On a family of Caldero-Chapoton algebras that have the Laurent phenomenon

TL;DR

This paper constructs a family of generalized cluster algebras as Caldero-Chapoton algebras associated with quivers derived from polygons with an orbifold point, establishing their algebraic and combinatorial properties.

Contribution

It introduces a new geometric realization of generalized cluster algebras via Caldero-Chapoton algebras linked to polygons with orbifold points, connecting algebraic and combinatorial structures.

Findings

Realization of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations.

Establishment of a bijection between cluster variables and isomorphism classes of certain representations.

Verification that exchange relations correspond to products of Caldero-Chapoton functions.

Abstract

We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc $j$ on the polygon with orbifold point a representation $M(j)$ of the referred quiver with relations, and by proving that for every triangulation $\tau$ and every arc $j\in\tau$, the product of the Caldero-Chapoton functions of $M(j)$ and $M(j')$, where $j'$ is the arc that replaces $j$ when we flip $j$ in $\tau$, equals the corresponding exchange polynomial of Chekhov-Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of $E$-rigid indecomposable decorated representations of $\Lambda$.

Figures

Equations

$$c_{i,j}=\left|\{a\in Q_1:h(a)=i,t(a)=j\}\right|-\left|\{a\in Q_1:h(a)=j,t(a)=i\}\right|.$$

$$\mathbb{C}⟨Q⟩=\underset{m\ge 0}{⨁}\mathbb{C}{C}_m,$$

$$\mathbb{C}⟨⟨Q⟩⟩=\prod _{m\ge 0}\mathbb{C}{C}_m,$$

$$(\sum _{l\ge 0}{b}_l)(\sum _{m\ge 0}{a}_m)=\sum _{k\ge 0}\sum _{l+m=k}{b}_l{a}_m.$$

$${\partial }_a(a_n{a}_{n-1}\cdots a_1)=\sum _{k=1}^n{\delta }_{a,a_k}{a}_{k-1}{a}_{k-2}\cdots a_1{a}_n{a}_{n-1}\cdots a_{k+2}{a}_{k+1},$$

$${\mathrm{rep}}_{\text{d}}(Q)=\prod _{a\in Q_1}{\mathrm{Hom}}_{\mathbb{C}}({\mathbb{C}}^{d_{t(a)}},{\mathbb{C}}^{d_{h(a)}}).$$

$$g\cdot M=((M_i{)}_{i\in Q_0},(g_{h(a)}{M}_a{g}_{t(a{)}^{-1}}{)}_{a\in Q_1}).$$

$${\mathrm{decrep}}_{\text{d},\text{v}}(\Lambda )={\mathrm{rep}}_{\text{d}}(\Lambda )\times \{({\mathbb{C}}^{v_1},\dots ,{\mathbb{C}}^{v_n})\}.$$

$$\mathcal{F}gh\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{F}}_{X,Y}:\underset{g\in G}{⨁}\mathcal{A}(X,g\cdot Y)\to \mathcal{B}(\mathcal{F}(X),\mathcal{F}(Y));(u_g{)}_{g\in G}\mapsto \sum _{g\in G}{\delta }_{g,Y}\mathcal{F}(u_g),$$

$${\mathcal{F}}^{X,Y}:\underset{g\in G}{⨁}\mathcal{A}(g\cdot X,Y)\to \mathcal{B}(\mathcal{F}(X),\mathcal{F}(Y));(v_g{)}_{g\in G}\mapsto \sum _{g\in G}\mathcal{F}(v_g){\delta }_{g,X}^{-1}.$$

$$g_i:=g_i(\mathcal{M})=-\dim {\mathrm{Hom}}_{\Lambda }(S_i,M)+\dim {\mathrm{Ext}}_{\Lambda }^1(S_i,M)+\dim (V_i).$$

$$0\to M\to I_0(M)\to I_1(M),$$

$$g_i=-a_i+b_i+\dim (V_i).$$

$$E_{\Lambda }(\mathcal{M},\mathcal{N})=\dim {\mathrm{Hom}}_{\Lambda }(M,N)+\sum _{i=1}^n\dim (M_i)g_i(\mathcal{N}).$$

$$E_{\Lambda }(\mathcal{M},\mathcal{N})=E_{{\Lambda }_p}(\mathcal{M},\mathcal{N})=\dim {\mathrm{Hom}}_{{\Lambda }_p}({\tau }_{{\Lambda }_p}^{-}(N),M)+\sum _{i=1}^n\dim (M_i)\dim (W_i).$$

$${\mathcal{C}}_{\Lambda }(\mathcal{M})={\text{x}}^{g_{\Lambda }(\mathcal{M})}\sum _{\text{e}\in \mathbb{N}}\chi ({\mathrm{Gr}}_{\text{e}}(M)){\text{x}}^{C_Q\text{e}},$$

$${\mathcal{C}}_{\Lambda }=\{{\mathcal{C}}_{\Lambda }(\mathcal{M}):\mathcal{M}\in \mathrm{decrep}(\Lambda )\}.$$

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$g_{\Lambda}(M_{\textbf{e}})_{k}=\left\{\begin{array}[]{lll}-1\mbox{ if }k=j,\\ \quad 1\mbox{ if }k=i-1,\\ \quad 0\mbox{ in other wise}.\end{array}\right.$$

$$\mathrm{Irr}(\Lambda )=\bigcup _{\text{d}}{\mathrm{Irr}}_{\text{d}}(\Lambda )\text{mbox}and\mathrm{decIrr}(\Lambda )=\bigcup _{(\text{d},\text{v})}{\mathrm{decIrr}}_{\text{d},\text{v}}(\Lambda )$$

$${\mathrm{decrep}}_{\text{d},\text{v}}(\Lambda )={\mathrm{rep}}_{\text{d}}(\Lambda )\times \{({\mathbb{C}}^{v_1},\dots ,{\mathbb{C}}^{v_n})\}.$$

$$c_{\Lambda }(Z)$$

$$e_{\Lambda }(Z)$$

$${\mathrm{ext}}_{\Lambda }^1(Z_1,Z_2)$$

$$c_{\Lambda }(Z)\le e_{\Lambda }(Z)\le E_{\Lambda }(Z)\text{mbox}and{\mathrm{ext}}_{\Lambda }^1(Z_1,Z_2)\le E_{\Lambda }(Z_1,Z_2).$$

$$C_{\text{d, v}}:{\mathrm{decrep}}_{\text{d, v}}(\Lambda )\to \mathbb{Z}[x_1^{\pm },\dots ,x_n^{\pm }],$$

$${\mathcal{A}}_{\Lambda }\subseteq \mathbb{C}[{\mathcal{C}}_{\Lambda }(Z_1{)}^{\pm },\dots {\mathcal{C}}_{\Lambda }(Z_t{)}^{\pm }].$$

$$z_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$b^{\prime}_{ij}=\left\{\begin{array}[]{ll}\quad-b_{ij}\mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if }k=i\mbox{ or }k=j,\\ b_{ij}+\frac{|b_{ik}|b_{kj}+b_{ik}|b_{k,j}|}{2}\mbox{ if }i\neq k\neq j.\end{array}\right.$$

$$v_k^{+}=\prod _{b_{l,k}>0}{x}_l^{b_{l,k}/d_k},v_k^{-}=\prod _{b_{l,k}<0}{x}_l^{-b_{l,k}/d_k}\text{mbox}and{\theta }_k(u,v)=\sum _{l=0}^{d_k}{u}^l{v}^{d_k-l}.$$

Full text

1 Introduction

The cluster algebras of Sergey Fomin and Andrei Zelevinsky have pervaded several areas of Mathematics and even Physics in the past 15 years. They are defined through a recursive process by iterating an algebraic-combinatorial operation called cluster mutation. Each cluster mutation produces a rational function by means of an exchange binomial dictated by a skew-symmetrizable matrix. Cluster algebras satisfy the remarkable Laurent phenomenon: all of the rational functions obtained during the process can be expressed as Laurent polynomials, cf [20, 21].

In [14], Leonid Chekhov and Michael Shapiro have defined the notion of generalized cluster algebra. The cluster mutation rule inside a generalized cluster algebra allows exchange polynomials to not be binomials, in contrast to cluster algebras, where all exchange polynomials are required to be binomials. Generalized cluster algebras, possess the remarkable Laurent phenomenon. Chekhov-Shapiro have shown that the “Lambda length coordinate ring” of a surface with marked points and arbitrary-order orbifold points carries a natural structure of generalized cluster algebra, thus generalizing previous works of Penner, Fomin-Shapiro-Thurston and Felikson-Shapiro-Tumarkin. Thomas Lam and Pavlo Pylyavskyy have shown that the generalized cluster algebras of Chekhov-Shapiro fit into a more general framework of Laurent phenomenon algebras (LP-algebras for short).

Works of Caldero-Chapoton, Caldero-Chapoton-Schiffler, Derksen-Weyman-Zelevinsky, Palu, Plamondon and Schiffler, among others, have established a very fruitful representation-theoretic approach to cluster algebras through representations of quivers. By now, it is very well known that every cluster monomial in a skew-symmetric cluster algebra can be expressed as the Caldero-Chapoton function of a representation of a suitable quiver, and that, for a non-degenerate quiver with potential $(Q,S)$, the Caldero-Chapoton algebra of the Jacobian algebra of $(Q,S)$ sits between the cluster algebra and the upper cluster algebra of $Q$, see [11, Proposición 7.1]. In this paper we prove that the generalized cluster algebra associated by Chekhov-Shapiro to a polygon with one orbifold point of order 3 is equal to the Caldero-Chapoton algebra of a quiver with relations naturally arising from the referred polygon.

Let us give a more specific context for the paper. Let $Q$ be a quiver of type ADE, and let $\Lambda=\mathbb{C}\langle Q\rangle$ be the path algebra of $Q$ over $\mathbb{C}$. In [9], Caldero-Chapoton defined a Laurent polynomial $\mathcal{C}_{\Lambda}(M)$ (Caldero-Chapoton function for us) for every representation $M$ of $\Lambda$ and show that there is a bijection, given by $\mathcal{C}_{\Lambda}(-)$, between the cluster variables of the cluster algebra $\mathcal{A}_{Q}$ and the indecomposable representations of $\Lambda$. Recall that the cluster algebras are defined by an inductive procedure called cluster mutation, see [20, 21], but the Caldero-Chapoton functions are defined without an inductive procedure. The coefficients of $\mathcal{C}_{\Lambda}(M)$ are given by the complex Euler charateristics of Grassmannians of submodules of quiver representations. The class of cluster algebras which cluster variables have been described with this idea has been extended, for example see [18, 22, 30, 32].

In [11], the authors define the Caldero-Chapoton algebra associated to a basic algebra $\Lambda$ and take account the *strongly reduced irreducible components * $\operatorname{decIrr^{s.r}}(\Lambda)$ of decorated representations of $\Lambda$. With this data they define a generic cluster structure on $\operatorname{decIrr^{s.r}}(\Lambda)$, namely they define the component clusters and $CC$-clusters of $\Lambda$, as generalizations of clusters and cluster variables in cluster algebras, respectively. The strongly reduced components were introduced in [22]. In [33] the strongly reduced components are parametrized in terms of the $g$-vector of irreducible components for finite-dimensional algebras. In [11] the results of [33] are generalized for basic algebras. A comment deserve to be done about basic algebras. The definition given in [11] is not the usual one, we kindly ask to the reader to see Definition 2.1 and be cautious.

These algebras with cluster structures have been related with surfaces in different contexts. Recently Charles Paquette and Ralf Schiffler introduced some notion of generalized cluster algebra that is different from the one mentioned above. In [19], cluster algebras are related with surfaces with marked points and orbifold points of order two. In [35], non-orientable surfaces are related with LP-algebras. In [14], some surfaces with arbitrary many orbifold points of arbitrary order are related with cluster generalized algebras. The interaction between the cluster algebraic structures and cluster combinatorial structures have been extensively studied, for instance [27, 34]. In this paper we show one more of those interactions.

In order to write our main result, see Theorem 11.6, let us introduce some notation. Let $\Sigma_{n}$ be a $(n+1)$-polygon with one orbifold point of order three. For any triangulation $\tau$ of $\Sigma_{n}$ we define an algebra $\Lambda(\tau)$. This algebra is given by a quiver associated to $\tau$ and relations given by the internal triangles of the triangulation. Our main result is:

Theorem 1.1

For any triangulation $\tau$ of $\Sigma_{n}$ the Caldero-Chapoton algebra of ${\Lambda(\tau)}$ is a generalized cluster algebra naturally associated to $\Sigma_{n}$.

The paper is organized as follows. In the first five sections we recall some facts about Caldero-Chapoton algebras, Galois $G$-covering functors, string algebras, cluster generalized algebras and surfaces with marked points and orbifold points of order 3 that we need for stating and proving our results. In Section 2 we recall the non-standard definition of basic algebra introduced in [11] and its representations. We also recall some facts about Galois $G$-coverings that will be crucial to deal with the $E$-invariant and $g$-vectors of arc representations with respect to an arbitrary triangulation of $\Sigma_{n}$. Section 3 is dedicated to recall the definition of Caldero-Chapoton algebra, see Example 3.5. In Section 5 we recall some results of [8] about string algebras, the concept of string is useful for us because the algebra $\Lambda(\tau)$ associated to a triangulation $\tau$ of $\Sigma_{n}$ is a string algebra.

In Section 6 we recall the definition of generalized cluster algebras introduced in [14], the reader can find interesting relations of these algebras with cluster algebras in [28, 29]. In Section 7 we recall some definitions and facts about surfaces with marked points and orbifold points. Also we define a natural Jacobian algebra for any triangulation denoted by $\Lambda(\tau)$.

The goal of Section 8 is to present the $\Lambda(\tau)$ as an orbit Jacobian algebra, see [31] and, give the definition of the arc representations of $\Lambda(\tau)$, which are (decorated) $\Lambda(\tau)$-modules that one can associate to arbitrary arcs on $\Sigma_{n}$. In Section 9 we study the arc representations of $\Lambda(\tau_{0})$ for a specific choice of triangulation $\tau_{0}$; we show that on these representations the action of the Auslander-Reiten translation of $\Lambda(\tau_{0})$ is given by rotation,111In sync with previous results of Brüstle-Qiu [7] and Caldero-Chapoton-Schiffler [10]. compute their $g$-vectors, show that they are $E$-rigid, and prove that their orbits are dense in their respective irreducible components.

In Section 10 we show that the results of Section 9 hold for any triangulation $\tau$ of $\Sigma_{n}$ and not only for the specific triangulation $\tau_{0}$. Here, Galois coverings, both of quivers and surfaces, come into play.

In Section 11 we state and prove our main result for any triangulation $\tau$. We close the paper with Section 12, which contains an example with explicit computations illustrating our main result; the example can be thought of as a complement to [11, Example 9.4.2].

Sections 9 and 10 deserve a few words: all the results in Section 9, about the $E$-invariant, are particular cases of results from Section 10. We have decided to not omit 9, and rather present a particular instance followed by the general treatment, because the Galois covering techniques used in Section 10 are not needed when establishing the same results for the specific triangulation $\tau_{0}$ chosen in 9.

2 Background

In this section we fix notation and we recall some basic definitions and facts about algebras and quiver representations that we will use throughout the work. The reader can find more details in [11].

A quiver $Q=(Q_{0},Q_{1},t,h)$ consists of a finite set of vertices $Q_{0}$, a finite set of arrows $Q_{1}$ and two maps $t,h:Q_{1}\rightarrow Q_{0}$ (tail and head). For each $a\in Q_{1}$ we write $a:t(a)\rightarrow h(a)$. If $Q_{0}=\{1,\ldots,n\}$, we define the skew-symmetric matrix $C_{Q}=(c_{i,j})\in\operatorname{Mat}_{n\times n}(\mathbb{Z})$ from $Q$ as follows

[TABLE]

We say that a sequence of arrows $\alpha=a_{l}a_{l-1}\cdots a_{2}a_{1}$, is a path of $Q$ if $t(a_{k+1})=h(a_{k})$, for $k=1,\ldots l-1$, in this case, we define the length of $\alpha$ as $l$. We say that $\alpha$ is a cycle if $h(a_{l})=t(a_{1})$. In this work we deal with quivers with loops, that is, quivers where there is an arrow $a\in Q_{1}$ such that $h(a)=t(a)$.

For $m\in\mathbb{N}$ let $C_{m}$ be the set of paths of length $m$ and let $\mathbb{C}C_{m}$ be the vector space with basis $C_{m}$. The *path algebra * of a quiver $Q$ is denoted by $\mathbb{C}\langle Q\rangle$ and it is defined as $\mathbb{C}$-vector space as

[TABLE]

where the product is given by the concatenation of paths. The completed path algebra of a quiver $Q$ is defined as vector space as

[TABLE]

where the elements are written as infinite sums $\sum_{m\geq 0}{x_{m}}$ with $x_{m}\in\mathbb{C}C_{m}$ and the product in $\mathbb{C}\langle\langle Q\rangle\rangle$ is defined as

[TABLE]

Let $\mathfrak{M}=\prod_{m\geq 1}{\mathbb{C}C_{m}}$ be the two-sided ideal of $\mathbb{C}\langle\langle Q\rangle\rangle$ generated by arrows of $Q$. Then $\mathbb{C}\langle\langle Q\rangle\rangle$ can be viewed as a topological $\mathbb{C}$-algebra with the powers of $\mathfrak{M}$ as a basic system of openneighborhoods of [math]. This topology is known as $\mathfrak{M}$-adic topology. Let $I$ be a subset of $\mathbb{C}\langle\langle Q\rangle\rangle$, we can calculate the closure of $I$ as $\overline{I}=\bigcap_{l\geq 0}{(I+\mathfrak{M}^{l})}$.

Definition 2.1**.**

A two-sided ideal $I$ of $\mathbb{C}\langle\langle Q\rangle\rangle$ is semi-admissible if $I\subseteq\mathfrak{M}^{2}$ and it is admissible if some power of $\mathfrak{M}$ is a subset of $I$. Following [11] we call an algebra $\Lambda$ basic if $\Lambda=\mathbb{C}\langle\langle Q\rangle\rangle/I$ for some quiver $Q$ and some semi-admissible ideal $I$.

A finite-dimensional representation of $Q$ over $\mathbb{C}$ is a pair $((M_{i})_{i\in Q_{0}},(M_{a})_{a\in Q_{1}})$ where $M_{i}$ is a finite-dimensional $\mathbb{C}$-vector space for each $i\in Q_{0}$ and $M_{a}:M_{t(a)}\rightarrow M_{h(a)}$ is a $\mathbb{C}$-linear map. Here the word representation means finite-dimensional representation.

The dimension vector of a representation $M$ of $Q$ is given by $\underline{\operatorname{dim}}(M)=(\operatorname{dim}(M_{1}),\ldots,\operatorname{dim}(M_{n}))$. We define $\operatorname{dim}(M)=\sum_{i=1}^{n}{\operatorname{dim}(M_{i})}$ as the dimension of $M$. We say $M$ is a nilpotent representation if there is an $n>0$ such that for every path $a_{n}a_{n-1}a\ldots a_{1}$ of length $n$ in $Q$ we have $M_{a_{n}}M_{a_{n-1}}\ldots M_{a_{1}}=0$. A subrepresentation of $M$ is an $n$-tuple of $\mathbb{C}$-vector spaces $N=(N_{i})_{i\in Q_{0}}$ such that $N_{i}\leq M_{i}$ for each $i\in Q_{0}$ and $M_{a}(N_{t(a)})\subseteq N_{h(a)}$ for every $a\in Q_{0}$.

We denote by $\operatorname{nil}_{\mathbb{C}}(Q)$ the category of nilpotent representations of $Q$, and by $\mathbb{C}\langle\langle Q\rangle\rangle$-$\operatorname{mod}$ the category of finite-dimensional left $\mathbb{C}\langle\langle Q\rangle\rangle$-modules. It is known that the category of representations of $Q$ and the category of $\mathbb{C}\langle Q\rangle$-modules are equivalent. In [18, Section 10] it was observed that $\operatorname{nil}_{\mathbb{C}}(Q)$ and $\mathbb{C}\langle\langle Q\rangle\rangle$-$\operatorname{mod}$ are equivalent.

Given a basic algebra $\Lambda=\mathbb{C}\langle\langle Q\rangle\rangle/I$ we define a representation of $\Lambda$ as a nilpotent representation of $Q$ which is annihilated by $I$. We consider the category $\operatorname{mod}(\Lambda)$ of finite-dimensional left modules as the category $\operatorname{rep}(\Lambda)$ of representations of $\Lambda$.

Let $\Lambda=\mathbb{C}\langle\langle Q\rangle\rangle/I$ be a basic algebra. We say $\mathcal{M}=(M,V)$ is a decorated representation of $\Lambda$ if $M$ is a representation of $\Lambda$ and $V=(V_{1},\ldots,V_{n})$ is an n-tuple of finite-dimensional $\mathbb{C}$-vector spaces. We can think of $V$ as a representation of a quiver with $n$-vertices and no arrows. That is a representation of the semisimple ring $\mathbb{C}^{Q_{0}}$. Let $\operatorname{decrep}(\Lambda)$ be the category of decorated representations of $\Lambda$. The objects of $\operatorname{decrep}(\Lambda)$ are the decorated representations of $\Lambda$ and its morphisms are given as follows. Let $(M,V)$ and $(N,W)$ be two decorated representations of $\Lambda$. We define the space of morhisms in $\operatorname{decrep}(\Lambda)$ by $\operatorname{Hom}_{\operatorname{decrep}(\Lambda)}((M,V),(N,W))=\operatorname{Hom}_{\operatorname{rep}(\Lambda)}(M,N)\times\operatorname{Hom}_{\operatorname{rep}(\mathbb{C}^{Q_{0}})}(V,W)$.

Let $\mathcal{M}=(M,V)$ be a decorated representation of $\Lambda$. If $V=0$, we write $M$ instead $\mathcal{M}$. For $i\in\{1,\ldots,n\}$ we define the negative simple representation of $\Lambda$ as $\mathcal{S}_{i}^{-}=(0,S_{i})$ where $(S_{i})_{j}$ is $\mathbb{C}$ if $j=i$ and $(S_{i})_{j}=0$ in other wise.

For a representation $M=((M_{i})_{i\in Q_{0}},(M_{a})_{a\in Q_{1}})$ of $\Lambda$ and a vector $\textbf{e}\in\mathbb{N}^{n}$ let $\operatorname{Gr}_{\textbf{e}}(M)$ be the quiver Grassmannian of subrepresentations $N$ of $M$ such that $\underline{\operatorname{dim}}(N)=\textbf{e}$. We denote the Euler characteristic of $\operatorname{Gr}_{\textbf{e}}(M)$ by $\chi(\operatorname{Gr}_{\textbf{e}}(M))$. About Euler characteristic we are going to need the following result, see [5],

Lemma 2.2** ((Biaĺynicki-Birula)).**

Let $T$ be an algebraic torus acting on an algebraic variety $X$. If we denote by $X^{T}$ the set of fixed points of the action, then $\chi(X^{T})=\chi(X)$.

The following definition plays a crucial role in some computations that will be involved later. It was introduced in [11, Section 2.4].

Definition 2.3**.**

Given a basic algebra $\Lambda=\mathbb{C}\langle\langle Q\rangle\rangle/I$ and $p\geq 2$ we define the $p$-truncation of $\Lambda$ by $\Lambda_{p}=\mathbb{C}\langle\langle Q\rangle\rangle/(I+\mathfrak{M}^{p})$.

We are going to need some basic definitions about quivers with potential, for all details the reader can see [17]. Let $Q$ be a quiver, we say that $S\in\mathbb{C}\langle\langle Q\rangle\rangle$ is a potential for $Q$ if $S$ is a, possibly infinite, $\mathbb{C}$-linear combination of cycles in $Q$ . Given two potentials $S$ and $W$ we say that they are cyclically equivalent and write $S\sim_{\operatorname{cyc}}W$, if $S-W$ is in the closure of the sub-vector space of $\mathbb{C}\langle\langle Q\rangle\rangle$ generated by all elements of the form $a_{1}a_{2}\cdots a_{n-1}a_{n}-a_{2}\cdots a_{n-1}a_{n}a_{1}$, with $a_{1}a_{2}\cdots a_{n-1}a_{n}$ a cycle on $Q$.

Definition 2.4**.**

We say $(Q,S)$ is a quiver with potential (QP) if $S$ is a potential for $Q$ and if any two different cycles appearing with non-zero coefficient in $S$ are not cyclically equivalent.

Given an arrow $a\in Q_{1}$ and a cycle $a_{n}a_{n-1}\cdots a_{1}$ in $Q$, define the cyclic derivative of $a_{n}a_{n-1}\cdots a_{1}$ with respect to $a$ as follows:

[TABLE]

we extend this definition by $\mathbb{C}$-linearity and continuity to all potentials for $Q$.

Definition 2.5**.**

Let $(Q,S)$ be a quiver with potential. We define the Jacobian ideal $\mathcal{J}(Q,S)$ as the closure of the ideal on $\mathbb{C}\langle\langle Q\rangle\rangle$ generated by all cyclic derivatives $\partial_{a}(S)$ with $a\in Q_{1}$. The quotient $\mathbb{C}\langle\langle Q\rangle\rangle/J(Q,S)$ is called the Jacobian algebra of $(Q,S)$ and is denoted as $\mathcal{P}(Q,S)$.

2.1 Varieties of representations

Let $\Lambda=\mathbb{C}\langle\langle Q\rangle\rangle/I$ and $\textbf{d}=(d_{1},\ldots,d_{n})\in\mathbb{N}^{n}$ be a basic algebra and a vector of non-negative integers. The representations $M$ of $Q$ with $\underline{\operatorname{dim}}(M)=\textbf{d}$ can be seen as points of the affine space

[TABLE]

Now, let $\operatorname{rep}_{\textbf{d}}(\Lambda)$ be the Zariski closed subset of $\operatorname{rep}_{\textbf{d}}(Q)$ given by the representations $N$ of $\Lambda$ with $\underline{\operatorname{dim}}(N)=\textbf{d}$.

In $\operatorname{rep}_{\textbf{d}}(\Lambda)$ we have the action of $G_{\textbf{d}}=\prod_{i=1}^{n}{GL(\mathbb{C}^{d_{i}})}$ by conjugation. If $g=(g_{1},\ldots,g_{n})\in G_{\textbf{d}}$ and $M=((M_{i})_{i\in Q_{0}},(M_{a})_{a\in Q_{1}})\in\operatorname{rep}_{\textbf{d}}(\Lambda)$, then

[TABLE]

From definitions follows that the isomorphism classes of representations of $\Lambda$ with dimension vector d are in bijection with the $G_{\textbf{d}}$-orbits in $\operatorname{rep}_{\textbf{d}}(\Lambda)$. If $M\in\operatorname{rep}_{\textbf{d}}(\Lambda)$, its $G_{\textbf{d}}$-orbit is denoted by $\mathcal{O}(M)$. For $(\textbf{d},\textbf{v})\in\mathbb{N}^{n}\times\mathbb{N}^{n}$ let $\operatorname{decrep}_{\textbf{d},\textbf{v}}(\Lambda)$ be the decorated representations variety of $\Lambda$. If $\textbf{v}=(v_{1},\ldots,v_{n})$, then

[TABLE]

We have an action of $G_{\textbf{d}}$ on $\operatorname{decrep}_{\textbf{d},\textbf{v}}(\Lambda)$ given by $g\cdot\mathcal{M}=(g\cdot M,V)$ where $\mathcal{M}=(M,V)\in\operatorname{decrep}_{\textbf{d},\textbf{v}}(\Lambda)$ and $g\in G_{\textbf{d}}$.

2.2 Galois G-covering

In this section we are going to make a reminder of Galois $G$-covering. For a nice review of this theory the reader can see the introductions of [1, 4]. For our convenience we are going to present some results from [4].

In this section $G$ will denote a finite group (in the general theory this assumption is not required). A category $\mathcal{A}$ is called $\mathbb{C}$-linear or $\mathbb{C}$-category if its sets of morphisms are $\mathbb{C}$-modules and the composition of morphisms is $\mathbb{C}$-linear. We assume that we have a morphism $\rho:G\rightarrow\operatorname{Aut}(\mathcal{A})$ from $G$ to the group of automorphisms of $\mathcal{A}$, not the group of auto-equivalences. That means we have an action of $G$ on $\mathcal{A}$. We will abuse of notation and we will write $g$ instead $\rho(g):\mathcal{A}\rightarrow\mathcal{A}$ for every $g\in G$. The action of $G$ on $\mathcal{A}$ is called free provided $g\cdot X$ is not isomorphic to $X$ for every non-trivial element $g\in G$ and for any indecomposable object $X$ of $\mathcal{A}$. The next definitions are due to Asashiba, see [1, Definition 1.1, Definition 1.7].

Definition 2.6**.**

Let $\mathcal{A}$ and $\mathcal{B}$ be $\mathbb{C}$-categories with $G$ acting on $\mathcal{A}$. A $\mathbb{C}$-linear functor $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ is called $G$-stable if there exist functorial isomorphisms $\delta_{g}:\mathcal{F}g\rightarrow\mathcal{F}$ such that $\delta_{h,X}\delta_{g,h\cdot X}=\delta_{gh,X}$ for any $g,h\in G$ and any object $X$ in $\mathcal{A}$. In this case $\delta=(\delta_{g})_{g\in G}$ is called a $G$-stabilizer. If $\delta_{g}=\operatorname{id}_{\mathcal{F}}$ for every $g\in G$, we say that $\mathcal{F}$ is $G$-invariant, see the next diagram.

[TABLE]

Definition 2.7**.**

Let $\mathcal{A},\mathcal{B}$ be $\mathbb{C}$-categories with a group $G$ acting on $\mathcal{A}$. Let $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ be a $G$-stable functor with stabilizer $\delta$.

1. (a)

We say that $\mathcal{F}$ is a $G$-precovering if the following maps are isomorphisms for any $X,Y$ objects in $\mathcal{A}$:

[TABLE]

[TABLE] 2. (b)

A $G$-precovering $\mathcal{F}$ is called a Galois $G$-covering if $\mathcal{F}$ has the following three conditions:

• (i)

The functor $\mathcal{F}$ is almost dense. It means that any indecomposable object $Y$ of $\mathcal{B}$ is isomorphic to $\mathcal{F}(X)$ for some object $X$ in $\mathcal{A}$.

• (ii)

If $X$ is indecomposable in $\mathcal{A}$, then $\mathcal{F}(X)$ is indecomposable in $\mathcal{B}$.

• (iii)

For any indecomposable objects $X,Y$ in $\mathcal{A}$ such that $\mathcal{F}(X)\cong\mathcal{F}(Y)$, there exist $g\in G$ such that $g\cdot X\cong Y$.

Remark 2.8**.**

In **[1, Proposition 1.6]** is proved that $\mathcal{F}^{X,Y}$ is an isomorphism if and only if $\mathcal{F}_{X,Y}$ is an isomorphism. Note that a $G$-precovering is a faithful functor, see **[4, Lemma 2.6]**. 2. 2.

In Krull-Schmidt categories a functor is almost dense if and only if it is dense.

The following lemma allows us to find examples of a Galois $G$-covering from a $G$-precovering between module categories, see [4, Lemma 2.9]

Lemma 2.9**.**

Let $\mathcal{A},\mathcal{B}$ be Krull-Schmidt $\mathbb{C}$-categories with a group $G$ acting freely on $\mathcal{A}$ and let $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ be a $G$-precovering. Assume $X$ is an object in $\mathcal{A}$ such that $\operatorname{End}_{\mathcal{A}}(X)$ is local and has nilpotent radical. Then $\operatorname{End}_{\mathcal{B}}(\mathcal{F}(X))$ is local with nilpotent radical and if $Y$ is an object in $\mathcal{A}$ such that $\mathcal{F}(X)\cong F(Y)$, then there exist $g\in G$ such that $g\cdot X\cong Y$.

The next theorem shows an interesting application of Galois $G$-covering in Auslander-Reiten theory, see [4, Theorem 3.7].

Theorem 2.10** ((Bautista-Liu, 2014)).**

Let $\mathcal{A},\mathcal{B}$ be Krull-Schmidt $\mathbb{C}$-categories with a group $G$ acting freely on $\mathcal{A}$ and let $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ be a Galois $G$-covering. Then

A short exact sequence $\eta$ in $\mathcal{A}$ is almost split if and only if $\mathcal{F}(\eta)$ is almost split. 2. 2.

An object $X$ in $\mathcal{A}$ is the starting or ending term of a almost split sequence if and only if $\mathcal{F}(X)$ is the starting or ending term of a almost split sequence, respectively.

Given a $\mathbb{C}$-algebra $\Lambda$ we consider it as a $\mathbb{C}$-category in the usual way, in other words, the set of objects of $\Lambda$ is a complete family of ortogonal and primitive idempotents and the set of morphisms is given by $\Lambda(e_{i},e_{j})=e_{j}\Lambda e_{i}$. In this context, the category of left $\Lambda$-modules $\Lambda\operatorname{-mod}$ is equivalent to the category of functors from $\Lambda$ to $\mathbb{C}\operatorname{-mod}$. An action of a group $G$ on $\Lambda$ induces an action of $G$ on $\Lambda\operatorname{-mod}$ in the following way. Given an $\Lambda$-module $M:\Lambda\rightarrow\mathbb{C}\operatorname{-mod}$ we define $g\cdot M:=Mg^{-1}$, remember that $g$ is thought as an automorphism of $\Lambda$; for a morphism $u:M\rightarrow N$ of $\Lambda\operatorname{-mod}$, we define $g\cdot u(x)=u(g^{-1}x)$ for $x$ an object of $\Lambda$. So, if we have a $G$-precovering $\pi:\Lambda\rightarrow A$, Bongarzt and Gabriel defined the push-down functor $\pi_{*}:\Lambda\operatorname{-mod}\rightarrow A\operatorname{-mod}$, see [6, Section 3.2]. In Remark 8.10 we define the push-down fuctor in our particular case.

We have a nice property for $\pi_{*}$, see [4, Lemma 6.3].

Lemma 2.11**.**

Let $\Lambda$ and $A$ be finite dimensional $\mathbb{C}$-algebras with a group $G$ acting on $\Lambda$. Assume the action of $G$ is free. If $\pi:\Lambda\rightarrow A$ is a Galois $G$-precovering, then the push-down functor $\pi_{*}$ admits a $G$-stabilizer $\delta$.

With the following lemma we can construct a $G$-precovering from a Galois $G$-covering, see [4, Theorem 6.5].

Lemma 2.12**.**

Let $\Lambda$ and $A$ be finite dimensional $\mathbb{C}$-algebras with a group $G$ acting on $\Lambda$. Assume the action of $G$ is free. If $\pi:\Lambda\rightarrow A$ is a Galois $G$-covering, then the push-down functor $\pi_{*}:\Lambda\operatorname{-mod}\rightarrow A\operatorname{-mod}$ is a $G$-precovering.

3 E-invariant and Caldero-Chapoton functions

In this section we recall some definitions that we work with. This definitions were introduced in [11, Section 3.4]. They were motivated by the theory of mutation of quivers with potential developed in [18] and the Caldero-Chapoton functions introduced in [9]. In this section let $\Lambda=\mathbb{C}\langle\langle Q\rangle\rangle/I$ a basic algebra.

3.1 $g$-vectors

For a decorated representation $\mathcal{M}=(M,V)$ of $\Lambda$ the $g$-vector of $\mathcal{M}$ is given by $g_{\Lambda}(\mathcal{M})=(g_{1},\ldots,g_{n})$ where

[TABLE]

It is clear that $g_{\Lambda}(\mathcal{M})\in\mathbb{Z}^{n}$. We denote by $I_{i}$ the injective envelope of the simple representation $S_{i}$ in $\Lambda\operatorname{-mod}$. We recall an interesting result to compute the $g$-vector of $\mathcal{M}$, see [11, Lemma 3.4] for a general version.

Lemma 3.1**.**

Let $\mathcal{M}=(M,V)$ be a decorated representation of a finite dimensional algebra $\Lambda$ and let $g_{\Lambda}(\mathcal{M})=(g_{1},\ldots,g_{n})$ be its $g$-vector. Assume we have a minimal injective presentation of $M$

[TABLE]

where $I_{0}(M)=\bigoplus_{i=1}^{n}{I_{i}^{a_{i}}}$ and $I_{1}(M)=\bigoplus_{i=1}^{n}{I_{i}^{b_{i}}}$. Then

[TABLE]

3.2 The $E$-invariant

For decorated representations $\mathcal{M}=(M,V)$ and $\mathcal{N}=(N,W)$ of $\Lambda$ let

[TABLE]

The $E$-invariant of $\mathcal{M}$ is defined as $E_{\Lambda}(\mathcal{M})=E_{\Lambda}(\mathcal{M},\mathcal{M})$.

In [11] it was shown that the $E$-invariant has a homological interpretation in terms of the Auslander-Reiten translation of truncations of $\Lambda$, see Definition 2.3.

Proposition 3.2** ([11, Proposition 3.5]).**

Let $\mathcal{M}=(M,V)$ and $\mathcal{N}=(N,W)$ be decorated representations of $\Lambda$. If $p>\operatorname{dim}(M),\operatorname{dim}(N)$, then

[TABLE]

This proposition is quite useful for us because the basic algebras we are considering satisfy $\Lambda_{p}=\Lambda$ for a sufficiently large $p$.

3.3 Caldero-Chapoton functions and algebras

Let $\mathcal{M}=(M,V)$ be a decorated representation of $\Lambda$. For $\textbf{f}=(f_{1},\ldots,f_{n})\in\mathbb{Z}^{n}$ by $\textbf{x}^{\textbf{f}}$ we mean $\prod_{i=1}^{n}{x_{i}^{f_{i}}}$. The Caldero-Chapoton function (CC function for short) associated to $\mathcal{M}$ of $\Lambda$ is the Laurent polynomial in $n$-variables $x_{1},\ldots,x_{n}$ defined by

[TABLE]

where $C_{Q}$ is defined as in the second paragraph of Section 2. From definitions we have $\mathcal{C}_{\Lambda}(\mathcal{M})\in\mathbb{Z}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$, Note $\mathcal{C}_{\Lambda}(\mathcal{S}_{i}^{-})=x_{i}$. The set of Caldero-Chapoton functions associated to $\Lambda$ is

[TABLE]

The next lemma was shown in [11]. It is convenient for computations of $g$-vectors and Caldero-Chapoton functions.

Lemma 3.3** ([11, Lemma 4.1]).**

If $\mathcal{M}=(M,V)$ and $\mathcal{N}=(N,W)$ are decorated representations of $\Lambda$, then the following hold:

$g_{\Lambda}(\mathcal{M}\oplus\mathcal{N})=g_{\Lambda}(\mathcal{M})+g_{\Lambda}(\mathcal{N})$. 2. 2.

$\mathcal{C}_{\Lambda}(\mathcal{M})=\mathcal{C}_{\Lambda}(M,0)\mathcal{C}_{\Lambda}(0,V)$. 3. 3.

$\mathcal{C}_{\Lambda}(\mathcal{M}\oplus\mathcal{N})=\mathcal{C}_{\Lambda}(\mathcal{M})\mathcal{C}_{\Lambda}(\mathcal{N})$.

Definition 3.4**.**

The Caldero-Chapoton algebra $\mathcal{A}_{\Lambda}$ associated to $\Lambda$ is the $\mathbb{C}$-subalgebra of $\mathbb{C}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$ generated by $\mathcal{C}_{\Lambda}$.

From Lemma 3.3(iii) follows that $\mathcal{C}_{\Lambda}$ generates $\mathcal{A}_{\Lambda}$ as $\mathbb{C}$-vector space, see [11, Lemma 4.2].

Example 3.5**.**

Let $Q$ be the quiver

[TABLE]

and let $\Lambda=\mathbb{C}\langle Q\rangle$. For each sub-interval $\textbf{e}=[i,j]$ of $[1,n]$ with $i\leq j$ we define an indecomposable representation $M_{\textbf{e}}$ of $\Lambda$ the following way. Let $(M_{\textbf{e}})_{k}=\mathbb{C}$ if $k\in\textbf{e}$ and $(M_{\textbf{e}})_{k}=0$ if $k\notin\textbf{e}$, for $k\in[1,n]$. For an arrow $a_{l}$ with $l\in[1,n-1]$ define $(M_{\textbf{e}})_{a_{l}}$ as $\operatorname{id}_{\mathbb{C}}$ if $t(a_{l}),h(a_{l})\in\textbf{e}$ and zero in other wise. Note that the dimension vector of $M_{\textbf{e}}$ can be identified with the sub-interval e. If $1<i$, then we have

[TABLE]

If $i=1$, then $g_{\Lambda}(M_{\textbf{e}})_{j}=-1$ and $g_{\Lambda}(M_{\textbf{e}})_{k}=0$ for $k\neq j$. We have $E_{\Lambda}(M_{\textbf{e}})=0$ for each sub-interval e of $[1,n]$. From [9, Theorem 3.4] we have $\mathcal{A}_{\Lambda}$ can be identified with the cluster algebra associated to $Q$.

4 Strongly reduced components

In this section we recall some facts about strongly reduced irreducible components which were introduced in [22, Section 1.5]. For our convenience we follow the exposition of [11], the reader can see [11, Sections 5 and 6] for a complete treatment about strongly reduced components in Caldero-Chapoton algebras.

Let $\Lambda$ be a basic algebra and consider dimension vectors $(\textbf{d},\textbf{v})\in\mathbb{N}^{n}\times\mathbb{N}^{n}$. We denote by $\operatorname{Irr}_{\textbf{d}}(\Lambda)$ and $\operatorname{decIrr}_{\textbf{d},\textbf{v}}(\Lambda)$ the set of irreducible components of $\operatorname{rep}_{\textbf{d}}(\Lambda)$ and $\operatorname{decrep}_{\textbf{d},\textbf{v}}(\Lambda)$ respectively. For $Z\in\operatorname{decIrr}_{\textbf{d},\textbf{v}}(\Lambda)$ we write $\underline{\operatorname{dim}}(Z)=(\textbf{d},\textbf{v})$. We define

[TABLE]

the corresponding sets of irreducible components. From Section 2.1 we have that

[TABLE]

It is clear that $T:\operatorname{decrep}_{\textbf{d},\textbf{v}}(\Lambda)\rightarrow\operatorname{rep}_{\textbf{d}}(\Lambda)$ with $(M,\mathbb{C}^{\textbf{v}})\mapsto M$ is an isomorphism of affin varieties. In this way results on the variety of representations can be transported to the variety of decorated ones. We introduce further notation in order to define strongly reduced components.

Let $Z,Z_{1},Z_{2}\in\operatorname{decIrr}(\Lambda)$ be irreducible components, define

[TABLE]

From the semi-continuity of the functions $\operatorname{dim}\operatorname{Hom}_{\Lambda}(-,?)$ and $\operatorname{dim}\operatorname{Ext}^{1}_{\Lambda}(-,?)$, see [15, Lemma 4.3], there exist an open set $U$ of $Z$ such that $E_{\Lambda}(\mathcal{M})=E_{\Lambda}(\mathcal{N})$ for all $\mathcal{M},\mathcal{N}\in U$. Then we define $E_{\Lambda}(Z)=E_{\Lambda}(\mathcal{M})$ for $\mathcal{M}\in U$. In a similar way we define $E_{\Lambda}(Z_{1},Z_{2})$.

The following lemma is proved in [11, Lemma 5.2].

Lemma 4.1**.**

Let $Z,Z_{1},Z_{2}\in\operatorname{decIrr}(\Lambda)$ be irreducible components. The following inequalities hold

[TABLE]

The following definition comes from [22].

Definition 4.2**.**

Let $Z\in\operatorname{decIrr}(\Lambda)$ be an irreducible component. We say that $Z$ is strongly reduced if $c_{\Lambda}(Z)=E_{\Lambda}(Z)$.

We say that an irreducible component $Z\in\operatorname{Irr}(\Lambda)$ (resp. $Z\in\operatorname{decIrr}(\Lambda)$) is indecomposable if there exist a dense open $U\subseteq Z$ which contains only indecomposable representations (resp. indecomposable decorated representations).

Crawley-Boevey and Schröer gave a canonical decomposition at the level of irreducible components. This seems something as the Krull-Schmidt property at that level, see [15, Theorems 1.1 and 1.2].

Theorem 4.3** ((Crawley-Boevey-Schröer)).**

Let $Z_{1},\ldots,Z_{t}$ be irreducible components in $\operatorname{Irr}(\Lambda)$. The following two statements are equivalent:

• •

$\overline{Z_{1}\oplus\cdots\oplus Z_{t}}$* is an irreducible component.*

• •

$\operatorname{ext}^{1}_{\Lambda}(Z_{i},Z_{j})=0$, for $i\neq j$ with $i,j\in\{1,\ldots,t\}$.

Moreover, the following hold

• •

If $W\in\operatorname{Irr}(\Lambda)$ is an irreducible component, then there exist indecomposable irreducible components $W_{1},\ldots,W_{t}$ in $\operatorname{Irr}(\Lambda)$ such that $W=\overline{W_{1}\oplus\cdots\oplus W_{t}}$ and this decomposition is unique up to a permutation on $\{1,\ldots,t\}$.

We wrote down the Crawley-Boevey-Schröer theorem in its original version for $\operatorname{rep}(\Lambda)$, but it is true for $\operatorname{decrep}(\Lambda)$ as is stated in [11, Theorem 5.3].

Now, we get something similar for strongly reduced components, see [11, Theorem 5.11].

Theorem 4.4** ((CI-LF-S)).**

Let $Z_{1},\ldots Z_{t}$ be irreducible components in $\operatorname{decIrr}(\Lambda)$. The following two statements are equivalent:

• •

$\overline{Z_{1}\oplus\cdots\oplus Z_{t}}$* is a strongly reduced irreducible component.*

• •

For any $Z_{i}$ we have that it is strongly reduced and $E_{\Lambda}(Z_{i},Z_{j})=0$ for all $i\neq j$ with $i,j\in\{1,\ldots,t\}$.

We want generic Caldero-Chapoton functions. For each $(\textbf{d},\textbf{v})\in\mathbb{N}^{n}\times\mathbb{N}^{n}$ we consider the function

[TABLE]

defined by $\mathcal{M}\mapsto C_{\Lambda}(\mathcal{M})$. Since this function is constructible, then its image is finite. It turns out that for any irreducible component $Z\in\operatorname{decIrr}_{\textbf{d, v}}(\Lambda)$ there exist a dense open subset $U\subseteq Z$ where $C_{\textbf{d, v}}$ is constant in $U$. We define $C_{\Lambda}(Z)=C_{\Lambda}(\mathcal{M})$, for any $\mathcal{M}\in U$.

The set of irreducible components is denoted by $\operatorname{decIrr^{s.r}}(\Lambda)$. We define the graph $\Gamma(\operatorname{decIrr^{s.r}}(\Lambda))$ of strongly reduced irreducible components as follows: it has a vertex for any indecomposable strongly reduced component and there is an edge between vertices $Y$ and $Z$ if $E_{\Lambda}(Y,Z)=0=E_{\Lambda}(Z,Y)$ . Note that $Y$ can be equal $Z$. The graph of irreducible components $\Gamma(\operatorname{Irr}(\Lambda))$ was defined in [15, Section 12.3]

Let $\Gamma$ be a graph. We consider graphs with single edges and loops. We denote by $\Gamma_{0}$ the set of vertices of $\Gamma$. By $\Gamma_{\mathcal{U}}$ we denote the full subgraph of $\Gamma$, whose set of vertices is $\mathcal{U}$.

We call $\Gamma_{\mathcal{U}}$ complete if for any vertices $i\neq j$ in $\mathcal{U}$ there is an edge between them. A complete $\Gamma_{\mathcal{U}}$ subgraph is called maximal if for any other complete subgraph $\Gamma_{\mathcal{U}^{\prime}}$ with $\mathcal{U}\subseteq\mathcal{U}^{\prime}$ we have $\mathcal{U}=\mathcal{U}^{\prime}$.

We define a component cluster of $\Lambda$ as the set of vertices of a maximal complete subgraph of $\Gamma(\operatorname{decIrr^{s.r}}(\Lambda))$. A component cluster $\mathcal{U}$ is $E$-rigid whenever $E_{\Lambda}(Z)=0$ for all $Z\in\mathcal{U}$.

The following notions were introduced in [11, Section 6.5].

Definition 4.5**.**

Let $\mathcal{U}=\{Z_{1},Z_{2},\ldots,Z_{t}\}$ be a component cluster. We define the $CC$-cluster of $\Lambda$ associated to $\mathcal{U}$ by $\mathcal{C}_{\mathcal{U}}=\{\mathcal{C}_{\Lambda}(Z_{1}),\mathcal{C}_{\Lambda}(Z_{2}),\ldots,\mathcal{C}_{\Lambda}(Z_{t})\}$.

The notation $CC$ comes from sets of Caldero-Chapoton functions. Cerulli Irelli, Labarini-Fragoso and Schröer introduced the notion of laurent phenomenon for Caldero-Chapoton algebras, see [11, Section 6.5].

Definition 4.6**.**

The Caldero-Chapoton algebra $\mathcal{A}_{\Lambda}$ has the Laurent phenomenon property provided for any $E$-rigid component cluster $\{Z_{1},\ldots,Z_{t}\}$ of $\Lambda$, we have

[TABLE]

5 String algebras

In this section we recall some definitions and results about string algebras exposed in [8]. Let $Q$ be a quiver.

Let $P$ be a subset of paths in $\mathbb{C}\langle Q\rangle$ and denote by $\langle P\rangle$ the ideal generated by $P$. The algebra $\Lambda=\mathbb{C}\langle Q\rangle/\langle P\rangle$ is called a string algebra if the following conditions hold:

Any vertex $i\in Q_{0}$ is the tail or head point of at most two arrows of $Q$, that is, $|\{a\in Q\colon t(a)=i\}|\leq 2$ and $|\{a\in Q\colon h(a)=i\}|\leq 2$. 2. 2.

For any arrow $a\in Q_{1}$ we have $|\{b\in Q_{1}\colon t(a)=h(b)\mbox{ and }ab\notin P\}|\leq 1$ and $|\{c\in Q_{1}\colon t(c)=h(a)\mbox{ and }ca\notin P\}|\leq 1$. 3. 3.

The ideal $\langle P\rangle$ is admissible on $\mathbb{C}\langle Q\rangle$.

To describe the finite-dimensional indecomposable $\Lambda$-modules we need the concept of string. We introduce an alphabet consisting of direct letters given by each arrow $a\in Q_{1}$ and inverse letters given by $a^{-1}$ for each arrow $a\in Q$. The head and tail functions extend to this alphabet in the obvious way, that is, $h(a^{-1})=t(a)$ and $t(a^{-1})=h(a)$ for every arrow $a\in Q_{1}$. For a letter $l$ in this alphabet we denote its inverse letter with $l^{-1}$ and we write $l$ instead $(l^{-1})^{-1}$. A word in this alphabet of length $r\geq 1$ is a sequence of letters $l_{r}\cdots l_{1}$ such that $t(l_{i+1})=h(l_{i})$ for $i=1,\ldots,r-1$. For a word $W=l_{r}\cdots l_{1}$ we denote its inverse word by $W^{-1}=l_{1}^{-1}\cdots l_{r}^{-1}$. It is clear we can extend the head and tail functions to words. A string of length $r\geq 1$ is a word $W=l_{r}\cdots l_{1}$ such that $W$ and $W^{-1}$ do not contain sub-words of the form $ll^{-1}$ for a letter $l$ and no sub-words of $W$ belongs to $P$.

We introduce strings of length [math] in the following way. For each vertex $i\in Q_{0}$ we have two strings of length [math] denoted by $1_{(i,u)}$ with $u\in\{1,-1\}$. In this case $h(1_{(i,u)})=i=t(1_{(i,u)})$. By definition $1_{(i,u)}^{-1}=1_{(i,-u)}$.

We recall the definition of two functions to deal with strings. In [8] it is shown we can choose two functions $\sigma,\epsilon\colon Q_{1}\rightarrow\{1,-1\}$ such that the following conditions are satisfied

If $a_{1}\neq a_{2}$ are arrows with $t(a_{1})=t(a_{2})$, then $\sigma(a_{1})=-\sigma(a_{2})$. 2. 2.

If $b_{1}\neq b_{2}$ are arrows with $h(b_{1})=h(b_{2})$, then $\epsilon(b_{1})=-\epsilon(b_{2})$. 3. 3.

If $a,b\in Q$ are arrows with $t(b)=h(a)$ and $ba\notin P$, then $\sigma(b)=-\epsilon(a)$.

For an arrow $a\in Q_{1}$ we have $\sigma(a^{-1})=\epsilon(a)$ and $\epsilon(a^{-1})=\sigma(a)$. For a string $W=l_{r}\cdots l_{1}$ we define $\sigma(W)=\sigma(l_{1})$ and $\epsilon(W)=\epsilon(l_{r})$. Besides we have $\sigma(1_{(i,u)})=-u$ and $\epsilon(1_{(i,u)})=u$. Note that if $W_{1}$ and $W_{2}$ are strings such that $W_{2}W_{1}$ is a string, then $\sigma(W_{2})=-\epsilon(W_{1})$. For $(i,u)\in Q_{0}\times\{1,-1\}$ let $\mathcal{W}_{(i,u)}$ be the set of all strings $W$ with $h(W)=i$ and $\epsilon(W)=u$. Let $\mathcal{W}$ be the set of all strings and define on $\mathcal{W}$ an equivalence relation given by $W_{1}\sim W_{2}$ if and only if $W_{2}\in\{W_{1},W_{1}^{-1}\}$. Let $\underline{\mathcal{W}}$ be a complete set of representatives of the corresponding equivalence classes.

Remark 5.1**.**

In this article we are not going to use this functions $\epsilon$ and $\sigma$, but it is useful to remember that for string algebras the strings can be thought of as sequence of signs.

In [8], it was also defined the set $\mathcal{B}$ of bands. A string $W\in\mathcal{W}$ belongs to $\mathcal{B}$ if length of $W$ is positive, $W^{n}\in\mathcal{W}$ for all $n\in\mathbb{N}$ and $W$ is not the power of some string of smaller length.

5.1 Indecomposable string modules

For a string $W$, in [8], it was defined a $\Lambda$-module $N(W)$, for convenience we repeat this definition. For the string $1_{(i,u)}$ we define $N(1_{(i,u)})$ as the simple representation $S_{i}$ at the vertex $i\in Q_{0}$. If $W=l_{r}\cdots l_{1}$, then $N(W)$ is a representation of $\mathbb{C}$-dimension $r+1$. For describe the structure of $\Lambda$-module let $p_{0}=t(l_{1})$ and $p_{k}=h(l_{k})$ for $k=1,\ldots,r$ vertices of $Q$. By definition $\operatorname{dim}(N(W)_{i})$ is $|\{k\in[1,r+1]\colon p_{k}=i\}|$. If $\{z_{0},\cdots,z_{r}\}$ is a basis of $N(W)$ with $z_{k}\in N(W)_{p_{k}}$ for $k=0,\ldots,r$, then the action of the arrows is given by the following way

[TABLE]

If $l_{k}$ is a direct letter, then $N(W)_{l_{k}}(z_{k-1})=z_{k}$; if $l_{k}$ is a inverse letter, then $N(W)_{l_{k}}(z_{k-1})=z_{k}$ with $k=1,\ldots,n$; if $a\in Q_{1}$ and $N(W)_{a}(z_{k})$ is not defined yet, then $N(W)_{a}(z_{k})=0$.

In [8], it was observed that $N(W)$ is isomorphic to $N(W^{-1})$. The modules $N(W)$ are called string modules. The next result is a special case of the more general result of Butler and Ringel proved in [8, Section 3].

Theorem 5.2** ((Butler-Ringel)).**

Let $\Lambda$ be a string algebra. If $\mathcal{B}=\emptyset$, then the $\Lambda$-modules $N(W)$ with $W\in\underline{\mathcal{W}}$ form a complete list of indecomposable, pairwise non-isomorphic $\Lambda$-modules.

5.2 Sub-strings

For a string of positive length $W=l_{r}\cdots l_{1}$ we define its support as $\operatorname{Supp}(W)=\{t(l_{1})\}\cup\{h(l_{k})\colon k=1,\cdots,r\}$. If $W=1_{(i,u)}$, then $\operatorname{Supp}(W)=\{i\}$. Given a string of positive length $W=l_{r}\cdots l_{1}$, we say that a string $V$ is a sub-string of $W$ if $V=l_{t+j}\cdots l_{t}$ is a subword of $W$ and there are no arrows $a,b\in Q_{1}$ such that $aV$ and $Vb^{-1}$ are subwords of $W$. For technical reasons we introduce the zero string [math] which is sub-string of any string. Now, given a string $W$ we denote by $\operatorname{Cam}(W)$ the set of all sub-strings of $W$.

Remark 5.3**.**

We use $\operatorname{Cam}$ from the word in spanish caminata for walk. On one hand the elements of $\operatorname{Cam}(W)$ can be thought as walks in the quiver. On the other hand, there are a lot of W’s in our notation.

6 Generalized cluster algebras

For convenience we recall the definition of generalized cluster algebras introduced by Chekhov and Shapiro in [14].

We say a matrix $B\in\operatorname{Mat}_{n\times n}(\mathbb{Z})$ is skew-symmetrizable if there exist positive integers $d_{1},\ldots,d_{n}$ such that $DB$ is skew-symmetric with $D=\operatorname{diag}(d_{1},\ldots,d_{n})$ a diagonal matrix. In this case we call $D$ a skew-symmetrizer of $B$.

Given a skew-simmetrizable matrix $B$ and an integer $k\in\{1,\ldots,n\}$, the mutation of $B$ with respect to $k$ is the matrix $\mu_{k}(B)$ with entries $b^{\prime}_{ij}$ defined as follows

[TABLE]

For us it is enough work without coefficients. For a study of generalized cluster algebras with principal coefficients, in parallel with the one for cluster algebras, the reader can see [28].

Let $\mathcal{F}$ be the field of the rational functions in $n$ algebraic independent variables with coefficients in $\mathbb{Q}$.

Definition 6.1**.**

A seed in $\mathcal{F}$ is a pair $(B,\textbf{x})$ where $B$ is a skew-symmetrizable matrix and $\textbf{x}=(x_{1},\ldots,x_{n})$ is an $n$-tuple of algebraic independent elements of $\mathcal{F}$ which generate it.

Now we assume that $B$ is skew-symmetric with skew-symmetrizer $D$ and $b_{ik}/d_{k}$ is an integer for all $i\in\{1,\ldots,n\}$.

For $k\in\{1,\ldots,n\}$ we define the polynomials

[TABLE]

Definition 6.2** ((Generalized cluster mutation)).**

For a seed $(B,\textbf{x})$ and $k\in\{1,\ldots,n\}$, the mutation of $(B,\textbf{x})$ with respect to $k$ is the pair $\mu_{k}(B,\textbf{x})=(\mu_{k}(B),\mu_{k}(\textbf{x})$ where $\mu_{k}(\textbf{x})=(x_{1}^{\prime},\ldots,x_{n}^{\prime})$ is the $n$-tuple of elements of $\mathcal{F}$ given by

[TABLE]

In this article the polynomials $\theta_{k}(v_{k}^{+},v_{k}^{-})$ are often called polynomials of Chekhov-Shapiro. For a seed $(B,\textbf{x})$, let

[TABLE]

Definition 6.3**.**

The generalized cluster algebra associated to $(B,\textbf{x})$, denoted by $\mathcal{A}(B)=\mathcal{A}(B,\textbf{x})$, is the subring of $\mathcal{F}$ generated by $\mathcal{X}$.

Note that exchange polynomials do not have to be binomials. In [14, Theorem 2.5], it was shown a desired property.

Theorem 6.4** ((The Laurent phenomenon)).**

Any generalized cluster variable can be expressed as a Laurent polynomial in the initial variables $x_{i}$.

Remark 6.5**.**

The reader should be cautious by comparing the definitions of [14] with this ones because there they take skew-symmetrizer by the right and here we did by the left.

Remark 6.6**.**

In [14, Lemma 3.1] the authors obtain exchange relations of the form $a^{2}+2\cos(\frac{\pi}{p})ab+b^{2}$ in the case of orbifold points of order $p$, with $p$ greater than one. We are going to obtain exchange relations of the form $a^{2}+ab+b^{2}$. Of course it is more natural to consider orbifold points of order three than order two as we made in the first version of this article.

7 Polygons with one orbifold point

We will work with polygons with one orbifold point of order three but for convenience we recall some definitions of surfaces with orbifold points. For more details about surfaces with orbifold points of order two or three and relations with generalized cluster algebras the reader can see [14] and references therein, for example [12, 13]. For an interesting and beautiful application of surfaces with orbifold points and group actions in some cluster structures the reader is kindly asked to look at [31].

7.1 Basic definitions

Here we review some combinatorial aspects about orbifolds, however orbifolds, in general, are spaces generalizing manifolds. Orbifolds are locally modeled by an Euclidean space modulo a finite group. The order of an orbifold point is the order of the non-trivial stabilizer at that point. In our setting is not necessary to recall more geometric aspects about orbifolds. In the next section, we will concentrate in our specific surface. In particular, we shall see that the orbifold point of order three we are talking about is the fixed point under a suitable action of $\mathbb{Z}_{3}$ on a polygon.

Let $\Sigma$ be a compact connected oriented 2-dimensional real surface with possible empty boundary. The pair $(\Sigma,\mathbb{M})$ where $\mathbb{M}$ is a finite subset of $\Sigma$ with at least one point from each connected component of the boundary of $\Sigma$ is called a bordered surface or just a surface. The points of $\mathbb{M}$ are called marked points and the points of $\mathbb{M}$ that lie in the interior of $\mathbb{M}$ are called punctures. A triple $(\Sigma,\mathbb{M},\mathbb{O})$ where $(\Sigma,\mathbb{M})$ is a bordered surface and $\mathbb{O}$ is a finite subset of $\Sigma\setminus(\mathbb{M}\cup\partial\Sigma)$ is called a marked surface with orbifold points. The points of $\mathbb{O}$ are called orbifold points and they will be denoted by a cross $\times$ in the surface.

7.2 Triangulations and flips

Let $(\Sigma,\mathbb{M},\mathbb{O})$ be a marked surface with orbifold points of order three, without punctures, with boundary and we assume $\mathbb{O}$ is not empty. An arc $i$ on $(\Sigma,\mathbb{M},\mathbb{O})$ is a curve $i\colon[0,1]\rightarrow\Sigma$ satisfying the following conditions

• •

the endpoints of $i$ are both contained in $\mathbb{M}$.

• •

$i$ does not intersect itself, except that its endpoints may coincide.

• •

$i$ does no intersect to $\mathbb{O}$ and $i$ does not intersect $\mathbb{M}$ except in its endpoints.

• •

if $i$ cuts out an monogon, then such monogon contains just one orbifold point. In this case $i$ is called a pending arc or the loop of a orbifold point.

Two arcs $i$ and $j$ are *isotopic relative *to $\mathbb{M}\cup\mathbb{O}$ if there exist a continuous function $H:[0,1]\times\Sigma\rightarrow\Sigma$ such that

• •

$H(0,x)=x$, for all $x\in\Sigma$;

• •

$H(1,i)=j$;

• •

$H(t,p)=p$ for all $p\in\mathbb{M}\cup\mathbb{O}$;

• •

For every $t\in[0,1]$ the function $H_{t}\colon\Sigma\rightarrow\Sigma$ with $x\mapsto H(t,x)$ is a homeomorphism.

We will consider arcs up to isotopy relative to $\mathbb{M}\cup\mathbb{O}$, parametrization and orientation.

Given an arc $i$, we denote by $\tilde{i}$ its isotopy class. Let $i$ and $j$ be two arcs. We say $i$ and $j$ are compatible if either $\tilde{i}=\tilde{j}$ or $\tilde{i}\neq\tilde{j}$ and there are arcs $i_{1}\in\tilde{i}$ and $j_{1}\in\tilde{j}$, such that $i_{1}$ a $j_{1}$ do not intersect in $\Sigma\setminus\mathbb{M}$.

Definition 7.1**.**

A triangulation of $(\Sigma,\mathbb{M},\mathbb{O})$ is a maximal collection of pairwise compatible arcs.

Given a triangulation $\tau$ of the surface we define a triangle of $\tau$ as the closure of a connected component of $(\Sigma\setminus\tau)\cup\{\mbox{ the pending arcs of }\tau\}$. An orbifold triangle is a triangle containing an orbifold point. A triangle without an orbifold point in its interior is called an ordinary triangle. If a triangle intersects to the boundary of the surface at most in a three points it is called an internal triangle.

Let $\tau$ be a triangulation of $(\Sigma,\mathbb{M},\mathbb{O})$. If $i$ is an arc of $\tau$, the flip of $i$ with respect to $\tau$ is the unique arc $i^{\prime}$ such that $\sigma=(\tau\setminus\{i\})\cup\{i^{\prime}\}$ is a triangulation of $(\Sigma,\mathbb{M},\mathbb{O})$. In this case we denote $i^{\prime}=\operatorname{flip}_{\tau}(i)$ and we say that $\sigma$ is obtained from $\tau$ by a flip of $i\in\tau$. In our case, flips act transitively on triangulations of $(\Sigma,\mathbb{M},\mathbb{O})$, see [19, Theorem 4.2].

7.3 Our surfaces

Let $\Sigma_{n}$ be the surface, with $n\geq 2$, given by a disk with boundary, $n+1$ marked points in its boundary, without punctures and one orbifold point of order three. In this work we often refer to $\Sigma_{n}$ as the $(n+1)$-polygon with one orbifold point, the marked points are called vertices and they are denoted by $\{v_{0},v_{1},\ldots,v_{n}\}$. We orient the vertices in counterclockwise order. In pictures the orbifold point is drawn with the symbol $\times$.

Let $\tau$ be a triangulation of $\Sigma_{n}$. We have that $|\tau|=n$, see [31, Lemma 4.1]. In this case we have two types of admissible triangles for $\tau$, see Figure 1

We associate a quiver $Q(\tau)$ to a triangulation $\tau$ of the orbifold $\Sigma_{n}$ in the following way: the set of vertices is given by the arcs of $\tau$ and the set of arrows is described as follows. For each triangle $\Delta$ of $\tau$ and arcs $i$ and $j$ in $\Delta$ we draw an arrow from $j$ to $i$ if $i$ succeeds $j$ in the clockwise orientation, with the understanding that no arrow incident to a boundary segment is drawn. Finally we draw a loop at the pending arc of $\tau$.

Remark 7.2**.**

In the classical context of marked Riemann surfaces without orbifold points this loop is not drawn. For instance the quiver of a triangulation $T$ of a polygon $P$ without punctures and without orbifold points will be denoted by $Q(T)$ and it is constructed as above but, as we said, it does not have loops.

Denote by $\mathcal{H}(\tau)$ the collection of all internal triangles $\Delta$ of a given triangulation $\tau$. Any element $\Delta$ of $\mathcal{H}(\tau)$ defines a 3-cycle $c_{\Delta}b_{\Delta}a_{\Delta}$ on $Q(\tau)$ up to cyclical equivalence. If we denote by $\varepsilon$ the loop of $\tau$, then the potential associated to $\tau$ is $S(\tau)=\sum_{\Delta\in\mathcal{H}(\tau)}{c_{\Delta}b_{\Delta}a_{\Delta}}+\varepsilon^{3}$.

Definition 7.3**.**

For any triangulation $\tau$ of $\Sigma_{n}$ we define the basic algebra $\Lambda(\tau)$ associated to $\tau$ as the Jacobian algebra $\Lambda(\tau)=\mathcal{P}(Q(\tau),S(\tau))$.

Example 7.4**.**

Consider the triangulation $\tau$ of Figure 2. We see that the algebra $\Lambda(\tau)$ is $\mathbb{C}\langle Q(\tau)\rangle/I$, where $I$ is the ideal generated by $ba$, $cb$, $ac$ and $\varepsilon^{2}$ (compare to [31, Example 2.3]).

Definition 7.5**.**

A weighted quiver is a pair $(Q,\textbf{d})$ where $Q$ is a quiver without loops and $\textbf{d}=(d_{i})_{i\in Q_{0}}$ is an $n$-tuple of positive integers.

Let $Q(\tau)^{*}$ be the quiver obtained from $Q(\tau)$ by deleting the loop. Now we denote by $(Q(\tau)^{*},\textbf{d}_{\tau})$ the weighted quiver associated to $\tau$ where $(\textbf{d}_{\tau})_{j}=2$ if $j$ is the pending arc and $(\textbf{d}_{\tau})_{j}=1$ in other wise.

Fix an $n$-tuple $\textbf{d}=(d_{1}\ldots,d_{n})$, in [26, Lemma 2.3] it was proved that there is a bijection between the set of 2-aclycic weighted quivers $(Q,\textbf{d})$ and the collection of skew-symmetrizable matrices $B$ with skew- symmetrizer given by $D=\operatorname{diag}(d_{1},\ldots,d_{n})$. Indeed, given a quiver $Q$, if $c_{ij}$ is as in Section 3, then $b_{ij}=d_{j}c_{ij}/\mbox{gcd}(d_{i},d_{j})$ define a matrix $B_{Q}$ skew-symmetrized by $D$.

Following [26, Lemma 2.3], we denoted by $B(\tau)$ the skew-symmetrizable matrix associated to $(Q(\tau),\textbf{d}_{\tau})$ and we call it the *adjacency matrix * associated to $\tau$.

8 $\Lambda(\tau)$ as an orbit algebra

In this section we shall note that $\Lambda(\tau)$ can be seen as an orbit Jacobian algebra. With this observation we are going to obtain some results about Galois coverings. For details and missing definitions about orbit Jacobian algebras the reader can see [31]. At the end of the section we will define the arcs representations of $\Lambda(\tau)$.

Let $\widetilde{\Sigma}_{n}$ be the regular $(3n+3)$-gon with $u_{1},u_{2},\ldots,u_{3n+3}$ vertices in counterclockwise orientation and let $\theta$ be the rotation by $120^{\circ}$ on $\widetilde{\Sigma}_{n}$ which sends a vertex $v_{i}$ to $v_{i+(n+1)}$ modulo $(3n+3)$. In the terminology of [31], $\Sigma_{n}$ is the $\mathbb{Z}_{3}$-orbit space of $\widetilde{\Sigma}_{n}$. We consider $\theta$ as a generator of $G=\mathbb{Z}_{3}$. We can see that $G$ acts freely on $\{u_{1},u_{2},\ldots,u_{3n+3}\}$, that is, if $g\in G\setminus\{e\}$, then $g\cdot u_{i}\neq u_{i}$ for $i\in[1,3n+3]$.

We say that an arc $\widetilde{j}$ of $\widetilde{\Sigma}_{n}$ is $G$-admissible or just admissible if $\widetilde{j}$ belongs to some $G$-invariant triangulation $T$ of $\widetilde{\Sigma}_{n}$.

Let $T$ be a triangulation of $\widetilde{\Sigma}_{n}$ and suppose that $T$ is $G$- invariant. Consider $Q(T)$ the quiver associated to $T$, see Remark 7.2. We can define a potential for $Q(T)$ as $S(T)=\sum_{\Delta\in\mathcal{H}(T)}{\gamma_{\Delta}\beta_{\Delta}\alpha_{\Delta}}$. Note that $G$ acts freely on $Q(T)_{0}$ and for any $\alpha_{\Delta}\beta_{\Delta}\gamma_{\Delta}$ we have that $g\cdot(\alpha_{\Delta}\beta_{\Delta}\gamma_{\Delta})$ is again a summand of $S(T)$ for all $g\in G$. We can define, [31, Section 2.1], the orbit quiver $Q(T)_{G}$ of $Q(T)$ in the obvious way. We define the potential $S(T)_{G}$ for $Q(T)_{G}$ as the image of $S(T)$ under the canonical morphism $\pi:\mathbb{C}\langle Q(T)\rangle\rightarrow\mathbb{C}\langle Q(T)_{G}\rangle$ induces by $\pi(i)=G\cdot i$ for $i\in Q(T)_{0}$ and $\pi(a)=G\cdot a$ for $a\in Q(T)_{1}$, note that $\pi$ is a Galois $G$-covering. We define the orbit Jacobian algebra of the orbit quiver with potential as $\mathcal{P}(Q(T),S(T))_{G}=\mathcal{P}(Q(T)_{G},S(T)_{G})$. We make the following convention $\Lambda(T)=\mathcal{P}(Q(T),S(T))$ and $\Lambda(T)_{G}=\mathcal{P}(Q(T),S(T))_{G}$, see Example 8.3. The following result shows that we get a Galois covering, see [31, Proposition 3.1].

Lemma 8.1** ((Paquette-Schiffler)).**

The Galois $G$-covering $\pi:\mathbb{C}\langle Q(T)\rangle\rightarrow\mathbb{C}\langle Q(T)_{G}\rangle$ induces a Galois $G$-covering $\pi:\Lambda(T)\rightarrow\Lambda(T)_{G}$.

Remark 8.2**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$ and let $T$ be the triangulation of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$. In $T$ there exist an unique triangle $\Delta_{T}$ such that it is $G$-invariant and the other triangles in $\mathcal{H}(T)$ have a trivial stabilizer. The triangle $\Delta_{T}$ corresponds to the pending arc of $\tau$ and the $G$-orbit of any triangle $\Delta$ different to $\Delta_{T}$ corresponds with a triangle of $\mathcal{H}(\tau)$. We conclude that $Q(\tau)=Q(T)_{G}$ and with the above observation we get that $\Lambda(\tau)=\Lambda(T)_{G}$.

Example 8.3**.**

Let $\tau$ be the triangulation of $\Sigma_{3}$ depicted on the right of Figure 3. Let $T$ be the corresponding triangulation on $\widetilde{\Sigma}_{3}$ depicted on the left of Figure 3. The quiver $Q(T)$ is drawn below

[TABLE]

Consider $S(T)=\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}$ the potential associated to $T$. Let $G=<\theta>$ be the cyclic group of order 3 with generator $\theta$. Then $G$ acts freely on $(Q,S)$ by increasing by one, module 3, the indices of the symbols. Passing to the orbit space of this action we get

[TABLE]

where $i=G\cdot i_{1}$, $j=G\cdot j_{1}$, $k=G\cdot k_{1}$, $\alpha=G\cdot\alpha_{1}$, $\beta=G\cdot\beta_{1}$ and $\varepsilon=G\cdot\varepsilon_{1}$. The orbit Jacobian algebra $\mathcal{P}(Q(T)_{G},S(T)_{G})$ is not but $\Lambda(\tau)=\mathbb{C}\langle Q(\tau)\rangle/\langle\varepsilon^{2}\rangle$.

Proposition 8.4**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$. Then $\Lambda(\tau)$ is finite dimensional.

Proof 8.5**.**

From Lemma 8.1 we have that $\pi:\Lambda(T)\rightarrow\Lambda(T)_{G}$ is a Galois $G$-covering. In particular we have isomorphism $\pi_{i,j}:\bigoplus_{g\in G}\Lambda(T)(e_{i},g\cdot e_{j})\rightarrow\Lambda(\tau)(\pi(e_{i}),\pi(e_{j}))$ for any idempotent $e_{i}$ and $e_{j}$ of $\Lambda(T)$. We know $\Lambda(T)$ is finite-dimensional (the reader can see the finite dimension of the Jacobian algebra associated to a triangulation in a more general context of surfaces with non-empty boundary in [25, Theorem 36]), so $\Lambda(\tau)$ is finite dimensional. The proof of the lemma is completed.

A string algebra $B=\mathbb{C}\langle Q\rangle/\langle P\rangle$ is a gentle algebra if the following conditions are satisfied

• (Gt1)

$P$ is generated by paths of length 2.

• (Gt2)

For any arrow $a\in Q_{1}$ we have $|\{b\in Q_{1}\colon t(a)=h(b)\mbox{ and }ab\in P\}|\leq 1$ and $|\{c\in Q_{1}\colon t(c)=h(a)\mbox{ and }ca\in P\}|\leq 1$.

Proposition 8.6**.**

For any triangulation $\tau$ of $\Sigma_{n}$ we have that $\Lambda(\tau)$ is a gentle algebra.

Proof 8.7**.**

Let $T$ be the triangulation of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$. The proof is an adaptation of proof [2, Lemma 2.5], from that lemma we have that $\Lambda(T)$ is gentle. By definition $\Lambda(\tau)=\mathbb{C}\langle Q_{G}\rangle/J(Q_{G},S_{G})$ and it is clear that $J(Q_{G},S_{G})$ is generated by paths of length two. Since we have Proposition 8.4, only remains to prove (Gt2), (S1) and(S2).

(S1). First, let $j$ be the pending arc of $\tau$. We consider $\tilde{j}$ an element in $\pi^{-1}(j)$. We have that $\tilde{j}$ is contained in two triangles of $T$. One of those triangles has the other preimages of $j$ as sides, say $\Delta(j)$, in other words, $\Delta(j)$ is invariant under $G$. By the definition of $S(T)_{G}$ we can conclude that there is a loop based at $j$ and there is at most one arrow starting at $j$ and one arrow ending at $j$. Now, one connected component of $\tilde{\Sigma}_{n}\setminus\{\tilde{j}\}$, precisely which no contain the other preimages of $j$, it is a fundamental region for the action of $G$ on $T$. If $k$ is not a pending arc of $\tau$, we can consider a preimage of $k$ in the fundamental region above, recall $\Lambda(T)$ is gentle, this implies (S1) for $k$. For this reason we just have to prove (S2) and (Gt2) in the orbifold triangle.

(S2) and (Gt2). This two properties follow from the fact that there is a loop based at the pending arc $j$ and at most one arrow with starting at $j$ and at most one arrow with ending at $j$. This conclude the proof.

By Lemma 2.12 and Lemma 8.1 we know the following.

Lemma 8.8**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$. Then the push-down functor $\pi_{*}:\Lambda(T)\operatorname{-mod}\rightarrow\Lambda(\tau)\operatorname{-mod}$ is a $G$-precovering.

We are going to define a string $W_{j}(\tau)$ of $\Lambda(\tau)$ for every arc $j\notin\tau$ of $\Sigma_{n}$. We denote by $\pi\colon\widetilde{\Sigma}_{n}\rightarrow\Sigma_{n}$ the canonical projection. We know that $\pi^{-1}(j)=\{\widetilde{j},\widetilde{j}_{\theta},\widetilde{j}_{\theta^{2}}\}$. Recall that $G=\langle\theta\rangle=\mathbb{Z}_{3}$. Let $T$ be the triangulation in $\widetilde{\Sigma}_{n}$ corresponding to $\tau$, see Figure 4.

Let $j$ be an arc of $\Sigma_{n}$ such that $j\notin\tau$. Choice $\alpha\in\pi^{-1}(j)$, by definition $\alpha$ is an arc of $\widetilde{\Sigma}_{n}$ and $\alpha$ joints two vertices $u_{l}$ and $u_{l+r}$ of $\widetilde{\Sigma}_{n}$. Every time $\alpha$ crosses two adjacent initial arcs $\gamma:\widetilde{i}_{s_{1}}\rightarrow\widetilde{i}_{s_{2}}$ of $\widetilde{\tau}$, we write the letter $G\cdot\gamma$ (a letter on $Q(\tau)$) if $\alpha$ crosses $\widetilde{i}_{s_{1}}$ first from $u_{l}$ to $u_{l+r}$ or we write the letter $(G\cdot\gamma)^{-1}$ in other wise, see Example 8.3 . This construction does not depend of the choice of $\alpha$ up to string equivalency, see Section 5. Denote by $W_{j}(\tau)$ the string of $\Lambda(\tau)$ obtained in this way. In Figure 4 we show an example of this construction.

Definition 8.9**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$. For any arc $j\notin\tau$ we define the arc representation $M(j,\tau)$ of $j$ with respect to $\tau$ as the string module associated to $W_{j}(\tau)$, i.e $M(j,\tau)=N(W_{j}(\tau))$, see Section 5.1. Since a string and its inverse generate isomorphic string modules we have that $M(j,\tau)$ is well defined up to isomorphism. Now, for any arc $j\in\tau$ we define $M(j,\tau):=\mathcal{S}_{j}^{-}$ as the corresponding negative simple representation of $\Lambda(\tau)$.

As long as there is no confusion we ease the notation and write $W_{j}:=W_{j}(\tau)$ and $M(j):=M(j,\tau)$. We finish this section with some remark about the push-down fuctor.

Remark 8.10**.**

For convenience we are going to define explicitly the push-down functor in our situation. Let $T$ be atriangulation of $\widetilde{\Sigma}_{n}$. Set $\Lambda=\Lambda(T)$ and consider $\pi:\Lambda\rightarrow\Lambda_{G}$ the canonical projection of the action, where $\Lambda_{G}=\Lambda(T)_{G}$. We define the push-down functor $\pi_{*}:\Lambda\operatorname{-mod}\rightarrow\Lambda_{G}\operatorname{-mod}$ as follows.

For objects*: let $M\in\Lambda\operatorname{-mod}$ be a $\Lambda$-representation. For $i\in Q_{0}$ we define $\pi_{*}(M)_{G\cdot i}=\bigoplus_{g\in G}{M_{g\cdot i}}$. Let $\alpha:i\rightarrow j$ be an arrow of $Q$. We are going to define $\pi_{*}(M)_{G\cdot\alpha}:\bigoplus_{g\in G}{M_{g\cdot i}}\rightarrow\bigoplus_{h\in G}{M_{h\cdot j}}$. Now, by definition, for any $h\in G$ we have an isomorphism $\pi_{j,h\cdot i}:\bigoplus_{g\in G}\Lambda(g\cdot i,h\cdot j)\rightarrow\Lambda_{G}(G\cdot i,G\cdot j)$. So, $G\cdot\alpha=\sum_{g\in G}{\pi(\alpha_{h,g})}$ for any $h\in G$ and we define $\pi_{*}(M)_{G\cdot\alpha}=(\alpha_{h,g})_{g,h\in G}$.*

For morphims:* let $f:M\rightarrow N$ be a morphism in $\Lambda\operatorname{-mod}$. For any $i\in Q_{0}$ we need to define $\pi_{*}(f)_{G\cdot i}:\bigoplus_{g\in G}{M_{g\cdot i}}\rightarrow\bigoplus_{h\in G}{N_{h\cdot i}}$. We set $\pi_{*}(f)_{G\cdot i}=\operatorname{diag}(f_{g\cdot i}:g\in G)$ as a diagonal map.*

9 The Caldero-Chapoton algebra for a specific triangulation

In this section we will study the Caldero-Chapoton algebra of a specific triangulation $\tau_{0}$ of $\Sigma_{n}$. We will see that the arc representations of $\Lambda(\tau_{0})$ play a central role. Fix the vertices of $\Sigma_{n}$ in counter clockwise order $\{v_{0}\ldots,v_{n}\}$. Let $i_{n}$ be the pending arc at $v_{0}$. We denote the pending at $v_{k}$ as $i^{\prime}_{k}$ for $k=0,\ldots,n$. With this notation we see that $i^{\prime}_{0}=i_{n}$. Let $i_{k}$ be the arc from $v_{0}$ to $v_{k+1}$ going in counterclockwise for $k=1,\ldots,n-1$. We define the special triangulation $\tau_{0}$ of $\Sigma_{n}$ as the collection of arcs $\{i_{1},\ldots,i_{n}\}$, see on the right hand of Figure 3.

For $\tau_{0}$ we have a nice description of the concepts introduced in Section 7, for instance, the weighted quiver associated to $\tau_{0}$ looks like

[TABLE]

The matrix $B(\tau_{0})$ is going to be our input to obtain the polynomials of Chekhov-Shapiro and we are going to describe a basic algebra associated to $\tau_{0}$.

Let $\Lambda:=\Lambda(\tau_{0})$ be the basic algebra associated to $\tau_{0}$, it is clear that $\Lambda$ is given by $\mathbb{C}\langle Q(\tau_{0})\rangle/I$ where $Q(\tau_{0})$ is the quiver

[TABLE]

and $I$ is the ideal generated by $\varepsilon^{2}$.

For every arc $j$ of $\Sigma_{n}$ we defined a decorated indecomposable representation $M(j)$ of $\Lambda$ with respect to $\tau_{0}$, see Definition 8.9.

Remark 9.1**.**

For any arc $j\notin\tau_{0}$ Definition 8.9 can be rewritten up to isomorphism by counting intersection numbers directly in $\Sigma_{n}$. We give this approach for convenience,

[TABLE]

Given an arrow $a_{l}$ with $l=1,\ldots,n-1$, we define $M(j)_{a_{l}}$ as follows:

• •

if $0<\operatorname{dim}(M(j)_{t(a_{l})})<\operatorname{dim}(M(j)_{h(a_{l})})$, then $M(j)_{a_{l}}=\left(\begin{smallmatrix}0\\ 1\end{smallmatrix}\right)$;

• •

if $0<\operatorname{dim}(M(j)_{t(a_{l})})=\operatorname{dim}(M(j)_{h(a_{l})})$, then $M(j)_{a_{l}}$ acts as the corresponding identity;

• •

$M(j)_{a_{l}}=0$* in otherwise.*

If $\operatorname{dim}(M(j)_{n})\neq 0$, then $M(j)_{\varepsilon}=\left(\begin{smallmatrix}0&1\\ 0&0\end{smallmatrix}\right)$.

Remark 9.2**.**

From Definition 8.9 and Theorem 5.2 we know $M(j)$ is indecomposable in $\operatorname{decrep}(\Lambda)$ for every arc $j$. Remark 9.1 allow us to compute $M(j)$ without $\widetilde{\Sigma}_{n}$.

For any arc $j\notin\tau_{0}$ we define the support of $M(j)$ as $\operatorname{Supp}M(j)=\{l\colon M(j)_{l}\neq 0\}$. The same argument of [10, Lemma 2.2] can be applied here to conclude that $\operatorname{Supp}M(j)$ is connected as a subset of $[1,n]$. So, we are going to think that $\operatorname{Supp}M(j)$ is an interval.

Example 9.3**.**

For $n=5$, we compute $M(j_{l})$ with $l=1,2$ and $3$, see Figure 5.

We have

[TABLE]

As illustration we have the Caldero-Chapoton function $\mathcal{C}_{\Lambda}(M(j_{2}))$

[TABLE]

9.1 AR translations, E-invariant and g-vectors of arc representations

Let $j$ be an arc of $\Sigma_{n}$. We introduce some notation. Given two vertices $v_{r}$ and $v_{l}$ of $\Sigma_{n}$ with $r+1<l$ and $r\in\{1,\ldots,n-2\}$ we have two arcs from $v_{r}$ to $v_{l}$ denoted by $[v_{r},v_{l}]^{+}$ and $[v_{r},v_{l}]^{-}$. Indeed,if $0<r$, then $[v_{r},v_{l}]^{+}$ does not intersect to $i_{n}$ in the interior of $\Sigma_{n}$ while $[v_{r},v_{l}]^{-}$ does. For example, in the Figure 5 we have $j_{1}=[v_{2},v_{5}]^{+}$ and $j_{2}=[v_{2},v_{4}]^{-}$. For $r=0$ we say that $i_{k}=[v_{0},v_{k+1}]^{-}$ and $[v_{0},v_{k+1}]^{+}$ is the another arc from $v_{0}$ to $v_{k+1}$ with $k=1,\ldots,n-1$. In the case $l=r+1$, we have $[v_{r},v_{r+1}]^{-}$ is not a boundary segment.

Remark 9.4**.**

If $n\geq 4$, with the above notation we can describe $W_{j}$ explicitly for any $j\notin\tau_{0}$.

• •

$W_{[v_{1},v_{n}]^{+}}=a_{n-3}\cdots a_{1}$* and $W_{[v_{1},v_{n}]^{-}}=\varepsilon\cdots a_{1}$;*

• •

$W_{[v_{i},v_{l}]^{+}}=a_{l-3}\cdots a_{i}$* and $W_{[v_{i},v_{l}]^{-}}=a_{l}^{-1}\cdots a_{n-1}^{-1}\varepsilon\cdots a_{i}$ for $0<i$ and $i+2<l<n$;*

• •

$W_{[v_{i},v_{l}]^{+}}=1_{(i,+)}$* and $W_{[v_{i},v_{l}]^{-}}=a_{l}^{-1}\cdots a_{n-1}^{-1}\varepsilon\cdots a_{i}$ for $l=i+2$ and $i\leq n-2$;*

• •

$W_{[v_{i},v_{l}]^{-}}=a_{l}^{-1}\cdots a_{n-1}^{-1}\varepsilon\cdots a_{i}$* for $l=i+1$ and $0<i<n-2$;*

• •

$W_{[v_{n-1},v_{n}]^{-}}=\varepsilon a_{n-1}$;

• •

$W_{[v_{0},v_{l}]^{+}}=a_{n-2}\cdots a_{l}$* for $1\leq l<n-1$;*

• •

$W_{i^{\prime}_{k}}=a_{k}^{-1}\cdots a_{n-1}^{-1}\varepsilon\cdots a_{k}$* for $1\leq k\leq n-1$;*

• •

$W_{i^{\prime}_{n}}=\varepsilon$.

The reader can compare the following lemma with the $A_{n}$ case, [10, Theorem 2.13]. Given an arc $j$ we denote by $r^{+}(j)$ ( in [10] it would be $r^{-}$) the arc of $\Sigma_{n}$ that we obtain by rotating $j$ in counterclockwise for an angle of $2\pi/(n+1)$. By $r^{-}(j)$ ( in [10] it would be $r^{+}$) we denote the arc obtained from $j$ by rotating $j$ in clockwise for an angle of $2\pi/(n+1)$.

Lemma 9.5**.**

Assume $j$ is not an initial arc of $\Sigma_{n}$.

• (a)

If $M(j)$ is not projective, then $\tau(M(j))=M(r^{+}(j))$, where $\tau$ denotes the Auslander-Reiten translation.

• (b)

If $M(j)$ is not injective, then $\tau^{-}(M(j))=M(r^{-}(j))$.

Proof 9.6**.**

The lemma can be proved by cases using Remark 9.4 and the classification of the Auslander-Reiten sequences containing string modules from [8, p.p 170-172]. However, we proved this result in Corollary 10.5.

Lemma 9.7**.**

Assume that $j$ is an arc of $\Sigma_{n}$. Then $E_{\Lambda}(M(j))=0$.

Proof 9.8**.**

We postpone this proof up to Corollary 10.9. It is worth mention that this lemma can be proved with the alluded results of [8].

Lemma 9.9**.**

Assume $j$ is an arc of $\Sigma_{n}$ such that $j\notin\tau_{0}$. Then $\operatorname{pd}M(j)\leq 1$ and $\operatorname{id}M(j)\leq 1$. Here $\operatorname{pd}M(j)$ (resp. $\operatorname{id}M(j)$) denotes the projective dimension of $M(j)$ (resp. the injective dimension of M(j)).

Proof 9.10**.**

The lemma follows from [23, Proposition 3.5] and the definition of $M(j)$. Indeed, in the language of [23], if we take

[TABLE]

and $\Omega=\{(i+1,i)\colon 1\leq i\leq n-1\}$, then we get $\Lambda=H(C,D,\Omega)$. By definition $M(j)$ is locally free $\Lambda$-module for every arc $j\notin\tau_{0}$ (see [23, Section 1.5]).

Remark 9.11**.**

In [24], Geiß-Leclerc-Schröer have proved that, in particular, for this algebra $\Lambda=H(C,D,\Omega)$ of the previous lemma, we can recover a classic cluster algebra by means of Caldero-Chapoton functions. However, they consider the quasiprojective variety $\operatorname{Gr}_{\mbox{l.f.}}(\textbf{r},M)$ of locally free submodules $N$ of $M$ with rank vector r instead $\operatorname{Gr}_{\textbf{e}}(M)$ as we made her, see Example [24, Section 13.1].

Remark 9.12**.**

Lemma 9.9 ensures that $\operatorname{id}M(j)\leq 1$, now it can be seen that we have the following minimal injective presentation of $M(j)$ for each arc $j\notin\tau_{0}$, this is a consequence of [8].

$j$* crosses to $i_{n}\colon$ in this case $W_{j}=a_{n_{j}}^{-1}\cdots\varepsilon\cdots a_{m_{j}}$ with $m_{j}\leq n_{j}$. Then the following exact sequence is a minimal injective presentation of $M(j)$,*

[TABLE]

Here we define $N(a_{r-2}\cdots a_{1})$ as zero if $r=1$ and it is the simple representation at $1$ if $r=2$. 2. 2.

$j$* does not cross to $i_{n}\colon$ in this case $W_{j}=a_{n_{j}}\cdots a_{m_{j}}$ with $n-2\geq n_{j}\geq m_{j}$. Then the following exact sequence is a minimal injective presentation of $M(j)$,*

[TABLE]

Proposition 9.13**.**

If $j_{1}$ and $j_{2}$ are not arcs of $\tau_{0}$, then the following hold:

• •

There exists a $\mathbb{C}$-linear isomorphism

[TABLE]

• •

There exists a $\mathbb{C}$-linear isomorphism

[TABLE]

Proof 9.14**.**

The proposition follows from Lemma 9.9, [3, Corollary (IV) 2.14(b)] and [3, Corollary (IV) 2.15(a)] respectively.

Albeit the previous proposition is true for any modules with projective and injective dimension at most 1 we stated it in that fashion for convenience.

Lemma 9.15**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$. If $j_{1}$ and $j_{2}$ are arcs of $\tau$, then $E_{\Lambda}(M(j_{1}),M(j_{2}))=0$.

Proof 9.16**.**

If $j_{2}\in\tau_{0}$ or $M(j_{2})$ is injective, then $E_{\Lambda}(M(j_{1}),M(j_{2}))=0$ for all arc $j_{1}\in\tau$ by definitions and Proposition 3.2. So we can suppose $j_{2}$ is not in $\tau_{0}$ and $M(j_{2})$ is not injective.

• Case 1.

$j_{1}=i_{k}$* for some $1\leq k\leq n\colon$ in this case $M(j_{1})$ is the negative simple representation of $\Lambda$ at $k$. It is clear that $E_{\Lambda}(M(j_{2}),M(i_{k}))=\operatorname{dim}M(j_{2})_{k}$ by Proposition 3.2, but $i_{k},j_{2}\in\tau$, then $\operatorname{dim}M(j_{2})_{k}=0$.*

• Case 2.

$j_{1}\notin\tau_{0}$* and $M(j_{1})$ is injective: then $E_{\Lambda}(M(j_{2}),M(j_{1}))=0$ for all $j_{2}\in\tau$. We have to prove $E_{\Lambda}(M(j_{1}),M(j_{2}))=0$. By Proposition 3.2 we get $E_{\Lambda}(M(j_{1}),M(j_{2}))=\operatorname{dim}\operatorname{Hom}_{\Lambda}(\tau^{-}(M(j_{2}),M(j_{1}))$. If $M(j_{1})$ is injective, then $j_{1}=[v_{1},v_{l}]^{+}$ with $2<l\leq n$, $j_{1}=i^{\prime}_{1}$ or $j=[v_{0},v_{1}]^{+}$. Since $j_{1},j_{2}\in\tau$ and $M(j_{2})$ is not injective, $\operatorname{Supp}(M(j_{1}))\cap\operatorname{Supp}(\tau^{-}M(j_{2}))=\emptyset$ and $\operatorname{Hom}_{\Lambda}(\tau^{-}(M(j_{2})),M(j_{1}))=0$.*

• Case 3.

$j_{1}\notin\tau_{0}$* and $M(j_{1})$ is not injective*$\colon$* we have to prove $E_{\Lambda}(M(j_{1}),M(j_{2}))=0$ and $E_{\Lambda}(M(j_{2}),M(j_{1}))=0$. For $l=1,2$, let $m_{l}$ be the minimum positive integer such that $M((r^{-})^{m_{l}}(j_{l}))$ is injective.*

If $m_{1}\leq m_{2}$ , then by **[3, Corollary (IV) 2.15 (c)]** and Proposition 3.2 we have

[TABLE]

[3, Corollary (IV) 2.14 (b)]** implies $\operatorname{dim}\operatorname{Ext}^{1}((\tau^{-})^{m_{1}}M(j_{1}),(\tau^{-})^{m_{1}}(M(j_{2})))=E_{\Lambda}(M(j_{1}),M(j_{2}))$. Since $(\tau^{-})^{m_{1}}(M(j_{1}))$ is injective, we get $E_{\Lambda}(M(j_{1}),M(j_{2}))=0$. Now,

[TABLE]

Since $m_{1}\leq m_{2}$, we apply **[3, Corollary (IV) 2.15 (c)]** to obtain $E_{\Lambda}(M(j_{2}),M(j_{1}))=0$. The case $m_{2}<m_{1}$ is similar.

This proves the lemma.

Lemma 9.17**.**

Given an arc $j\notin\tau_{0}$ we have $\operatorname{Ext}^{1}(M(j),M(j))=0$.

Proof 9.18**.**

By Proposition 3.2 we have $E_{\Lambda}(M(j))=\operatorname{dim}\operatorname{Hom}_{\Lambda}(\tau^{-}(M(j)),M(j))$. Proposition 9.13 implies $E_{\Lambda}(M(j))=\operatorname{dim}\operatorname{Ext}^{1}(M(j),M(j))$. The lemma follows from Lemma 9.7.

The following result is a consequence of Voigt’s Lemma, see [16, Sections 1.6 and 1.8].

Lemma 9.19**.**

Assume $j\notin\tau_{0}$. Then the $G_{\textbf{d}}$-orbit $\mathcal{O}(M(j))$ is open in $\operatorname{rep}_{\textbf{d}}(\Lambda)$.

Proof 9.20**.**

By Lemma 9.17 we have $\operatorname{Ext}^{1}(M(j),M(j))=0$, this implies that $\mathcal{O}(M)$ is open, for example see [16, 1.7 Corollary 3].

By Lemma 9.7 we have examples of $E$-rigid indecomposable $\Lambda$- modules i.e.$E_{\Lambda}(M(j))=0$. The next result shows that we already know all $E$-rigid $\Lambda$-modules, this result can be seen as a consequence of Lemma 10.9, but we have been mentioned that for $\tau_{0}$ the proof of some result may be made with explicit computations, here we give an example.

Proposition 9.21**.**

If $N$ is a indecomposable $\Lambda$-module and $N$ is not of the form $M(j)$ for some arc $j$ of $\Sigma_{n}$, then $E_{\Lambda}(N)>0$.

Proof 9.22**.**

Let $W$ be the string associated to $N$ and assume that $N$ is not $E$-rigid. Given a non-initial arc $j$ of $\Sigma_{n}$ we have the string $W_{j}$ is one of the following

[TABLE]

Therefore $W$ is different to $W_{j}$ for any arc $j$ of $\Sigma_{n}$, here we use Remark 9.4. If $W=a_{n-1}\cdots a_{l}$ with $l>1$ we have $N(W)$ looks like

[TABLE]

By [8] we get $\tau^{-1}(N(W))=N(\varepsilon^{-1}a_{n-1}\cdots a_{l-1})$. Since

[TABLE]

we have $\operatorname{Hom}_{\Lambda}(\tau^{-1}(N(W)),N(W))\neq 0$. Proposition 3.2 implies $E_{\Lambda}(N(W))>0$. The case when $l=1$ is similar and follows from [8].

If $W=a_{n_{W}}^{-1}\cdots\varepsilon^{-1}\cdots a_{m_{W}}$ with $1<m_{W}<n_{W}$, then

[TABLE]

and by [8] we get $\tau^{-1}(N(W))=N(a_{n_{W}-1}Wa_{m_{W}-1})$. From definitions we get $\operatorname{Hom}_{\Lambda}(\tau^{-1}(N(W)),N(W))\neq 0$, so $E_{\Lambda}(N(W))>0$. The case when $m_{W}=1$ is similar and follows from [8]. Since the indecomposable $\Lambda$-modules are parametrized by strings, the proposition follows from Theorem 5.2.

Now we interpret the $g$-vector of a representation $M(j)$ in terms of intersection numbers. The three lemmas below follow from Remark 9.12 and Lemma 3.1. For instance

Lemma 9.23**.**

For a pending arc $i_{k}^{\prime}$ with $k\in\{1,2,\ldots,n\}$ we have

[TABLE]

Proof 9.24**.**

We start with the pending arc $i^{\prime}_{k}$, from Remark 9.12 we obtain $I_{n}=N(a_{1}^{-1}\cdots\varepsilon\cdots a_{1})$ and $N(a_{n_{j}-2}\cdots a_{1})=N(a_{m_{j}-2}\cdots a_{1})$ because $n_{j}=k=m_{j}$, remember that the string associated to $i^{\prime}_{k}$ is $a_{k}^{-1}\cdots\varepsilon\cdots a_{k}$. By the other hand $I_{k-1}=N(a_{k-2}\cdots a_{1})$. The result follow from Lemma 3.1.

Lemma 9.25**.**

Let $j$ be an arc. If $j$ is not a initial arc, it is not a pending arc and it intersects to $i_{n}$ in the interior of $\Sigma_{n}$, then we have

[TABLE]

Lemma 9.26**.**

Let $j$ be an arc. If $j$ is not an initial arc, it is not a pending arc and it does not intersect to $i_{n}$, then we have

[TABLE]

Proposition 9.27**.**

The set

[TABLE]

is linearly independent over $\mathbb{C}$.

Proof 9.28**.**

From the three lemmas above we have that the $g$-vectors $g_{\Lambda}(M(j))$ are pairwise different. For $n$ even the result follows directly from [11, Proposition 4.3] since $\ker(C_{Q})=0$. For $n$ arbitrary we can adapt the argument in proof of [11, Proposition 4.3] as follows. Define

[TABLE]

We can define two partial orders in $\mathbb{Z}^{n}$. Let $\textbf{a},\textbf{b}\in\mathbb{Z}^{n}$ be vectors. We say $\textbf{a}\leq\textbf{b}$ if there exist some $\textbf{e}\in\mathbb{Q}^{n}_{\mbox{succ}}$ such that $\textbf{a}=\textbf{b}+C_{Q}\textbf{e}$ and $\textbf{a}\preceq\textbf{b}$ if there exist some $\textbf{f}\in\mathbb{Q}^{n}_{0}$ such that $\textbf{a}=\textbf{b}+C_{Q}\textbf{f}$. These two orders induce two partial orders on the set of Laurent monomials in $n$ variables $x_{1},x_{2},\ldots,x_{n}$. We say $\textbf{x}^{\textbf{a}}\leq\textbf{x}^{\textbf{b}}$ if $\textbf{a}\leq\textbf{b}$ and $\textbf{x}^{\textbf{a}}\preceq\textbf{x}^{\textbf{b}}$ if $\textbf{a}\preceq\textbf{b}$. We define the degree of $\textbf{x}^{\textbf{a}}$ as $\deg(\textbf{x}^{\textbf{a}})=\textbf{a}$.

If $\operatorname{soc}(M(j))=S_{n}$ (the socle of $M(j)$) , then $\mathcal{C}_{\Lambda}(M(j))$ has an unique monomial of maximal degree with respect to $\leq$, namely $g_{\Lambda}(M(j))$. If $\operatorname{soc}(M(j))=S_{i}$ with $i\neq n$, then $\mathcal{C}_{\Lambda}(M(j))$ has an unique monomial of maximal degree with respect to $\preceq$ given by $g_{\Lambda}(M(j))$. Since the $g$-vectors are pairwise different we have that the Caldero-Chapoton functions are pairwise different. Now assume $\lambda_{1}\mathcal{C}_{\Lambda}(M(j_{1}))+\cdots+\lambda_{t}\mathcal{C}_{\Lambda}(M(j_{t}))=0$ for some $\lambda_{l}\in\mathbb{C}$. We can assume that $\lambda_{l}\neq 0$ for all $l$.

It can be seen that if $\operatorname{soc}(M(j))=S_{n}$, then $\textbf{x}^{g_{\Lambda}(M(j))}$ does not occur as a summand of any $\mathcal{C}_{\Lambda}(M(k))$ with $\operatorname{soc}(M(k))=S_{i}$ and $i<n$. If there exist an index $s_{0}$ such that $M(j_{s_{0}})$ has socle $S_{n}$, then there exist an index $s$ such that $\textbf{x}^{g_{\Lambda}(M(j_{s}))}$ is $\leq$-maximal in the set of $\{\textbf{x}^{g_{\Lambda}(M(j_{l}))}\colon\operatorname{soc}(M(j_{l}))=S_{n}\}$. Since the $g$-vectors are pairwise different we can conclude that $\lambda_{s}=0$. Indeed, $\textbf{x}^{g_{\Lambda}(M(j_{s}))}$ does not occur as a summand of any $\mathcal{C}_{\Lambda}(M(j_{l}))$ with $l\neq s$, which is a contradiction.

If $\operatorname{soc}(M(j_{l}))\neq S_{n}$ for all $l$, then there exist an index $r$ such that $\textbf{x}^{g_{\Lambda}(M(j_{r}))}$ is $\preceq$-maximal in the set of $\{\textbf{x}^{g_{\Lambda}(M(j_{l}))}\colon 1\leq l\leq t\}$. Since the $g$-vectors are pairwise different we have that $\textbf{x}^{g_{\Lambda}(M(j_{r}))}$ does not occur as a summand of any of the $\mathcal{C}_{\Lambda}(M(j_{l}))$ with $l\neq r$. Thus $\lambda_{r}=0$, a contradiction. Therefore $\mathcal{C}_{\Lambda}(M(j_{1})),\ldots,\mathcal{C}_{\Lambda}(M(j_{t}))$ are linearly independent.

9.2 Generic version

In this section we obtain generic version of the results of the last section. Given a triangulation $\tau$ of $\Sigma_{n}$ we construct an strongly reduced irreducible component of $\operatorname{decrep}(\Lambda)$, see Section 4.

Denote by $Z_{j}$ the irreducible component of $\operatorname{decrep}(\Lambda)$ that contains $\mathcal{O}(M(j))$. We know $\mathcal{O}(M(j))$ is open, so it is dense in $Z_{j}$. Then $E_{\Lambda}(Z_{j})=E_{\Lambda}(M(j))=0$. In the notation of Section 4 this means, in particular, that $Z_{j}$ is a strongly reduced irreducible component of $\operatorname{decrep}(\Lambda)$. We can think that some generic homological data of $Z_{j}$ is encoded in the homological data of $M(j)$.

Proposition 9.29**.**

Given a triangulation $\tau$ of $\Sigma_{n}$ and two arcs $j_{1},j_{2}\in\tau$ we have $E_{\Lambda}(Z_{j_{1}},Z_{j_{2}})=0$.

Proof 9.30**.**

By Lemma 9.15 we know $E_{\Lambda}(M(j_{1}),M(j_{2}))=0$. It can be seen that the set $\mathcal{O}(M(j_{1}))\times\mathcal{O}(M(j_{2}))$ is open in $Z_{j_{1}}\times Z_{j_{2}}$. Indeed, from Lemma 9.19 we know that $\mathcal{O}(M(j_{l}))$ is open in $Z_{j_{l}}$ for $l=1,2$. The claim follows from the equality of sets

[TABLE]

because $Z_{j_{1}}\times Z_{j_{2}}\setminus[\mathcal{O}(M(j_{1}))\times\mathcal{O}(M(j_{2}))]$ would be the union of two closed sets.

If $(M,N)\in\mathcal{O}(M(j_{1}))\times\mathcal{O}(M(j_{2}))$, then $E_{\Lambda}(M,N)=0$. Since $Z_{j_{1}}$ and $Z_{j_{2}}$ are irreducible, we have $\mathcal{O}(M(j_{1}))\times\mathcal{O}(M(j_{2}))$ is dense in $Z_{j_{1}}\times Z_{j_{2}}$. Then $E_{\Lambda}(Z_{j_{1}},Z_{j_{2}})=0$.

The next result is a consequence of Theorem 4.4 and Proposition 9.29.

Proposition 9.31**.**

Given a triangulation $\tau=\{j_{1},\ldots,j_{n}\}$ of $\Sigma_{n}$, the closed set

[TABLE]

is a strongly reduced irreducible component of $\operatorname{decrep}(\Lambda)$.

We introduce some notation that, albeit non-standard, shall be useful to us. Given a vector $v=(v_{1},\ldots,,v_{n})^{t}\in\mathbb{Z}^{n}$ we write $v=v_{1}[1]+\cdots+v_{n}[n]$. Moreover, we write $[n_{1},n_{2}]\subseteq[1,n]$ to simplify $1[n_{1}]+\cdots+1[n_{2}]$. For example, with this notation, $2[n_{1},n_{2}]$ means $2[n_{1}]+\cdots+2[n_{2}]$.

The next proposition generalizes [11, Proposition 9.4].

Proposition 9.32**.**

The set

[TABLE]

generates the Caldero-Chapoton algebra $\mathcal{A}_{\Lambda}$ as $\mathbb{C}$-algebra, where $\operatorname{decIrr^{s.r}}(\Lambda)$ denotes the strongly reduced irreducible components of $\operatorname{decrep}(\Lambda)$.

Proof 9.33**.**

Only remains to prove that the Caldero-Chapoton functions of the non-$E$-rigid representations can be expressed in terms of the Caldero-Chapoton functions of the $E$-rigid representations. Let $L_{1}=a_{n_{1}}^{-1}\cdots\varepsilon\cdots a_{m_{1}}$ with $m_{1}<n_{1}$ be a string and let $m_{2}\leq n$ be an integer. A direct calculation yields the following equations

[TABLE]

Here we set $W_{\emptyset}=0$ and $\mathcal{S}_{0}^{-}:=0$. The proposition follows from Proposition 9.21. Indeed, let us verify the first equality: set $N_{1}=N(L_{1})$, $N_{1}^{\prime}=N(L_{1}^{\prime})$ and $M=N(W_{[m_{1},n_{1}-2]})$. With that convention we obtain that $g_{\Lambda}(N_{1})=[n_{1}-1]+[m_{1}-1]-[n]=g_{\Lambda}(N_{1}^{\prime})$ and $g_{\Lambda}(M)=-[n_{1}-2]+[m_{1}-1]$.

For $M$: the dimension vector of sub-representations are given by $\textbf{h}_{0}=\textbf{0}$ and $\textbf{h}_{i}=[n_{1}-i-1,n_{1}-2]$, where $i\in[1,n_{1}-m_{1}-1]$. From this we get the vectors $C_{Q}\textbf{h}_{0}=\textbf{h}_{0}$ and $C_{Q}\textbf{h}_{i}=[n_{1}-1,n_{1}-2]-[n_{1}-2-i,n_{1}-1-i]$, where $i\in\{1,\ldots,n_{1}-m_{1}-1\}$. Therefore,

[TABLE]

Note that all the dimension vectors of $N_{1}$ are dimension vector of $N_{1}^{\prime}$ and the only vectors that are dimension vectors of $N_{1}^{\prime}$ but not of $N_{1}$ are $\textbf{e}_{j}=[n-j+1,n]$, where $j\in[n-n_{1}+2,n-m_{1}+1]$. Besides, if we consider a as dimension vector of $N_{1}$, then $\chi(\operatorname{Gr}_{\textbf{a}}(N_{1}))=\chi(\operatorname{Gr}_{\textbf{a}}(N_{1}^{\prime}))$ and $\chi(\operatorname{Gr}_{\textbf{e}_{i}}(N_{1}^{\prime}))=1$, for all $i\in\{1,\ldots,n_{1}-m_{1}-1\}$.

In the following sum, a runs over all the dimension vectors of $N_{1}$:

[TABLE]

The first equality is completed. Now we verify the second equality, recall the notation before this proposition. Set $N_{2}=N(W_{[m_{2},n-1]})$ and $N_{2}^{\prime}=N(W_{[m_{2},n]})$. From definitions we get that $g_{\Lambda}(N_{2})=[m_{2}-1]-[n-1]$ and $g_{\Lambda}(N_{2}^{\prime})=[m_{2}-1]$.

For $N_{2}$: the dimension vector are given by $\textbf{f}_{0}=\textbf{0}$ and $\textbf{f}_{j}=[n-j,n-1]$, where $j\in\{1,2,\ldots,n-m_{2}\}$. Therefore $C_{Q}\textbf{f}_{0}=\textbf{f}_{0}$ and $C_{Q}\textbf{f}_{j}=-[n-j-1,n-j]+[n-1]+[n]$. In this case we know that $\chi(\operatorname{Gr}_{\textbf{f}_{j}}(N_{2}))=1$ for all $j\in[0,n-m_{2}]$. Then

[TABLE]

For $N_{2}^{\prime}$: the dimension vector are given by $\textbf{e}_{0}=\textbf{0}$ and $\textbf{e}_{i}=[n-(i-1),n]$, for $i\in\{1,2,\ldots,n-m_{2}+1\}$. From this, we get that $C_{Q}\textbf{e}_{0}=\textbf{e}_{0}$ and $C_{Q}\textbf{e}_{i}=-[n-i,n-i+1]+[n]$ where $i\in\{1,2,\ldots,n-m_{2}+1\}$. Then

[TABLE]

The second equality is completed.

10 The Caldero-Chapoton algebra for an arbitrary triangulation

In this section we will extend the main results of the previous section. Let $\tau$ be any triangulation of $\Sigma_{n}$ and let $\Lambda(\tau)$ be the algebra associated to $\tau$. We can find a triangulation $T$ of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$. By Lemma 8.8 we have that the push-down functor $\pi_{*}:\Lambda(T)\operatorname{-mod}\rightarrow\Lambda(\tau)\operatorname{-mod}$ is a $G$-precovering. Recall that $G=\mathbb{Z}_{3}$ acts on $\widetilde{\Sigma}_{n}$ by an appropriate rotation. In this section we are going to prove that $\pi_{*}$ is a $G$-covering. We are going to use this to characterize the $E_{\Lambda(\tau)}$-rigid representations.

We are following the notation of [31, Section 5]. For $\tau=\{t_{1},t_{2},\ldots,t_{n}\}$ we write, unless we say something else, the triangulation $T$ according to its orbits, namely $T=\{t_{1,1},t_{1,2},t_{1,3},\ldots,t_{n,1},t_{n,2},t_{n,3}\}$. If we denote by $x_{i,j}$ the initial cluster variable associated with the arc $t_{i,j}$, then the initial cluster will be $\textbf{x}_{0}=(x_{1,1},x_{1,2},x_{1,3},\ldots,x_{n,1},x_{n,2},x_{n,3})$. If we associated the variable $z_{i}$ to the arc $t_{i}$, then we obtain a morphism of algebras $\pi:\mathbb{C}[x_{i,j}^{\pm}]\rightarrow\mathbb{C}[z_{i}^{\pm}]$, given by $\pi(x_{i,j})=z_{i}$ for $i=1,\ldots,n$ and $j=1,2,3$. The action of $\mathbb{Z}_{3}$ on $T$ allow us to define the following function

[TABLE]

We hope that the reader is not confused with our unfortunately choice of $\pi$ for different maps. For us the arc $t_{n}\in\tau$ will denote the pending of the triangulation and $t_{n,1},t_{n,2},t_{n,3}$ are the three sides of the triangle invariant under the action of $\mathbb{Z}_{3}$ on $\widetilde{\Sigma}_{n}$. Therefore the matrix $C_{Q(T)}$ has a decomposition in blocks of size $3\times 3$. Recall that by definition $C_{Q(T)}$ is skew-symmetric, moreover $C_{Q(T)}$ is skew-symmetric by blocks and every block is a multiple of the identity of size 3, except for the block $n,n$ that corresponds to the adjacency of the $3$-cycle of $Q(T)$ with vertices $t_{n,1},t_{n,2},t_{n,3}$.

Lemma 10.1**.**

The push down functor $\pi_{*}:\Lambda(T)\operatorname{-mod}\rightarrow\Lambda(\tau)\operatorname{-mod}$ is a Galois $G$-covering.

Proof 10.2**.**

By Lemma 2.9 we only need to prove that $\pi_{*}$ is dense. Well, by Proposition 8.4 and Proposition 8.6 we know that $\Lambda(\tau)$ is a finite-dimensional gentle algebra. From Theorem 5.2 we have that the strings parametrize the indecomposable modules of $\Lambda(\tau)$. Since we work in Krull-Schmidt categories we need to prove that the Galois $G$-covering $\pi:\Lambda(T)\rightarrow\Lambda(\tau)$ induces a surjective function between the set of all string of those algebras and that $\pi_{*}(N(\tilde{W}))\cong N(W)$ where $\tilde{W}$ is a string of $\Lambda(T)$ such that $G\cdot\tilde{W}=W$, recall that $N(W)$ denotes the string module associated to $W$. The last fact follows from definitions.

Suppose the pending arc of $\tau$ is based at $v_{i}$. Assume $W=W_{2}\varepsilon^{k(W)}W_{1}$ is a string for $\Lambda(\tau)$ with $W_{i}$ a string without the letter $\varepsilon$ for $i=1,2$ and $k(W)\in\{-1,0,+1\}$. Recall that $\varepsilon$ is the loop based at the pending arc of $\tau$. It is clear that if $W_{1}$ does not contain the letter $\varepsilon$, then $W_{1}$ can be lifted to a string with letters contained in one of the three fundamental region divided by the dashed blue lines, see Figure 6, say that is contained in the region that contains to $[u_{i},u_{i+n+1}]$. Note that the final letter of $W_{1}$ must be $a_{1}$ or $b_{1}^{-1}$, see Figure 6. Now, if $k(W)=0$, then $W$ itself can be lifted to a word in that region. If $k(W)=1$, we choose $\varepsilon_{3,1}$ and $W_{2}$ can be lifted to a string in the third fundamental region containing $[u_{i},u_{i+2(n+1)}]$. If $k(W)=-1$, we put the letter $\varepsilon_{2,1}$ and it is clear that $W_{2}$ can be lifted to a string of $\Lambda(T)$ with letter of the second fundamental region containing $[u_{i+n+1},u_{i+2(n+1)}]$. Therefore the string $W$ can be lifted to one string $\tilde{W}$ of $\Lambda(T)$. Note that $\tilde{W}$ depends on where we lifted the tail point of $W_{1}$. The proof of the lemma is completed.

Lemma 10.3**.**

The push down functor $\pi_{*}:\Lambda(T)\operatorname{-mod}\rightarrow\Lambda(\tau)\operatorname{-mod}$ induces a Galois $G$-covering $\pi_{*}:\operatorname{decrep}(\Lambda(T))\rightarrow\operatorname{decrep}(\Lambda(\tau))$.

Proof 10.4**.**

Let $R=\mathbb{C}^{Q(T)_{0}}$ be the vertex span of $\Lambda(T)$. We can see that $R_{G}=\mathbb{C}^{Q(\tau)_{0}}$ is the vertex span of $\Lambda(\tau)$. With this notation it is clear that a decorated representation $(M,V)$ is a pair where $M\in\Lambda(T)\operatorname{-mod}$ and $V\in R\operatorname{-mod}$. For $V\in R\operatorname{-mod}$ we can define $\pi_{*}(V)\in R_{G}\operatorname{-mod}$ as in Remark 8.10. We put $\pi_{*}(M,V)=(\pi_{*}(M),\pi_{*}(V))$ and the lemma follows from the fact that $\operatorname{Hom}_{\operatorname{decrep}(\Lambda(T))}((M,V),(N,W))=\operatorname{Hom}_{\Lambda(T)}(M,N)\bigoplus\operatorname{Hom}_{R}(V,W)$.

Now, we can extend Lemma 9.5 to other triangulations. Recall that we denote by $r^{+}(j)$ (resp. $r^{-}(j)$) the arc of $\Sigma_{n}$ that we obtain by rotating $j$ in counterclockwise (resp. clockwise) for an angle of $2\pi/(n+1)$.

Corollary 10.5**.**

Let $\sigma$ be a triangulation of $\Sigma_{n}$ and let $j$ be an arc of $\Sigma_{n}$ not in $\sigma$.

• (a)

Assume $M(j,\sigma)$ is not projective, then $\tau(M(j,\sigma))=M(r^{+}(j),\sigma)$.

• (b)

Assume $M(j,\sigma)$ is not injective, then $\tau^{-}(M(j,\sigma))=M(r^{-}(j),\sigma)$.

Proof 10.6**.**

This is a consequence of the $A_{n}$ case from [10, Theorem 2.13] and Theorem 2.10. Let $\widetilde{j}$ be a lifting of $j$ in $\widetilde{\Sigma}_{n}$ and let $\widetilde{\sigma}$ be the lifting of $\sigma$.

(a). From [10, Theorem 2.13] we see that $\tau(M(\widetilde{j},\widetilde{\sigma}))=M(r^{+}(\tilde{j}),\widetilde{\sigma})$ and by applying $\pi_{*}$, from [4, Theorem 4.7 (1)], we get what we want, $\tau(M(j,\sigma))=M(r^{+}(j),\sigma)$. The proof of (b) is similar.

The reader can compare the next proposition and Proposition [31, Proposition 7.15].

Proposition 10.7**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$ and let $\Lambda(\tau)$ be the algebra associated to $\tau$. Suppose $T$ is the triangulation of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$. If $M$ is an indecomposable representation of $\Lambda\operatorname{-mod}$, then $\pi_{*}(M)$ is $E_{\Lambda(\tau)}$-rigid if and only if $E_{\Lambda(T)}(M,g\cdot M)=0$ for any $g\in G$.

Proof 10.8**.**

The proposition follows from Proposition 3.2 and the following equalities

[TABLE]

Since $\pi_{*}$ is a Galois $G$-covering, Lemma 10.1, the first equality follows from Theorem 2.10, see [4, Theorem 4.7 (1)]. The second line follows from definition of $G$-precovering and the third line is a consequence that $g\cdot-$ is an isomorphism of categories. This conclude the proof.

Lemma 10.9**.**

Let $N$ be an indecomposable representation of $\Lambda(\tau)$. Then $N$ is $E_{\Lambda}$-rigid if and only if $N=M(j,\tau)$ for some arc $j$ of $\Sigma_{n}$.

Proof 10.10**.**

Let $M$ be a representation of $\Lambda(\tau)$ such that $\pi_{*}(M)=N$, recall that $\pi_{*}$ is dense. From [10, Corollary 2.12] we know that there is a bijection between the indecomposable representations of $\Lambda(T)$ and the diagonals of $\widetilde{\Sigma}_{n}$ not in $T$. Then $M=M(\widetilde{j},T)$ for some arc $\widetilde{j}$ of $\widetilde{\Sigma}_{n}$. By Proposition 10.7 we know that $N$ is $E_{\Lambda(T)}$-rigid if and only if $E_{\Lambda(T)}(M,g\cdot M)=0$. We need to analyze $\operatorname{dim}\operatorname{Hom}_{\Lambda(T)}(\tau^{-1}(g\cdot M),M)$. Suppose $\widetilde{j}=[u_{l},u_{l+k}]$ for some $l\in[0,3n+1]$ , then $g\cdot M=M([u_{l-(n+1)},u_{l+k-(n+1)}],T)$ and $\tau^{-1}(g\cdot M)=M([u_{l-n-2},u_{l+k-n-2}],T)$. This means, in particular, that $g\cdot M$ is an arc representation. By [10, Lemma 2.5], the Auslander-Reiten formulas and [10, Remark 2.15] if $E_{\Lambda(T)}(M,g\cdot M)=0$ for any $g\in\mathbb{Z}_{3}$, then we can conclude that $\widetilde{j}$ has to be an admissible arc of $\widetilde{\Sigma}_{n}$, therefore $G\cdot\widetilde{j}=j$ is an arc of $\Sigma_{n}$ and $N=\pi_{*}(M(\widetilde{j},T))=M(j,\tau)$. The proof of the lemma is completed.

Remark 10.11**.**

Let $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ be a Galois $G$-precovering. Then $\mathcal{F}$ is faithful, see [4, Lemma 2.6 (2)].

Lemma 10.12**.**

Let $I_{l}$ be the indecomposable injective at $l\in Q(T)$. Then $\pi_{*}(I_{l})\cong I_{G\cdot l}$, where $I_{G\cdot l}$ is the indecomposable injective at $G\cdot l$.

Proof 10.13**.**

Let $D=\operatorname{Hom}_{\mathbb{C}}(-,\mathbb{C})$ be the standar $\mathbb{C}$-dual functor. Remember that $I_{l}(k)=D\operatorname{Hom}(e_{k},e_{l})$ and $\pi_{*}(I_{l})({G\cdot k})=\bigoplus_{g\in\mathbb{Z}_{3}}{D\operatorname{Hom}(g\cdot e_{k},e_{l})}$. By definition there exist an isomorphism $\pi_{*}^{k,l}:\bigoplus_{g\in\mathbb{Z}_{3}}{\operatorname{Hom}(g\cdot e_{k},e_{l})}\rightarrow\operatorname{Hom}(G\cdot e_{k},G\cdot e_{l})$. In other words, for any $k\in Q(T)_{0}$ we get an isomorphism $\overline{\pi}_{*}^{k,l}:I_{G\cdot l}(G\cdot k)\rightarrow\pi_{*}(I_{l})({G\cdot k})$. Let $\alpha:k_{1}\rightarrow k_{2}\in Q(T)$ be an arrow. It can be shown that $\overline{\pi}_{*}^{k_{2},l}\circ(I_{G\cdot l})_{G\cdot\alpha}=\pi_{*}(I_{l})_{G\cdot\alpha}\circ\overline{\pi}_{*}^{k_{1},l}$.

Lemma 10.14**.**

If $f:M\rightarrow N$ is injective in $\Lambda(T)\operatorname{-mod}$, then $\pi_{*}(f):\pi_{*}(M)\rightarrow\pi_{*}(N)$ is injective in $\Lambda(\tau)\operatorname{-mod}$.

Proof 10.15**.**

Remember that $f=(f_{i})_{i\in Q(T)_{0}}$ is a $3n$-tuple of linear transformations and by hypothesis we have that $\operatorname{dim}\operatorname{rank}f_{i}=\operatorname{dim}M_{i}$. The lemma follows from Remark 8.10 and that $\operatorname{dim}\operatorname{rank}\pi_{*}(f)_{G\cdot i}=\sum_{g\in\mathbb{Z}_{3}}{\operatorname{dim}\operatorname{rank}f_{g\cdot i}}$.

Lemma 10.16**.**

Let $\widetilde{j}$ be an arc of $\widetilde{\Sigma}_{n}$. For $M:=M(\widetilde{j})\in\Lambda(T)\operatorname{-mod}$, let

[TABLE]

be a minimal injective presentation of $M$. Then

[TABLE]

is a minimal injective presentation of $\pi_{*}(M(\alpha))$.

Proof 10.17**.**

The lemma follows from the fact that $\pi$ is dense, Lemma 10.14 and Lemma 10.12.

We shall discuss about the Caldero-Chapoton algebra associated to different triangulation. Let $T_{1}$ and $T_{2}$ be triangulations of $\widetilde{\Sigma}_{n}$. Denote by $\mathcal{A}_{i}:=\mathcal{A}_{\Lambda(T_{i})}$ the Caldero-Chapoton algebra corresponding for $i=1,2$. Let $\mathcal{D}_{i}$ the $\mathbb{C}$-subalgebra of $\mathcal{A}_{i}$ generated by $\mathcal{C}_{\Lambda(T_{i})}(M(\widetilde{j},T_{i}))$ for any admissible arc $\widetilde{j}$ of $\widetilde{\Sigma}_{n}$ for $i=1,2$.

Lemma 10.18**.**

With the above notation $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are isomorphic as $\mathbb{C}$-algebras.

Proof 10.19**.**

Let $\varphi:\mathcal{A}_{1}\rightarrow\mathcal{A}_{2}$ be the corresponding isomorphism of cluster algebras. This isomorphism sends a cluster variable to the corresponding Laurent Polynomial in the initial seed associated to $T_{2}$, i.e $x_{\widetilde{i}}\mapsto\mathcal{C}_{\Lambda(T_{2})}(M(\tilde{i},T_{2}))$ for an arc $\widetilde{i}$ in $T_{1}$. Suppose that we can get $T_{2}$ from $T_{1}$ by the flip sequence $(s_{l},s_{l-1},\ldots,s_{1})$. By [18] we conclude that $\mathcal{C}_{\Lambda(T_{1})}(M(\widetilde{j},T_{1}))=\mathcal{C}_{\Lambda(T_{2})}(\mu_{s_{1}}\mu_{s_{2}}\cdots\mu_{s_{l}}(M(\widetilde{j},T_{2})))$, then

[TABLE]

That means that $\varphi$ can be restricted to $\mathcal{D}_{1}$ and we obtain the isomorphism desired. The lemma is completed.

Remark 10.20**.**

Let $W$ be a string of $\Sigma_{n}$. Consider a lifting $\widetilde{W}$ of $W$ on $\widetilde{\Sigma}_{n}$. If f is a dimension vector of some sub-representation of $N(\widetilde{W})$, then $\pi(\textbf{y}^{C_{Q(T)}\cdot\textbf{f}})=\textbf{z}^{C_{Q(\tau)}\cdot\pi(\textbf{f})}$. Indeed, we need to prove that

[TABLE]

for any $i\in[1,n]$, here $C_{Q(T)_{i,j}}$ denote the $(i,j)$-th row of the matrix $C_{Q(T)}$. For $i<n$ the calculation is straightforward and we will concentrate in the case when $i=n$. Assume $i=n$, we orient the arcs $t_{n,1},t_{n,2},t_{n,3}$ in counter clockwise on $\widetilde{\Sigma}_{n}$. This orientation determines the $(n,n)$ block of $C_{Q(T)}$. In order to obtain (10.1) we need that

[TABLE]

The observation is that (10.2) is true in case f is the dimension vector of an indecomposable representation of $\Lambda(T)$.

Lemma 10.21**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$ and let $\Lambda(\tau)$ be the algebra associated to $\tau$. Let $W$ be a string on $Q(\tau)$ and let $\widetilde{W}$ be a lifting of $W$ in $Q(T)$, where $T$ is the triangulation of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$. Then for any dimension vector e of $N(W)$ we get

[TABLE]

Proof 10.22**.**

First, we are going to introduce some notation. We write $j:=j(W)$ for the arc determined by $W$, note that this arc can have self intersections. We will denote $M(j):=N(W)$. Suppose $j$ connects $v_{k}$ and $v_{l}$ with $k\leq l$. So, we orient $j$ from $v_{k}$ to $v_{l}$. Let $x_{p_{1}}$ be the first intersection point between $j$ and the pending arc $p(\tau)$ of $\tau$. Let $x_{p_{2}}$ be the second intersection point between $j$ and $p(\tau)$. We divide the arc $j$ in three parts;

• •

The top part $j_{1,0}=[v_{k},x_{p_{1}}]$.

• •

The center part $j_{1,1}=[x_{p_{1}},x_{p_{2}}]$.

• •

The buttom part $j_{0,1}=[x_{p_{2}},v_{l}]$.

Let $\operatorname{up}_{j}=\{x_{i}\colon x_{i}=j_{1,0}\cap i\mbox{ with }i\in\tau\}$ be the upper points of $j$. Let $\operatorname{bp}_{j}=\{y_{i}\colon y_{i}=j_{0,1}\cap i\mbox{ with }i\in\tau\}$ be the below points of $j$. For convention if $j$ does not cross $p(\tau)$, then $x_{p_{1}}=v_{l}$, $x_{p_{2}}=v_{k}$ and $\operatorname{bp}_{j}=\operatorname{up}_{j}$, see Figure 7.

Let $L\in\operatorname{Gr}_{\textbf{e}}(M(j))$ be a sub-representation of $M(j)$ with dimension vector e. We are going to define an action of $\mathbb{C}^{*}$ on $\operatorname{Gr}_{\textbf{e}}(M(j))$. For $t\in\mathbb{C}^{*}$ we define $t\cdot L$ as follows:

[TABLE]

Indeed, this define an action of $\mathbb{C}^{*}$ on $\operatorname{Gr}_{\textbf{e}}(M(j))$. By Lemma 2.2 we know that $\chi(\operatorname{Gr}_{\textbf{e}}(M(j))^{\mathbb{C}^{*}})=\chi(\operatorname{Gr}_{\textbf{e}}(M(j)))$. In this case $\operatorname{Gr}_{\textbf{e}}(M(j))^{\mathbb{C}^{*}}$ is a finite set, then the Euler characteristic is its cardinality. Denote by $Q(j)$ the full sub-quiver of $Q(\tau)$ defined by $j$. We consider the lifting $\widetilde{j}$ of $j$ on $\widetilde{\Sigma}_{n}$. On $\widetilde{\Sigma}_{n}$ we can also define the corresponding top, center and bottom part of $\widetilde{j}$.

Note that if the arc $j$ does not cross $p(\tau)$, then $\widetilde{j}$ is completely contained in one fundamental region of the action and $\pi$ acts as a bijection between dimension vectors of sub-representations of $M(\widetilde{j})$ and dimension vector of sub-representations of $M(j)$. In other words, there exist an unique f such that $\pi(\textbf{f})=\textbf{e}$. Therefore $\chi(\operatorname{Gr}_{\textbf{f}}(N(\widetilde{W})))=\chi(\operatorname{Gr}_{\textbf{e}}(N(W)))$.

Let $L\in\operatorname{Gr}_{\textbf{e}}(M(j))^{\mathbb{C}^{*}}$ be a sub-representation of $M(j)$. It is clear that $L$ define a walk in $Q(j)$ and also an unique subset $F(L)$ of $\operatorname{bp}_{j}\cup\operatorname{up}_{j}$. Indeed, the action we have defined allows to identify every subspace $L_{i}$ with points of $F(L)\subset\operatorname{bp}_{j}\cup\operatorname{up}_{j}$ in the following way. If $L_{i}$ is generated by $(1,0)^{t}$, then we take the corresponding upper point of $j$. If $L_{i}$ is generated by $(0,1)^{t}$, then we take the corresponding below point of $j$. In case $L_{i}$ is 2 dimensional, then we take both, the upper and below point of $j$. It is clear that $F(L)$ determines an unique vector $\textbf{f}_{L}$ of $M(\widetilde{j})$ such that $\pi(\textbf{f}_{L})=\textbf{e}$. This implies that

[TABLE]

For any vector f of some sub-representation of $N(\widetilde{W})$ with $\pi(\textbf{f})=\textbf{e}$ we can find a subset $D_{\textbf{f}}$ of $\operatorname{bp}_{j}\cup\operatorname{up}_{j}$ corresponding to a sub-representation of $N(W)$, namely we obtain the image under $\pi_{*}$ of the representation given by f. We make this by cuting the arc $\widetilde{j}$ on $\widetilde{\Sigma}_{n}$ along the boundary of the fundamental regions that it crosses and gluing that parts on $\Sigma_{n}$ according to the orientation we fixed on $\widetilde{j}$. This subset corresponds to a sub-representation $L_{\textbf{f}}$ of $N(W)$. By the definition of the action we can conclude that $L_{\textbf{f}}\in\operatorname{Gr}_{\textbf{e}}(M(j))^{\mathbb{C}^{*}}$. This shows the another inequality. Hence the lemma is completed.

The next proposition follows from Lemma 10.16, Remark 10.20 and Lemma 10.21. The reader can compare this result with the discussion of [31, Remark 7.9].

Proposition 10.23**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$ and let $\Lambda(\tau)$ be the algebra associated to $\tau$. Assume $T$ is the triangulation of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$. Then for any string $W$ of $\Lambda(\tau)$ the following equation is true

[TABLE]

Proof 10.24**.**

By expanding $\mathcal{C}_{\Lambda(T)}(N(\widetilde{W}))$ and rearranging its monomials as in Lemma 10.21 we are able to apply Lemma 10.16 and Remark 10.20 to any monomial. Note that from Lemma 10.16 we have that $\pi(g_{\Lambda(T)}(N(\widetilde{W})))=g_{\Lambda(\tau)}(N(W))$. The proposition is completed.

11 The Caldero-Chapoton algebra is a generalized cluster algebra

In this section we will state and prove our main result. Before that, we need some previous propositions.

Proposition 11.1**.**

Let $\tau$ be a triangulation of $\Sigma_{n}$ and let $\Lambda(\tau)$ be the algebra associated to $\tau$. Assume $T$ is the triangulation of $\widetilde{\Sigma}_{n}$ such that $G\cdot T=\tau$ and $j\notin\tau$. Then the $G_{\textbf{d}}$-orbit $\mathcal{O}(M(j))$ is open in $\operatorname{rep}_{\textbf{d}}(\Lambda)$.

Proof 11.2**.**

By [16, 1.7 Corollary 3] we need to prove that for any arc $j$ of $\Sigma_{n}$ we have that $\operatorname{Ext}_{\Lambda(\tau)}(M(j),M(j))=0$. This is clear by the Auslander-Reiten formulas since $E(M(j))=0=\operatorname{dim}\operatorname{Hom}(\tau^{-}(M(j)),M(j))$.

If we denote by $Z(j)$ the irreducible component containing $M(j)$, then $\mathcal{O}(M(j))$ is dense in $Z(j)$. Therefore $M(j)$ is generic and all its homological data is generic in $Z(j)$. We can take generic versions of the results of the above section as in Section 9.

Repeating the arguments of Proposition 10.7 and applying what we know for the $A_{n}$ case, for instance see [10, Remark 2.15], we have

Proposition 11.3**.**

Given a triangulation $\sigma$ of $\Sigma_{n}$ and two arcs $j_{1},j_{2}\in\sigma$ we have $E_{\Lambda(\tau)}(Z_{j_{1}},Z_{j_{2}})=0$.

The next proposition shows that the $E$-rigid representations generate the corresponding Caldero-Chapoton algebra.

Proposition 11.4**.**

The set

[TABLE]

generates the Caldero-Chapoton algebra $\mathcal{A}_{\Lambda(\tau)}$ as $\mathbb{C}$-algebra.

Proof 11.5**.**

As in Proposition 9.32 we are going to prove that the Caldero-Chapoton functions of $E$-rigid representations generate the remaining Caldero-Chapoton functions. Let $\widetilde{j}$ be an arc of $\widetilde{\Sigma}_{n}$ such that it does not belong to any triangulation invariant under the action of $\mathbb{Z}_{3}$. By Proposition 10.7 from these arcs come all the non-$E$-rigid representations of $\Lambda(\tau)$, so what we need to do is to prove the result in this case. In other words, we are going to prove that the Caldero-Chapoton function of $\pi_{*}(M(\widetilde{j}))$ can be expressed in terms of the Caldero-Chapoton functions of $E$-rigid representations. For $\widetilde{j}$ we construct a quadrilateral in the following way; first, we choose an ending point of $\widetilde{j}$, say $u_{i}$. Then we draw the triangle invariant under the $\mathbb{Z}_{3}$-action incident to $u_{i}$ with sides given by $\widetilde{j}_{1}$, $\widetilde{j}^{\prime}$ and $\widetilde{j}_{4}$. Finally, we complete the quadrilateral with the other ending point of $\widetilde{j}$ such that $\widetilde{j}$ and $\widetilde{j}^{\prime}$ are the respective diagonals. We label the remaining sides with $\widetilde{j}_{2}$ and $\widetilde{j}_{3}$. This construction is depicted in Figure 8. From [9, 10] we have that

[TABLE]

Note that $\widetilde{j}_{1}$, $\widetilde{j}^{\prime}$ and $\widetilde{j}_{4}$ are in the same orbit. By applying the algebras homomorphism $\pi$ to the above equation, from Proposition 10.23, we obtain

[TABLE]

Since we are in an integral domain, we have the desired relation

[TABLE]

The proposition is completed.

From the above proposition we obtain our main result.

Theorem 11.6**.**

For any triangulation $\tau$ of $\Sigma_{n}$ we have that the Caldero-Chapoton algebra $\mathcal{A}_{\Lambda(\tau)}$ is isomorphic to the generalized cluster algebra $\mathcal{A}(B(\tau_{0}))$.

Proof 11.7**.**

Let $T_{1}$ and $T_{2}$ be triangulations of $\widetilde{\Sigma}_{n}$. Denote by $\mathcal{A}_{i}:=\mathcal{A}_{\Lambda(T_{i})}$ the Caldero-Chapoton algebra corresponding for $i=1,2$. Let $\mathcal{D}_{i}$ the $\mathbb{C}$-subalgebra of $\mathcal{A}_{i}$ generated by $\mathcal{C}_{\Lambda(T_{i})}(M(\widetilde{j},T_{i}))$ for any admissible arc $\widetilde{j}$ of $\widetilde{\Sigma}_{n}$ for $i=1,2$. By Lemma 10.18 we have that $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are isomorphic. The previous Proposition and Proposition 10.23 show that the Caldero-Chapoton algebra associated to $\tau_{i}=G\cdot T_{i}$ is $\pi(\mathcal{D}_{i})$ for $i=1,2$. We conclude that the Caldero-Chapoton algebras $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are isomorphic. Now, from [31] we know that the image of $\Lambda(\tau_{0})$ under $\pi$ is a Chekhov-Shapiro generalized cluster algebra with initial seed $(B(\tau_{0}),\textbf{d}_{\tau_{0}})$. Indeed, from [31, Lemma 5.6] and [31, Lemma 5.7] we know that the exchange polynomial are those of Chekhov-Shapiro. The theorem is completed.

12 Example

The example in this section is meant to illustrate our main result. It can also be considered a complement to [11, Example 9.4.2]. Let $\tau_{0}$ be the special triangulation of $\Sigma_{3}$ and let $\tau$ be the triangulation of Example 7.4, see Figure 9.

It is clear that $\tau$ and $\tau_{0}$ are related by a flip at one arc. To ease the notation we set $\Lambda=\Lambda(\tau)$, see Example 7.4. From Theorem 5.2 we know that the indecomposable $\Lambda$-modules are parametrized by the strings of $\Lambda$. We say that a string $W$ is $E$-rigid if its string module $N(W)$ is $E$-rigid. There are 12 indecomposable $E$-rigid decorated representations of $\Lambda$ of which 9 are given by the $E$-rigid strings $1_{1}$, $1_{2}$, $\varepsilon$, $a$, $\varepsilon b$, $c\varepsilon$, $c\epsilon b$, $b^{-1}\varepsilon b$ and $c\varepsilon c^{-1}$; and the remaining three are the negative simple representations of $\Lambda$. The non-$E$-rigid strings are $1_{3}$, $b$, $c$, $b^{-1}\varepsilon$, $\varepsilon c^{-1}$ and $b^{-1}\varepsilon c^{-1}.$

By definition $\mathcal{C}_{\Lambda}(\mathcal{S}_{i}^{-})=y_{i}$ for $i=1,2,3$. In Figure 10 we write the string module corresponding to every arc of $\Sigma_{3}$. The Caldero-Chapoton functions associated to the 9 $E$-rigid strings of $\Lambda$ are

[TABLE]

[TABLE]

The Caldero-Chapoton functions associated to the non-$E$-rigid strings of $\Lambda$ are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Remark 12.1**.**

According to Proposition 11.4 we have that the Caldero-Chapoton functions of indecomposable $E$-rigid representations generate the remaining Caldero-Chapoton functions. In this case we get the following relations

[TABLE]

These relations correspond to the procedure of the proof of Proposition 11.4. With the notation of [11, Example 9.4.2] we can define the following isomorphism of the corresponding Caldero-Chapoton algebras $\varphi:\mathcal{A}_{\Lambda(\tau)}\rightarrow\mathcal{A}_{\Lambda(\tau_{0})}$. We set $y_{1}\mapsto x_{1}$, $y_{2}\mapsto C_{\Lambda(\tau_{0})}(2)=\frac{x_{1}+x_{3}}{x_{2}}$ and $y_{3}\mapsto x_{3}$. This morphism sends the Caldero-Chapoton function of the arc representation associated to one arc of $\tau$ to the Caldero-Chapoton function with respect to $\Lambda(\tau_{0})$ of the same arc.

Acknowledgements.

We owe thanks to Andrew T. Carroll, Christof Geiß, Jan Geuenich, Pierre-Guy Plamondon and Jan Schröer for helpful discussions. The first author was supported by grants CONACyT-238754 and PAPIIT-IA102215. The second author was supported by CONACyT-269197 scholarship.