n}$ with $r_{i}\leq m_{i}$ for $1\leq i\leq n$. Set ${\mathbf{j}}_{\mathbf{r}}:=(i_{1}^{r_{i_{1}}},\ldots,i_{n}^{r_{i_{n}}},i_{1}^{m_{i_{1}}-r_{i_{1}}},\ldots,i_{n}^{m_{i_{n}}-r_{i_{n}}})$, where $i_{k}^{a}$ means $i_{k}$ repeated $a$ times. The map $\pi:{\mathcal{F}}_{{\mathbf{j}}_{\mathbf{r}}}(M)\to{\rm Gr}_{\mathrm{l.f.}}(r,M)$ given by $\pi(\mathfrak{f})=M_{r_{1}+\cdots+r_{n}}$ is a surjective morphism whose fibers all have the same Euler characteristic $r_{1}!(m_{1}-r_{1})!\cdots r_{n}!(m_{n}-r_{n})!$. Hence

[TABLE]

Proof.

Let us introduce the intermediate variety ${\mathcal{F}}^{\prime}_{{\mathbf{j}}_{\mathbf{r}}}(M)$ of filtrations

[TABLE]

where $M^{\prime}_{k}/M^{\prime}_{k-1}\cong E_{i_{k}}^{r_{k}}$ and $M^{\prime}_{k+n}/M^{\prime}_{k+n-1}\cong E_{i_{k}}^{m_{k}-r_{k}}$ for $1\leq k\leq n$. We can factor $\pi=\pi^{\prime}\circ\pi^{\prime\prime}$, where $\pi^{\prime}:{\mathcal{F}}^{\prime}_{{\mathbf{j}}_{\mathbf{r}}}(M)\to{\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)$ is defined by $\pi^{\prime}(\mathfrak{f}^{\prime})=M^{\prime}_{n}$ and $\pi^{\prime\prime}:{\mathcal{F}}_{{\mathbf{j}}_{\mathbf{r}}}(M)\to{\mathcal{F}}^{\prime}_{{\mathbf{j}}_{\mathbf{r}}}(M)$ is defined by

[TABLE]

Let $N$ be a locally free submodule of $M$ with rank ${\mathbf{r}}$. Since $i_{1}$ is a sink, $N$ has a unique submodule $M^{\prime}_{1}$ isomorphic to $E_{i_{1}}^{r_{i_{1}}}$. More generally, the assumption on ${\mathbf{i}}$ implies that there is a unique filtration

[TABLE]

with $M^{\prime}_{k}/M^{\prime}_{k-1}\cong E_{i_{k}}^{r_{i_{k}}}$ for $1\leq k\leq n$. Similarly, there is a unique filtration

[TABLE]

with $M^{\prime}_{n+k}/M^{\prime}_{n+k-1}\cong E_{i_{k}}^{m_{i_{k}}-r_{i_{k}}}$ for $1\leq k\leq n$. Therefore $\pi^{\prime}$ is an isomorphism.

Consider now the fiber $(\pi^{\prime\prime})^{-1}(\mathfrak{f}^{\prime})$ of $\pi^{\prime\prime}$ over $\mathfrak{f}^{\prime}$. To determine $\mathfrak{f}$ in this fiber, we must first choose a submodule $M_{1}$ of $M^{\prime}_{1}$ isomorphic to $E_{i_{1}}$. Since $M^{\prime}_{1}\cong E_{i_{1}}^{r_{i_{1}}}$, we claim that the variety of submodules of $M^{\prime}_{1}$ isomorphic to $E_{i_{1}}$ has a natural fibration over the projective space $\mathbb{P}(\mathbb{C}^{r_{i_{1}}})$ with fibers isomorphic to affine spaces (Indeed the kernel $K$ of the structure map $\varepsilon_{i_{1}}$ on $M^{\prime}_{1}$ is an $r_{i_{1}}$-dimensional subspace, and the kernel of the restriction of $\varepsilon_{i_{1}}$ to the submodule $M_{1}$ is a one-dimensional line $L$ inside $K$. The map $M_{1}\mapsto L$ is the required fibration, with fibers isomorphic to $\mathbb{C}^{(c_{i_{1}}-1)(r_{i_{1}}-1)}$.) More generally, the variety of filtrations

[TABLE]

with $M_{k}/M_{k-1}\cong E_{i_{1}}$ has a natural fibration over the flag variety $\operatorname{Fl}(\mathbb{C}^{r_{i_{1}}})$, with fibers isomorphic to affine spaces. Repeating this for every step of $\mathfrak{f}^{\prime}$, we get that $(\pi^{\prime\prime})^{-1}(\mathfrak{f}^{\prime})$ has a fibration with basis the product of flag varieties

[TABLE]

and fibers isomorphic to affine spaces. The proposition now follows from the fact that $\chi(\operatorname{Fl}(\mathbb{C}^{k}))=k!$. ∎

Let $G$ be a complex simple and simply connected algebraic group with Cartan matrix $C$. Let $N$ be a maximal unipotent subgroup of $G$ with Lie algebra $\mathfrak{n}$. It is well-known that the algebra $\mathbb{C}[N]$ of polynomial functions on $N$ can be regarded as the graded dual of the Hopf algebra $U(\mathfrak{n})$. Hence we get a natural isomorphism $\upsilon:{\mathcal{M}}(H)_{\rm gr}^{*}\to\mathbb{C}[N]$.

For a locally free $H$-module $M$, let $\varphi_{M}=\upsilon(\delta_{M})$ denote the polynomial function on $N$ corresponding to the linear form $\delta_{M}$. It can be described more explicitly as follows. Let $e_{i}\ (1\leq i\leq n)$ be the Chevalley generators of $U(\mathfrak{n})$, and let

[TABLE]

be the corresponding one-parameter subgroups of $N$. Then, for arbitrary sequences $(j_{1},\ldots,j_{k})\in\{1,\ldots,n\}^{k}$ and $(t_{1},\ldots,t_{k})\in\mathbb{C}^{k}$ we have

[TABLE]

(Of course, only finitely many tuples $(a_{1},\ldots,a_{k})$ can give a nonzero contribution to this sum.) Using Proposition 9.1, we then deduce immediately the next proposition.

Proposition 9.2**.**

Let ${\mathbf{i}}=(i_{1},\ldots,i_{n})$ be a fixed $(+)$-admissible sequence. For every locally free $H$-module $M$ we have

[TABLE]

In view of this proposition, in order to prove (7.5) we need to understand better the polynomial functions $\varphi_{M(\beta)}$ for $\beta\in\Delta^{+}(C)$. This will be done in the next section. Before that we can already prove:

Corollary 9.3**.**

For every locally free $H$-module $M$, one can write $X_{M}$ as a polynomial in the functions $X_{M(\beta)}$ with $\beta\in\Delta^{+}(C)$.

Proof.

Using Theorem 1.3 and the isomorphism $\upsilon\colon{\mathcal{M}}(H)_{\rm gr}^{*}\to\mathbb{C}[N]$, we see that $\varphi_{M}$ is a polynomial in the functions $\varphi_{M(\beta)}$. Moreover $\mathbb{C}[N]$ is $\mathbb{N}^{n}$-graded and $\varphi_{M}$ is homogeneous of multi-degree $\underline{\operatorname{rank}}(M)$. Hence by Proposition 9.2, $F_{M}$ is a polynomial in the functions $F_{M(\beta)}$. Finally, since $g_{M}$ is a linear function of $\underline{\operatorname{rank}}(M)$, it follows from Lemma 7.1 that $X_{M}$ is a polynomial in the functions $X_{M(\beta)}$. ∎

Corollary 9.4**.**

If $C$ is symmetric, Theorem 1.2 (a) holds.

Proof.

If $C$ is symmetric, we already know by Corollary 4.2 that the functions $X_{M(\beta)}$ with $\beta\in\Delta^{+}(C)$ are the cluster variables of ${\mathcal{A}}$. Hence Corollary 9.3 shows that for every locally free $H$-module $M$, we have $X_{M}\in{\mathcal{A}}$. ∎

10. $\varphi_{M(\beta)}$ as a generalized minor

Let $w_{0}$ be the longest element of the Weyl group $W$ associated with $C$. Let ${\mathbf{j}}=(j_{1},\ldots,j_{r})$ be a reduced word for $w_{0}$ such that $j_{1}$ is a source of $Q^{\circ}(C,\Omega)$, and $j_{k}$ is a source of $Q^{\circ}(s_{j_{k}}\cdots s_{j_{1}}(\Omega))$ for $2\leq k\leq r$. We call such a word ${\mathbf{j}}$ a $(-)$-adapted reduced word for $w_{0}$. It is well known that the positive roots can be enumerated as follows:

[TABLE]

Lemma 10.1**.**

Such an ordering satifies the condition :

[TABLE]

Proof.

For $1\leq k\leq r$, let us write for short

[TABLE]

We also put $\langle-,-\rangle_{0}:=\langle-,-\rangle_{H(C,D,\Omega)}$. It is easy to see that we then have

[TABLE]

Thus, for $k<l$ we get

[TABLE]

Here, the last inequality holds since $j_{l}$ is by definition a source for $s_{j_{l-1}}\cdots s_{j_{1}}(\Omega)$, and $s_{j_{l-1}}\cdots s_{j_{k+1}}(\alpha_{j_{k}})$ is a positive root. (We agree that $s_{j_{l-1}}\cdots s_{j_{k+1}}(\alpha_{j_{k}}):=\alpha_{j_{k}}$ in case $l=k+1$.) A similar argument shows that $\langle\beta_{k},\beta_{l}\rangle_{H}\geq 0$ if $k\geq l$.

Note, that we can identify $\langle-,-\rangle_{H}$ with the Ringel bilinear form attached to the tensor $F$-algebra $T$ of Section 2.3. Recall that the indecomposable $T$-modules are denoted by $X(\beta)\ (\beta\in\Delta^{+}(C))$. Suppose that $\operatorname{Hom}_{T}(X(\beta_{k}),X(\beta_{l}))\neq 0$ for $k<l$. Then, by the Auslander-Reiten formula we get

[TABLE]

This contradicts the fact that the Auslander-Reiten quiver of the representation-finite hereditary $F$-algebra $T$ has no cycles, see for example [ARS, VIII.5]. Hence we have

[TABLE]

and using Proposition 2.3, we deduce that

[TABLE]

∎

Hence this total ordering of the roots satisfies the assumption of Sections 5 and 6, so Theorem 6.1 shows that the root vectors of the dual PBW-basis $\delta_{\mathbf{a}}$ of ${\mathcal{M}}(H)^{*}_{\rm gr}$ defined by this ordering are the delta functions $\delta_{M(\beta_{k})}\ (1\leq k\leq r)$. Therefore, using the isomorphism $\upsilon:{\mathcal{M}}(H)_{\rm gr}^{*}\to\mathbb{C}[N]$, we can identify the polynomial functions $\varphi_{M(\beta_{k})}\ (1\leq k\leq r)$ with the root vectors of the dual PBW-basis of $\mathbb{C}[N]$ defined by this root ordering.

Recall the generalized minors $\Delta_{u(\varpi_{k}),v(\varpi_{k})}\in\mathbb{C}[G]$ introduced by Fomin and Zelevinsky [FZ1]. These are distinguished polynomial functions on $G$ defined using Gaussian decomposition. Here $\varpi_{k}\ (1\leq k\leq n)$ denote the fundamental weights of $G$, and $u,v$ are elements of the Weyl group $W$. We will denote by $D_{u(\varpi_{k}),v(\varpi_{k})}$ the restriction of $\Delta_{u(\varpi_{k}),v(\varpi_{k})}$ to $N$.

Proposition 10.2**.**

For $1\leq k\leq r$, we have

[TABLE]

Proof.

This follows immediately from [GLS2, Proposition 7.4]. ∎

Combining Proposition 10.2 with Proposition 9.2 we obtain:

Corollary 10.3**.**

Let ${\mathbf{i}}=(i_{1},\ldots,i_{n})$ be a $(+)$-admissible sequence, and let ${\mathbf{j}}=(j_{1},\ldots,j_{r})$ be a $(-)$-adapted reduced word for $w_{0}$. For every $1\leq k\leq r$ we have

[TABLE]

11. Comparison with [YZ]

In this section we will deduce (7.5) from the results of [YZ]. By Corollary 4.2, it is sufficient to consider the case when $C$ is not symmetric, that is, of type $B_{n}$, $C_{n}$, $F_{4}$ or $G_{2}$.

Let ${\mathbf{i}}=(i_{1},\ldots,i_{n})$ be a $(+)$-admissible sequence. Then $c=s_{i_{1}}\cdots s_{i_{n}}$ is a Coxeter element. Let $h$ denote the Coxeter number. Since $C$ is not symmetric, $h$ is even and $r=nh/2$. Moreover $w_{0}=(c^{-1})^{h/2}$ [B, Chapter 5, §6, Cor. 3]. Hence

[TABLE]

is a $(-)$-adapted reduced word for $w_{0}$. From now on, we fix ${\mathbf{i}}$ and ${\mathbf{j}}$ as above.

Lemma 11.1**.**

For $1\leq k\leq r$, we write $k=un+v$ with $0\leq u<h/2$ and $1\leq v\leq n$. Then

[TABLE]

Proof.

Clearly, $j_{k}=i_{n-v+1}$. Moreover, for $1\leq i,j\leq n$ we have

[TABLE]

In particular, if $j\not=i$ then $s_{i}(\varpi_{j})=\varpi_{j}$. Hence $s_{j_{1}}\cdots s_{j_{k-1}}(\varpi_{j_{k}})=c^{-u}(\varpi_{i_{n-v+1}})$ and $s_{j_{1}}\cdots s_{j_{k}}(\varpi_{j_{k}})=c^{-u-1}(\varpi_{i_{n-v+1}})$. The lemma follows then immediately from Corollary 10.3. ∎

Let us now quote from [YZ] a formula for the $F$-polynomials $F_{\beta}$ of the cluster variables $x(\beta)$ of ${\mathcal{A}}$. Let $\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{n}_{-}$ be the triangular decomposition of $\mathfrak{g}$, and let $f_{i}\ (1\leq i\leq n)$ denote the Chevalley generators of $\mathfrak{n}_{-}$ corresponding to the negative simple roots $-\alpha_{i}$. Let

[TABLE]

be the corresponding one-parameter subgroups in $G$.

Theorem 11.2** ([YZ, Theorem 1.12]).**

Every $\beta\in\Delta^{+}(C)$ can be written

[TABLE]

for a unique fundamental weight $\varpi_{i}$ and $1\leq j\leq h/2$. Then

[TABLE]

Note that in [YZ], the principal minors $\Delta_{\varpi_{i},\varpi_{i}}$ correspond to the initial cluster variables $u_{1},\ldots,u_{n}$. We can now show

Theorem 11.3**.**

For $\beta\in\Delta^{+}(C)$ we have $F_{M(\beta)}=F_{\beta}$.

Proof.

We first slightly rewrite the formula of Theorem 11.2. Following [YZ, Section 3], let $\iota$ denote the involutive antiautomorphism of $G$ such that

[TABLE]

It satisfies [YZ, (3.7)]

[TABLE]

for any generalized minor $\Delta_{\gamma,\delta}$, and any $x\in G$. Also, denoting by $i\mapsto i^{*}$ the involution on $\{1,\ldots,n\}$ defined by $\varpi_{i^{*}}=-w_{0}(\varpi_{i})$, we have

[TABLE]

Hence, Theorem 11.2 can be restated as

[TABLE]

where $\beta=c^{-h/2+j}(\varpi_{i^{*}})-c^{-h/2+j-1}(\varpi_{i^{*}})$. In other words, we can write

[TABLE]

for a unique $k\in\{1,\ldots,n\}$ and $0\leq u<h/2$, and we have

[TABLE]

On the other hand, by Lemma 11.1 and [YZ, (3.2)], we have

[TABLE]

Recall the following commutation relations in $G$. By [FZ1, (2.13)], we have

[TABLE]

Also, denoting by $x_{\beta}(t)$ the one-parameter subgroup of $N$ attached to $\beta\in\Delta^{+}(C)$ (so that $x_{i}(t)=x_{\alpha_{i}}(t)$), we have that, if $s_{i}(\beta)\in\Delta^{+}(C)$ then

[TABLE]

for some $t^{\prime}\in\mathbb{C}$. Finally, we also have

[TABLE]

Using these relations we get

[TABLE]

for some $w_{1}\in\mathbb{C}$. Iterating this procedure, we obtain

[TABLE]

for some $w_{1},\ldots,w_{n}\in\mathbb{C}$. So, writing

[TABLE]

we see that ${\mathbf{x}}(t)={\mathbf{z}}(t)\mathbf{a}$, with $\mathbf{a}\in N$. Hence it follows from the definition of the generalized minors that

[TABLE]

Moreover, writing $\gamma_{k}=s_{i_{1}}\cdots s_{i_{k-1}}(\alpha_{i_{k}})\ (1\leq k\leq n)$, we have

[TABLE]

for some $w^{\prime}_{1},\ldots,w^{\prime}_{n}\in\mathbb{C}$. Hence, writing $\mathbf{b}=x_{c(\gamma_{1})}(w^{\prime}_{n})x_{c(\gamma_{2})}(w^{\prime}_{n-1})\cdots x_{c(\gamma_{n})}(w^{\prime}_{1})$, we have that

[TABLE]

with $\mathbf{b}\in N$ (because $c(\gamma_{1}),\ldots,c(\gamma_{n})$ are positive roots). Again, using the definition of the generalized minors, this implies that

[TABLE]

Since for $0\leq u<h/2$ we have $c^{u}(\gamma_{k})\in\Delta^{+}(C)$ for all $1\leq k\leq n$, we can iterate this procedure and show that

[TABLE]

So we have proved that the right-hand sides of (11.1) and (11.2) are equal. This finishes the proof. ∎

Putting together Proposition 8.2 and Theorem 11.3, we have now a complete proof of Theorem 1.2 (c). Moreover, using Corollary 9.3, this finishes also the proof of Theorem 1.2 (a).

12. Proof of Theorem 1.2 (d)

In [FZ2], when the orientation $\Omega$ is such that every vertex of $Q^{\circ}(C,\Omega)$ is either a source or a sink, the clusters of $\mathcal{A}$ were described by means of the compatibility degree $(\alpha\|\beta)$ of positive roots $\alpha,\beta\in\Delta^{+}(C)$. When $C$ is symmetric, this was extended in [MRZ] to all orientations $\Omega$ by introducing an $\Omega$-compatibility degree $(\alpha\|\beta)_{\Omega}$ using decorated representations of the Dynkin quiver $Q^{\circ}(C,\Omega)$. This description was eventually extended by Zhu [Z] to all Cartan matrices $C$ using the cluster category of a modulated graph of the same type as $C$ and orientation $\Omega$.

Recall from Section 2.3 the tensor $F$-algebra $T$ attached to such a modulated graph, and its indecomposable modules $X(\gamma)$ with dimension vectors $\gamma\in\Delta^{+}(C)$. It follows from [Z] that a subset of cluster variables $\{x(\gamma_{1}),\ldots,x(\gamma_{k})\}$ of $\mathcal{A}$ (other than $u_{1},\ldots,u_{n}$) is $\Omega$-compatible if and only if for every $i,j$ we have $\operatorname{Ext}^{1}_{T}(X(\gamma_{i}),X(\gamma_{j}))=0$.

Therefore Theorem 1.2 (d) is now an immediate consequence of Theorem 1.2 (c) and Proposition 2.3. This finishes the proof of Theorem 1.2.

13. Examples

13.1. Type $B_{2}$

We use the same notation as in [GLS3, §13.6]. Thus $H=H(C,D,\Omega)$ is given by the quiver

[TABLE]

with unique relation $\varepsilon_{1}^{2}=0$. The skew symmetrizable matrix $B$ is given by

[TABLE]

Hence, using the notation of Section 7, we have $z_{1}=u_{2}^{-2}$ and $z_{2}=u_{1}$.

(a) For $M=M(\alpha_{1})=E_{1}$, we have ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)\not=\emptyset$ only if ${\mathbf{r}}=(0,0)$ or ${\mathbf{r}}=(1,0)$. Clearly in both cases ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)$ is just a point, so $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))=1$. Hence we have $F_{M}(t_{1},t_{2})=1+t_{1}$. Moreover the injective coresolution of $E_{1}$ is $0\to E_{1}\to I_{1}\to I_{2}^{2}\to 0$. Hence $g_{M}=(-1,2)$, and

[TABLE]

(b) For $M=M(\alpha_{2})=E_{2}=S_{2}=I_{2}$, we have ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)\not=\emptyset$ only if ${\mathbf{r}}=(0,0)$ or ${\mathbf{r}}=(0,1)$. Again in both cases $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))=1$. So $F_{M}(t_{1},t_{2})=1+t_{2}$. Moreover $M=I_{2}$ is injective, so $g_{M}=(0,-1)$

[TABLE]

(c) For $M=M(\alpha_{1}+\alpha_{2})=P_{2}$, we have ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)\not=\emptyset$ only if ${\mathbf{r}}=(0,0)$, ${\mathbf{r}}=(1,0)$, or ${\mathbf{r}}=(1,1)$. Again in all cases $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))=1$, so $F_{M}(t_{1},t_{2})=1+t_{1}+t_{1}t_{2}$. The injective coresolution of $P_{2}$ is $0\to P_{2}\to I_{1}\to I_{2}\to 0$. Hence $g_{M}=(-1,1)$ and

[TABLE]

(d) For $M=M(\alpha_{1}+2\alpha_{2})=I_{1}$, we have ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M)\not=\emptyset$ only if ${\mathbf{r}}=(0,0)$, ${\mathbf{r}}=(1,0)$, ${\mathbf{r}}=(1,1)$ or ${\mathbf{r}}=(1,2)$. If ${\mathbf{r}}=(1,1)$ then ${\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))$ is isomorphic to $\mathbb{P}^{1}(\mathbb{C})$ and $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))=2$. In all other cases $\chi({\rm Gr}_{\mathrm{l.f.}}({\mathbf{r}},M))=1$. So $F_{M}(t_{1},t_{2})=1+t_{1}+2t_{1}t_{2}+t_{1}t_{2}^{2}$. Since $M=I_{1}$ is injective, $g_{M}=(-1,0)$, and

[TABLE]

(e) Hence we have

[TABLE]

This shows that the 6 cluster variables of $\mathcal{A}$ are equal to

[TABLE]

(f) It is easy to see that if $M$ denotes the non rigid locally free indecomposable module, we have $X_{M}=X_{P_{2}}$.

(g) Finally, it is easy to check using the Auslander-Reiten formula and the Auslander-Reiten quiver of $H$ displayed in [GLS3] that the only locally free rigid $H$-modules are of the form

[TABLE]

These are in one-to-one correspondence with the cluster monomials of $\mathcal{A}$ which do not contain $u_{1},u_{2}$. (The clusters containing $u_{1}$ or $u_{2}$ are $(u_{1},u_{2})$, $(u_{1},X_{E_{2}})$, and $(u_{2},X_{E_{1}})$.)

13.2. Type $G_{2}$

We use the same notation as in [GLS3, §13.9]. Thus $H=H(C,D,\Omega)$ is given by the quiver

[TABLE]

with unique relation $\varepsilon_{2}^{3}=0$. The skew symmetrizable matrix $B$ is given by

[TABLE]

Hence $z_{1}=u_{2}$ and $z_{2}=u_{1}^{-3}$.

(a) Easy calculations similar to those in type $B_{2}$ yield:

[TABLE]

[TABLE]

(b) Let $M=M(3\alpha_{1}+2\alpha_{2})$ be the largest indecomposable locally free rigid module. One calculates

[TABLE]

which yields

[TABLE]

(c) As can be seen from the Auslander-Reiten quiver of $H$ (see [GLS3, Figure 11]), there are two non-rigid indecomposable locally free $H$-modules $M_{1}$ and $M_{2}$ with rank vector $(3,2)$. They can be identified via their graded dimension vectors:

[TABLE]

Similar calculations show that

[TABLE]

Thus, $X_{M_{2}}$ is not a cluster variable.

(d) Dually, one can also calculate some PBW basis vectors of the algebra ${\mathcal{M}}(H)$. As mentioned in the introduction, the support of $\theta_{\mathbf{a}}$ is in general not reduced to the orbit of $M_{\mathbf{a}}$ in its representation variety (see e.g. the type $B_{2}$ case in [GLS5, Section 8.1]).

Using the calculation of (c), one can check that the orbit of $M_{2}$ is contained in the support of $\theta_{3\alpha_{1}+2\alpha_{2}}$, and also in the support of $\theta_{\alpha_{1}+\alpha_{2}}*\theta_{2\alpha_{1}+\alpha_{2}}$. This shows that the supports of two PBW basis vectors may also intersect non-trivially.

13.3. Type $B_{3}$

We use the same notation as in [GLS3, §13.7]. Here $H=H(C,D,\Omega)$ is given by the quiver

[TABLE]

with relations $\varepsilon_{1}^{2}=\varepsilon_{2}^{2}=0$, and $\varepsilon_{1}\alpha_{12}=\alpha_{12}\varepsilon_{2}$. The skew symmetrizable matrix $B$ is given by

[TABLE]

so $z_{1}=u_{2}^{-1}$, $z_{2}=u_{1}u_{3}^{-2}$, and $z_{3}=u_{2}$.

Let $M=M(\alpha_{1}+2\alpha_{2}+2\alpha_{3})$. One calculates

[TABLE]

which yields

[TABLE]

Note that $z_{2}=z_{1}z_{2}z_{3}$, so the two rank vectors $(0,1,0)$ and $(1,1,1)$ contribute to the same monomial with coefficient $1+2=3$.

Acknowledgements.  The authors thank the Mathematisches Forschungsinstitut Oberwolfach (MFO) for two weeks of hospitality in February/March 2017 where most of this work was done. The second named author thanks Kiyoshi Igusa and Gordana Todorov for interesting discussions at the Centre de Recerca Matemática (CRM) Barcelona which arose his interest in this problem.