TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0

QUIVERS WITH RELATIONS ARISING FROM CLUSTERS (A_{n} CASE)

P. CALDERO, F. CHAPOTON, R. SCHIFFLER

Abstract. Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_{n}. We associate to each cluster C of U an abelian category C_{C} such that the indecomposable objects of C_{C} are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of C_{C}. Then, we generalize the “denominator Theorem” of Fomin and Zelevinsky to any cluster.

0. Introduction

Cluster algebras were introduced in the work of A. Berenstein, S. Fomin, and A. Zelevinsky, [2, 5, 6, 7]. This theory appeared in the context of dual canonical basis and more particularly in the study of the Berenstein-Zelevinsky conjecture. Cluster algebras are now connected with many topics: double Bruhat cells, Poisson varieties, total positivity, Teichmüller spaces. The main results on cluster algebras are on the one hand the classification of finite cluster algebras by root systems and on the other hand the realization of algebras of regular functions on double Bruhat cells in terms of cluster algebras.

Recall some facts about cluster algebras. Cluster algebras U of rank n form a class of algebras defined axiomatically in terms of a distinguished set of generators {u_{1}, …, u_{n}}. A cluster is a set of “cluster variables” {w_{1}, …, w_{n}} obtained combinatorially from {u_{1}, …, u_{n}}. The so-called Laurent phenomenon asserts that each cluster variable is a Laurent polynomial in the set of variables given by a cluster. For each cluster C, one can define combinatorially an oriented quiver Q_{C}.

Suppose from now on that U is a finite cluster algebra, i.e. there only exists a finite number of cluster variables. It is known that U can be described by the data of a root system X_{n}. Moreover, there exists a cluster Σ such that Q_{Σ} is the alternating quiver on X_{n}. S. Fomin and A. Zelevinsky give in [6] a more precise description of the Laurent phenomenon: there exists a one-to-one correspondence α ↦ w_{α} between the set of almost positive roots of X_{n}, i.e. positive roots and simple negative roots, and the set of cluster variables, such that the denominator of w_{α} as a Laurent polynomial in Σ is given by the decomposition of α in the basis of simple roots. Via the Gabriel’s Theorem, this property suggests a link between cluster algebras and representation theory of artinian rings. This relation was already investigated in [10] and is the core subject of the recent papers [3, 4].

In this paper, we give conjectural relations R_{C} on the quiver Q_{C}, such that for any cluster C, the denominators of the cluster variables as Laurent polynomial in C are described by indecomposables of the category C_{C} of representations of Q_{C} with relations R_{C}.

The main result of this article is the proof of this conjecture in the A_{n} case.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors 25 February 2004, last revision 10 May 2004.

2000 Mathematics Subject Classification. 16G20, 16G70, 05E15, 20F55.

Another important result of this paper is a geometric realization of the category C_{C} in the A_{n} case. Recall that the algebra of regular functions on the 2-grassmannian of ℂ^{n+3} is a finite cluster algebra U of type A_{n}. Via this realization, the cluster variables are in natural bijection with the diagonals of a regular (n + 3) polygon. Moreover, a result of Fomin and Zelevinsky asserts that this bijection gives a one-to-one correspondence between the set of clusters of U and the set of diagonal triangulations of the polygon. Theorem 4.4 gives a simple realization of the category C_{C} in terms of the diagonals of the (n + 3) polygon. There also exists a more canonical category associated to a finite cluster algebra and also studied in [3]. We give in the A_{n} case a geometric realization of this category, see Theorem 5.2.

Let us mention that so-called cluster-tilted rings, introduced in [3, 4] as endomorphism rings of tilting objects in this canonical category, are conjectured there to be isomorphic to the path algebras of our quivers with relations. The view-point of these articles should provide a well-suited categorical background to generalize our constructions.

The authors would like to thank Robert Marsh and Bernhard Keller for useful conversations. Our thanks also go to the LACIM at UQAM in Montreal, where this work was started, for its great hospitality.

1. Quivers of cluster type

Let X_{n} be a simply-laced Dynkin diagram of rank n and finite type. First we need to recall some material on clusters.

Each cluster C of a cluster algebra of rank n is associated with a sign-skew-symmetric square matrix B_{C} whose rows and columns are indexed by the cluster variables of the cluster C. In the simply-laced case, coefficients of the matrices B_{C} belong to the set {−1, 0, 1}. Hence it is easy and convenient to depict these matrices using oriented graphs (once a convention is chosen for the orientation). This oriented graph is called the quiver associated to the cluster C and is denoted by Q_{C}. It is known that all triangles (and more generally cycles) in these quivers are oriented in a cyclic way [6, Proposition 9.7].

The mutation procedure of clusters contains in particular a mutation rule for the associated matrices, which can be translated as a mutation rule for the associated quivers. In the simply-laced case, the mutation rule can be further simplified. The result is as follows.

Let C be a cluster in a cluster algebra of simply-laced type. The mutation μ_{i}(Q_{C}) of the quiver Q_{C} at a vertex i is described as follows. First, all arrows incident to i in Q_{C} are reversed in the mutated quiver. Then, for each pair of one incoming arrow j → i and one outgoing arrow i → k in Q_{C}, the arrow j → k is in the mutated quiver if and only if the arrow k → j is not in Q_{C}. The other arrows of Q_{C} are kept unchanged in the mutated quiver.

By definition, a shortest path in the quiver Q_{C} is an oriented path (with no repeated arrow) contained in an induced subgraph of Q_{C} which is a cycle.

Definition 1. Let Q_{C} be the quiver associated to the cluster C. For each arrow i → j in Q_{C}, a relation Rel_{i,j} is defined as follows. Consider the set of shortest paths from j to i:

• If there are exactly two distinct paths c and c′ then c = c′.

• If there exists only one path c then c = 0.

• If there is no such path, there is no relation.

QUIVERS WITH RELATIONS ARISING FROM CLUSTERS  (A_{n} CASE)  3

To each cluster C, one defines the (abelian) category Mod Q_{C} as the category of modules over the algebra A_{C} which is the quotient of the path algebra of the quiver Q_{C} by the ideal generated by the relations Rel_{i,j} for all arrows i → j of Q_{C}.

Remark: it is clear in type A that there is at most one shortest path for each arrow. One could show using the geometric model of type D cluster algebras that there are at most two such paths in this case. In order for the conjecture to make sense also in the exceptional cases E, it remains to prove that this is also true in these cases. We will not consider this question here.

Obvious remark: there is a natural one-to-one correspondence between the vertices of the quiver Q_{C} and the simple modules in Mod Q_{C}. Therefore, the isomorphism class of the simple module associated to the vertex i will be denoted α_{i}.

Let Ind(Q_{C}) be the set of isomorphism classes of indecomposables of Mod Q_{C}.

Conjecture 1.1. Let C = {u_{1}, …, u_{n}} be a cluster in a cluster algebra of simplylaced type and rank n. Let V be the set of all cluster variables for this cluster algebra. There exists a bijection b : Ind(Q_{C}) → V \C, α ↦ w_{α} such that w_{α} = \frac{P (u_{1},...,u_{n})}{∏_{i} u_{i}^{n_{i}}}, where P is a polynomial prime to u_{i} for all i and where n_{i} = n_{i}(α) is the multiplicity of the simple module α_{i} in the module α.

Remark 1.2. Through Gabriel’s celebrated Theorem relating indecomposables and positive roots, this conjecture generalizes the Theorem of Fomin and Zelevinsky ([6, Theorem 1.9]) which corresponds to the case of the alternating quiver.

The main aim of the present article is to prove Conjecture 1.1 in the case of cluster algebras of type A_{n}. This will be done using the geometric realization in terms of triangulations given by Fomin and Zelevinsky [7, §3.5].

2. Categories of diagonals

In this section, we introduce some terminology on triangulations and define a category C_{T} for each triangulation T in a geometric way.

2.1. Triangulations and diagonals. Let us fix a nonnegative integer n and a triangulation T of a regular polygon with n + 3 vertices. The diagonals of this polygon will be called roots and designed by Greek letters. Let us call negative the roots belonging to T and positive the other roots. Let Φ_{+} be the set of positive roots with respect to T. Let I be the set of negative roots. By convention, the negative root corresponding to i ∈ I will be called −α_{i}. The support Supp α ⊆ I of a positive root α is the set of negative roots which cross α. Note that a positive root α is determined by its support. Indeed it is possible to recover the vertices of a positive diagonal from the sequence of crossed negative diagonals. A positive root α is related to a positive root α′ by a pivoting elementary move if the associated diagonals share a vertex on the border (the pivot), the other vertices of α and α′ are the vertices of a border edge of the polygon and the rotation around the pivot is positive (for the trigonometric direction) from α to α′. Let P_{v} denote the pivoting elementary move with pivot v. A pivoting path from a positive root α to a positive root α′ is a sequence of pivoting elementary moves starting at α and ending at α′.

2.2. Categories of diagonals. One can define a combinatorial ℂ-linear additive category C_{T} as follows. The objects are positive integral linear combinations of positive roots. By additivity, it is enough to define morphisms between positive roots. The space of morphisms from a positive root α to a positive root α′ is a quotient of the vector space over ℂ spanned by pivoting paths from α to α′.

Figure 1. mesh relation P_{v′_{2}}P_{v_{1}} = P_{v′_{1}}P_{v_{2}}

The subspace which defines the quotient is spanned by the so-called mesh relations (see Figure 1). For any couple α, α′ of positive roots such that α is related to α′ by two consecutive pivoting elementary moves with distinct pivots, we define the mesh relation P_{v′_{2}}P_{v_{1}} = P_{v′_{1}}P_{v_{2}}, where v_{1}, v_{2} (respectively v′_{1}, v′_{2}) are the vertices of α (respectively α′) such that P_{v′_{1}}P_{v_{2}}(α) = α′. That is, any two consecutive pivoting elementary moves using different pivots can in some sense be “exchanged”.

In these relations, negative roots or border edges are allowed, with the following conventions.

(i) If one of the intermediate edges is a border edge, the corresponding term in the mesh relation is replaced by zero.

(ii) If one of the intermediate edges is a negative root, the corresponding term

in the mesh relation is replaced by zero. More generally, a mesh relation is an equality between two pivoting paths which

differ only in two consecutive pivoting elementary moves by such a change.

We can now define the set of morphisms from a positive root α to a positive root α′ to be the quotient of the vector space over ℂ spanned by pivoting paths from α to α′ by the subspace generated by mesh relations.

Therefore, the image of a pivoting path in the space of morphisms is either the zero morphism or only depends on the class of the pivoting path modulo the equivalence relation on the set of pivoting paths generated by the mesh relations with no vanishing terms.

The following Lemma will be useful later.

Lemma 2.1. The vector space Hom_{C_{T}} (α, α′) is not zero if and only if there exists i ∈ Supp α ∩ Supp α′ such that the relative positions of α, α′ and −α_{i} are as in Figure 2. That is, let v_{1}, v_{2} be the endpoints of −α_{i} and u_{1}, u_{2} (respectively u′_{1}, u′_{2}) be the endpoints of α (respectively α′). Then ordering the vertices of the polygon in the positive trigonometric direction starting at v_{1}, we have v_{1} < u_{1} ≤ u′_{1} < v_{2} < u_{2} ≤ u′_{2}. In this case, Hom_{C_{T}} (α, α′) is of dimension one.

Figure 2. Relative position

Figure 3. Tree t_{T} and Graph Q_{T} defined by a Triangulation T

Proof. Suppose that Hom_{C_{T}} (α, α′) is not zero. Let P ≠ 0 be a sequence α = α^{0} ⟶^{P_{1}} α^{1} ⟶^{P_{2}} … ⟶^{P_{m}} α^{m} = α′ where the α^{i} are positive roots and the P_{i} are pivoting elementary moves. Because of the mesh relations, we may suppose that the first k moves P_{1}, …, P_{k} have as pivot one of the vertices of α and the last (m − k) moves P_{k+1}, …, P_{m} have as pivot the vertex which is the intersection of α′ and α^{k}. Denote by V_{1} (respectively V_{2}) the set of vertices of α^{1}, …, α^{k} (respectively α^{k+1}, …, α^{m}) other than the pivot of P_{1} (respectively P_{k+1}). Since P ≠ 0 this implies that α and α′ intersect and that all diagonals with one vertex in V_{1} and the other in V_{2} are positive. Thus the vertices of α and α′ form a quadrilateral in the polygon without any diagonals of the triangulation T crossing it from V_{1} to V_{2}. Because T is a triangulation we must have a −α_{i} ∈ T crossing this quadrilateral in the other direction and we get the situation in the diagram. On the other hand, in the situation of the diagram it is clear that there is a non-zero morphism from α to α′. Finally, the dimension of Hom_{C_{T}} (α, α′) is at most one, since any two non-zero pivoting paths from α to α′ are in the same class. □

3. Quivers and triangulations

In this section, we define a quiver directly from a triangulation and introduce the category of modules on this quiver with some relations.

3.1. Graphs and trees. Let T be a triangulation. Then one can define a planar tree t_{T} as follows. Its vertices are the triangles of T and its edges are between adjacent triangles (see left part of Figure 3). Vertices of t_{T} have valency 1, 2 or 3. It is clear that there is always at least one vertex of valency 1.

From T, one can also define a graph Q_{T} as follows. The vertices of Q_{T} are the inner edges of T and are related by an edge if they bound part of the same triangle (see right part of Figure 3).

In fact, it is possible to define the graph Q_{T} starting from the planar tree t_{T}. Vertices of Q_{T} are the edges of t_{T}. Two vertices of Q_{T} are related by an edge in Q_{T} if the corresponding edges of t_{T} share a vertex in t_{T}. The equivalence with the previous definition is obvious.

A leaf is an edge e of t_{T} such that at least one of its vertices has valency 1. As there is always a vertex of valency 1 in t_{T}, there always exists a leaf.

Recall that the mutation of a triangulation at one of its diagonal is the unique triangulation which can be obtained by replacing this diagonal with another one.

3.2. Quivers and triangulations. Let T be a triangulation. Let us define a quiver Q_{T} with underlying graph the graph Q_{T} defined in the previous section. Recall that its vertices are in bijection with I. Put a point at the middle of each negative root in T and draw an edge between points in two negative roots −α_{i}, −α_{j} which bound part of the same triangle. The orientation of the edge is defined as follows: Denote by x the common vertex of −α_{i}, −α_{j}, then −α_{i} → −α_{j} if the

Figure 4. Definition of the arrows of Q_{T}

rotation with minimal angle around x that sends the line through −α_{i} to the line through −α_{j} is in positive trigonometric direction (see Figure 4). From this description, it follows that all triangles in Q_{T} are oriented.

Lemma 3.1. The mutation of quivers, as defined in §1, corresponds to the mutation of triangulations described above.

Proof. Left to the reader. □

One can define a ℂ-linear abelian category Mod Q_{T} as follows. This is the category of modules over the quiver Q_{T} with the following relations, called triangle relations.

(1)  In any triangle, the composition of two successive maps is zero.

These relations are exactly the relations prescribed by Definition 1.

The next two Lemmas are steps for the proof of Lemma 3.4 which will be used later.

Lemma 3.2. The support Supp α of a positive root α is connected as a subset of the quiver Q_{T}.

Proof. Let −α_{i}, −α_{j} be two distinct diagonals in Supp α. We will show that there is an unoriented path from i to j in Q_{T} ∩ Supp α. The diagonals −α_{i} and −α_{j} cut the polygon into three parts. Denote by R_{ij} the part that contains both −α_{i} and −α_{j}. We proceed by induction on m the number of negative roots in R_{ij}. If m = 1 then −α_{i} = −α_{j} and there is nothing to prove. Let us assume that m > 1. Let Δ be the unique triangle in T that contains −α_{i} and lies in R_{ij}. Since α crosses both −α_{i} and −α_{j}, it has to cross exactly one of the two sides different from −α_{i} in Δ. This side cannot be a border edge of the polygon, hence it has to be a negative root, call it −α_{k}. Thus there is an edge between −α_{i} and −α_{k} in Q_{T} and i, k ∈ Supp α. We may suppose by induction that there is an unoriented path in Supp α from k to j and we are done. □

Lemma 3.3. Let α, α′ be positive roots, then Supp α ∩ Supp α′ is connected.

Proof. Suppose the contrary. Write S = Supp α and S′ = Supp α′ for short. Let i, k ∈ S ∩ S′ be two vertices that belong to different connected components of S ∩ S′. Since S and S′ are connected (Lemma 3.2) we may choose two minimal paths p : i = i_{1}, i_{2}, …, i_{p} = k in S and p′ : i = j_{1}, j_{2}, …, j_{q} = k in S′. Let m be the smallest integer such that i_{m+1} ≠ j_{m+1}. In the triangulation T, each of the diagonals −α_{i_{m−1}}, −α_{i_{m+1}}, −α_{j_{m+1}} has a vertex in common with −α_{i_{m}}. Since a positive root can only cross two sides of a triangle, we get that −α_{i_{m}}, −α_{i_{m+1}} and −α_{j_{m+1}} form a triangle Δ in T. Moreover i_{m+1} ∈ S \ S′ and j_{m+1} ∈ S′ \ S. Now cutting out the triangle Δ divides the polygon into three parts: R_{i_{m}}, R_{i_{m+1}} and R_{j_{m+1}} such that R_{l} contains −α_{l}, l = i_{m}, i_{m+1}, j_{m+1}. Clearly all −α_{i_{l}}, l ≥ m + 1 lie in R_{i_{m+1}} and all −α_{j_{l}}, l ≥ m + 1 lie in R_{j_{m+1}}. But this contradicts the fact −α_{j_{q}} = −α_{k} = −α_{i_{p}} and we have shown that S ∩ S′ is connected. □

Lemma 3.4. The vector space Hom_{Mod Q_{T}} ((M^{α}, f^{α}), (M^{α′}, f^{α′})) is not zero if and only if the following conditions hold. Let S = Supp α and S′ = Supp α′ for short.

(i) S ∩ S′ is not empty, (ii) There is no arrow from S \ S′ to S ∩ S′ in Q_{T}, (iii) There is no arrow from S ∩ S′ to S′ \ S in Q_{T}.

In this case, Hom_{Mod Q_{T}} ((M^{α}, f^{α}), (M^{α′}, f^{α′})) is of dimension one.

Proof. Let P be a non-zero element of Hom_{Mod Q_{T}} ((M^{α}, f^{α}), (M^{α′}, f^{α′})). Then condition (i) is clearly true. Let us show conditions (ii) and (iii). Suppose that condition (ii) is not true, thus there is an arrow i → j in Q_{T} with i ∈ S \ S′, j ∈ S ∩ S′ and such that the following diagram commutes.

[figure: M_{i}^{α}; id_{ℂ}; M_{j}^{α}; 0; P_{i}; M_{j}^{α′}; P_{j}]

Thus P_{i} and P_{j} are both zero. Now let k be any vertex in S ∩S′, We will show that P_{k} is zero. By Lemma 3.3, there is an unoriented path k = k_{0} k_{1} ⋯ k_{m} = i in Q_{T} such that each k_{i} ∈ S ∩ S′. We proceed by induction on m. The case m = 0 is done above, suppose m > 0. By induction P_{k_{1}} is zero and the commutativity of the diagram

\begin{matrix}M_{k}^{α} & \overline{id_{ℂ}} & M_{k_{1}}^{α} \\ M_{k}^{α′} & \begin{aligned}P_{k} \\ & \end{aligned} & \overline{id_{ℂ}} & M_{k_{1}}^{α′} & {0 \atop } \end{matrix}

implies that P_{k} is zero for both possibilities of orientation of k k_{1} in Q_{T}. By contradiction, this shows (ii). Condition (iii) is proved by a similar argument.

In order to show the converse statement, let α, α′ be such that (i),(ii) and (iii) hold. Define P ∈ Hom_{Mod Q_{T}} ((M^{α}, f^{α}), (M^{α′}, f^{α′})) by P_{i} = id_{ℂ} whenever i ∈ S∩S′ and P_{i} = 0 otherwise. Then (i) implies that P is non-zero. We only have to check that P is a morphism of quiver modules, i.e. that the diagram

[figure: M_{i}^{α}; f_{ij}^{α}; M_{j}^{α}; M_{i}^{α′}; P_{i}; f_{ij}^{α′}; M_{j}^{α′}; P_{j}]

commutes for all i → j in Q_{T}. But this is true because of conditions (ii) and (iii). Finally, the dimension of Hom_{Mod Q_{T}} ((M^{α}, f^{α}), (M^{α′}, f^{α′})) is at most one, since all vector spaces M_{i}^{α}, M_{i}^{α′} are of dimension zero or one and the intersection of the supports is connected (Lemma 3.3). □

4. Equivalence of categories

In this section we define a functor and prove that it is an equivalence of categories.

4.1. The Θ functor. Let us define a ℂ-linear additive functor Θ from C_{T} to Mod Q_{T}. On objects, it is sufficient by additivity to define Θ on positive roots. The image of the positive root α is the module (M^{α}, f^{α}) = (M_{i}^{α}, f_{ij}^{α}) defined by

M_{i}^{α} = \{ \begin{matrix}ℂ & if i ∈ Supp α, \\ 0 & otherwise \end{matrix} and f_{ij}^{α} = \{ \begin{matrix}id_{ℂ} & if M_{i}^{α} = ℂ = M_{j}^{α}, \\ 0 & otherwise. \end{matrix}

This is indeed an object in Mod Q_{T} because a positive root α can only cross two sides of a triangle in T, which implies that in each triangle in Q_{T} there is at most one arrow i → j such that f_{ij}^{α} ≠ 0 and hence the triangle relations (1) hold. Now we define the functor Θ on morphisms. By additivity, it is sufficient to define the functor on morphisms from a positive root to a positive root. Our strategy is to define first the functor on pivoting elementary moves, then check that the mesh relations hold. For any pivoting elementary move P : α → α′, define the morphism Θ(P) from (M^{α}, f^{α}) to (M^{α′}, f^{α′}) to be id_{ℂ} whenever possible and 0 else. Let us now check that this is indeed a morphism in Mod Q_{T}. For a given arrow j → i in Q_{T}, we have to check the commutativity of the following diagram:

[figure: M_{j}^{α}; f_{ji}^{α}; M_{i}^{α}; M_{j}^{α′}; Θ(P)_{j}; f_{ji}^{α′}; M_{i}^{α′}; Θ(P)_{i}]

This is obvious if M_{j}^{α} = 0 or M_{i}^{α′} = 0 and also if both M_{i}^{α} and M_{j}^{α′} are 0. Suppose M_{j}^{α} ≠ 0 and M_{i}^{α′} ≠ 0. If M_{i}^{α} ≠ 0 and M_{j}^{α′} ≠ 0 then all four maps are id_{ℂ} and the diagram commutes. The only remaining case is if exactly one of M_{i}^{α}, M_{j}^{α′} is not zero. We will show that this cannot happen. Suppose that M_{j}^{α′} = 0 and M_{i}^{α} ≠ 0, that is j, i ∈ Supp α, i ∈ Supp α′ and j ∉ Supp α′. Since α → α′ is a pivoting elementary move we get that −α_{j} crosses α, that −α_{j} and α′ have a common point on the boundary of the polygon and that −α_{i} crosses α and α′. This contradicts the orientation j → i in the quiver Q_{T}. The other case can be excluded by a similar argument. To show that the functor is well defined, it only remains to check the

mesh relations. Let α ⟶^{P^{1}} β, β ⟶^{P^{2}} γ, α ⟶^{P^{3}} β′, β′ ⟶^{P^{4}} γ be pivoting elementary moves with α, β, γ positive roots and β ≠ β′. Note that we can exclude the case where β and β′ are both negative roots or border edges because in this case either α or γ has to be negative too, since T is a triangulation. Suppose first that β′ is positive. One has to check the commutativity of the diagram

[figure: M_{i}^{α}; Θ(P^{1})_{i}; M_{i}^{β}; M_{i}^{β′}; Θ(P^{3})_{i}; Θ(P^{4})_{i}; M_{i}^{γ}; Θ(P^{2})_{i}]

for all i. The only non trivial case is when i ∈ Supp α ∩ Supp γ. In this case, we also have i ∈ Supp β ∩ Supp β′ because any diagonal crossing both α and γ must also cross β and β′. Thus all maps are id_{ℂ} and the diagram commutes. Suppose now that β′ is negative or a border edge. We have to show that the composition

M_{i}^{α} ⟶^{Θ(P^{1})_{i}} M_{i}^{β} ⟶^{Θ(P^{2})_{i}} M_{i}^{γ} is zero for all i. But in this case no negative root can cross both α and γ. So Supp α ∩ Supp γ is empty, therefore the composition is zero. Hence the mesh relations hold, with the conventions made in its definition.

4.2. Theorem of equivalence. First we need a Lemma.

Lemma 4.1. The conditions in Lemma 2.1 and in Lemma 3.4 are equivalent.

Proof. Suppose α, α′ are as in Lemma 2.1. Then i ∈ S ∩ S′ which implies (i). Suppose there is an arrow j → k in Q_{T} such that j ∈ S \ S′ and k ∈ S ∩ S′. Then −α_{k} crosses both α and α′ while −α_{j} crosses only α. Since j → k, we know that −α_{j} and −α_{k} are related as in figure 4. This is impossible because of the way that −α_{i} intersects α and α′. We have shown that condition (ii) holds; condition (iii) can be shown similarly. This proves one direction of the Lemma.

Suppose now that α, α′ satisfy the conditions (i),(ii) and (iii) of Lemma 3.4. By (i), there exists −α_{i} in S ∩ S′. Let v_{1}, v_{2} be the endpoints of −α_{i}. Consider the two parts of the polygons R_{l} and R_{r} delimited by −α_{i}. Each of them contains exactly one vertex of α and exactly one vertex of α′. Consider the positive roots α, α′ as a paths running from R_{r} to R_{l}. The sequence of negative roots given by the successive intersections of the path α (respectively α′) with elements of T yields an ordering of Supp α (respectively Supp α′). Let S_{l} = {−α_{i} = −α_{i_{1}}, −α_{i_{2}}, …, −α_{i_{p}}} (respectively S′_{l} = {−α_{i} = −α_{j_{1}}, −α_{j_{2}}, …, −α_{j_{q}}}) be the set of negative roots in R_{l} crossing α (respectively α′) in that order. Let m be the greatest integer such that −α_{i_{m}} = −α_{j_{m}}. We will distinguish four cases.

1. m = p = q, then on the boundary of R_{l}, going from an endpoint of −α_{i} in positive direction, we meet α and α′ at the same time.

2. m = p < q, then −α_{j_{m+1}} and −α_{j_{m}} bound part of the same triangle in T. The corresponding edge in Q_{T} is oriented j_{m+1} → j_{m} by (iii). This implies that going from an endpoint of −α_{i} in positive direction on the boundary of R_{l}, we meet α first and then α′.

3. m = q < p, then −α_{i_{m+1}} and −α_{i_{m}} bound part of the same triangle in T. The corresponding edge in Q_{T} is oriented i_{m+1} ← i_{m} by (ii). This implies again that going from an endpoint of −α_{i} in positive direction on the boundary of R_{l}, we meet α first and then α′.

4. m < p and m < q, then −α_{i_{m+1}}, −α_{i_{m}} and −α_{j_{m+1}} are three different diagonals that bound part of the same triangle in T. The corresponding edges in Q_{T} are oriented i_{m+1} ← i_{m} by (ii) and j_{m+1} → j_{m} by (iii). This implies once more that going from an endpoint of −α_{i} in positive direction on the boundary of R_{l}, we meet α first and then α′.

By symmetry, we obtain the same results in the other part R_{r}. This implies that the relative positions of α, α′ and α_{i} are exactly as described in Lemma 2.1. □

Proposition 4.2. The functor Θ is fully faithful.

Proof. Using Lemmas 2.1, 3.4 and 4.1, it only remains to show that the image of a non-zero morphism is a non-zero morphism. It is sufficient to show this for all nonzero morphisms between positive roots. Let P ∈ Hom(α, α′) be such a morphism. Then P is given by a sequence of pivoting elementary moves α = α^{1} → … → α^{m} = α′. This sequence being a non-zero morphism implies that there exists a negative root −α_{i} crossing all the α^{k}, k = 1, …, m, by Lemma 2.1. By definition, Θ(P)_{i} is id_{ℂ}, hence non-zero. □

Remark 4.3. If i is a leaf (see §3.1) and α a positive root, then i ∈ Supp α if and only if one endpoint of α is the vertex x of the polygon that is cut off by −α_{i}. Therefore there exists one positive root α^{P r_{i}} such that the set of all positive roots with i in their supports is equal to the set

{α^{P r_{i}}, P_{x}(α^{P r_{i}}), P_{x}(P_{x}(α^{P r_{i}})), …, P_{x}^{n−1}(α^{P r_{i}})},

where P_{x} is the pivoting elementary move with pivot x.

Theorem 4.4. The functor Θ gives an equivalence of categories from C_{T} to Mod Q_{T}.

Proof. It only remains to show that the functor Θ is essentially surjective, i.e. that each indecomposable module in Mod Q_{T} is the image of a positive root under Θ. In fact, we will characterize the indecomposable modules of Mod Q_{T} with the help of the Auslander-Reiten theory, and this will enable us to conclude by proving that there are \frac{n(n+1)}{2} indecomposable Q_{T} -modules.

In the following, we refer to [8], see also [1] for definitions, notation and results in representation theory of finite dimensional algebras. In this proof, we use the following notations in Mod Q_{T}: P_{i} (respectively I_{i}) is the i−th projective (respectively injective) indecomposable module, with the convention that P_{∅} = I_{∅} = 0. Hence (P_{i})_{l} = ℂ if there is an oriented path in Q_{T} modulo the triangle relations from i to l and (P_{i})_{l} = 0 otherwise. Similarly, (I_{i})_{l} = ℂ if there is an oriented path in Q_{T} modulo the triangle relations from l to i and (I_{i})_{l} = 0 otherwise. The maps of P_{i} and of I_{i} are id_{ℂ} whenever possible and zero otherwise. In particular, these modules are multiplicity free. Fix a triangulation T. In the sequel, we set Q = Q_{T} when no confusion occurs and we denote by Q_{0} its set of vertices. Given a subset of vertices S of Q_{0}, a full subquiver of Q with vertices S will be the set S together with the set of all arrows (with relations) of Q joining vertices of S. We say that a Q_{T} -module M is of type A if the full subquiver of Q_{T} on the support of M is of type A_{k} for some k ∈ ℕ.

Lemma 4.5. Let M be an indecomposable Q-module of type A and let N be any indecomposable Q-module. If Hom_{Q}(N, M) or Hom_{Q}(M, N) contains an irreducible morphism, then N is of type A.

Proof. The proof is based on the construction of irreducible morphisms via the Nakayama functor [8, §4.4]. The dual functor gives an (anti)-equivalence between Mod Q and Mod Q^{opp} = Mod Q_{T^{∗}}, where T^{∗} is the triangulation obtained from T by a reflection w.r.t a line containing the center of the regular polygon. Using this (contravariant) functor, we can easily reduce the proof to the case where Hom_{Q}(N, M) contains an irreducible.

Suppose that the support of M is given by the set Q′_{0} := {1, …, m}. Let Q′ be the full subquiver of Q given by Supp M. By assumption, Q′ is of type A_{m} with extremal vertices 1 and m and we can suppose that the edges link i with i ± 1. Remark that, as M is indecomposable of type A, M is multiplicity free.

In order to go further, we need some more precise description of the quiver Q′ inside Q. The reader may like to follow the argument on the example provided by

Figure 5. Quiver Q and subquiver Q′ of type A

Figure 5. Suppose that the module M is not simple. Note that there exists at most one vertex which will be denoted by 0, resp. m + 1, in Q_{0}\Q′_{0} such that 1 → 0, resp. m → m + 1, and such that there exist no other edges between Q′_{0} and 0, resp. m + 1. Note also that for all k in Q′_{0}, there exist at most two vertices i(k)^{±} such that i(k)^{±} ∉ Q′_{0} and i(k)^{±} → k is an arrow of Q. We can define i(k)^{+}, resp. i(k)^{−}, to be the vertex such that k + 1 → i(k)^{+}, resp. k − 1 → i(k)^{−}. By convention, if the vertices i(k)^{+}, i(k)^{−}, 0, m + 1 do not exist, we define the corresponding symbol to be the empty set.

Let S_{0}, resp. S_{m+1}, be the support of the injective module associated to 0, resp. m + 1, in the full subquiver of Q with set of vertices Q\1, resp. Q\m. It is clear from the tree structure of Q, see §3.1, that the set Q′′_{0} := S_{0} ∪ Q′_{0} ∪ S_{m+1} is the set of vertices of a full subquiver Q′′ of Q of type A.

For each source k or for k = 1, m, let S_{k}^{±} be the support of the injective module associated to i(k)^{±} in the full subquiver of Q with set of vertices Q_{0}\Q′_{0}.

Note that if M is projective, then the module N is a direct summand of the radical of M and the lemma is true in this case. Suppose now that M is not projective. Then, we can calculate the Auslander-Reiten translate τM via the Nakayama functor. We have a minimal projective presentation of M: P^{1} → P^{0} → M → 0 by setting P^{0} := ⊕_{i∈R}P_{i}, where i runs over the set R of sources of Q′, and P^{1} := ⊕_{j∈R′}P_{j}, where j runs over the set R′ given by the union of {0, m + 1} with the sinks of Q′_{0}\{1, m}.

By [8, Remark 3.1], with the help of the Nakayama functor, we obtain a minimal injective representation of the Auslander-Reiten translate τM of M: 0 → τM → I^{1} → I^{0}, with I^{0} := ⊕_{i∈R}I_{i}, and I^{1} := ⊕_{j∈R′}I_{j}. For k in R′, let k^{+}, resp. k^{−}, be the source of Q′ succeeding, resp. preceding, k, with the convention that k^{+} = m, resp. k^{−} = 1, if there is no such source.

Then, for each sink k of Q′_{0}, the support of I_{k} is given by Supp(I_{k}) = S_{k^{−}}^{−} ∪ S_{k^{+}}^{+} ∪ (Supp(I_{k}) ∩ Q′_{0}). The support of I_{0}, resp. I_{m+1}, is S_{l}^{+} ∪ (Supp(I_{0}) ∩ Q′′_{0}), resp. S_{g}^{−} ∪(Supp(I_{m+1})∩Q′′_{0}), where l, resp. g, is the lowest, resp. greatest, source of Q′_{0}. The support of I_{k}, k ∈ R contains S_{k}^{±}.

This implies that the support of τM is a subset of Q′′_{0}.

Now, let X be the middle term in the Auslander-Reiten sequence 0 → τM → X → M → 0. We have Supp X ⊂ Supp τM ∪ Supp M ⊂ Q′′_{0}. Hence, X is of type A. The Auslander-Reiten Theorem asserts that the module N of the Lemma is a direct summand of X. So, we obtain the Lemma in this case. The case where M is simple is very similar and left to the reader. □

We can now prove the Theorem. Let M be an indecomposable Q-module of type A. By the Lemma, the component of M in the Auslander-Reiten quiver contains only modules of type A, therefore can only be finite. Hence, by [8, Proposition 6.3], every indecomposable module is of type A and in particular is multiplicity free. So, there exists a one-to-one correspondence between indecomposable Q-modules and full subquivers of Q of type A. Let i be a leaf of Q and j be any vertex of Q. By §3.1, there exists a unique full subquiver of type A of Q whose extreme vertices are i and j. This implies by induction that the number of such subquivers is \frac{n(n+1)}{2}. Hence, there are \frac{n(n+1)}{2} indecomposable Q-modules as required.

□

Corollary 4.6. The category C_{T} is abelian.

Corollary 4.7. There exists a bijection φ between Ind(Q_{T}) and the diagonals of the polygon not in T. Moreover, for M in Ind(Q_{T}) and any vertex i of Q_{T}, the multiplicity of the simple module S_{i} in the module M is 1 if φ(M) crosses the i^{th} diagonal of T and 0 if not. In particular, for two isoclasses M, M′ in Ind(Q_{T}), we have M = M′ if and only if n_{i}(M) = n_{i}(M′) for all i.

5. The orbit category

This section is not used in the sequel. We give here a description of the category C_{T}, using the equivalence of categories proved above. Then, we prove that the orbit category introduced by [3] has a nice geometric realization in the A_{n} case. Let r^{+}, resp. r^{−}, be the elementary rotation of the polygon in the positive, resp. negative, direction.

Theorem 5.1. Let T be a triangulation of the n + 3 polygon, and let C_{T} be the corresponding category, then:

(i) The irreducible morphisms of C_{T} are direct sums of the generating morphisms given by pivoting elementary moves.

(ii) The mesh relations of C_{T} are the mesh relations [1] of the Auslander-Reiten

quiver of C_{T}. (iii) The Auslander-Reiten translate is given on diagonals by r^{−}. (iv) The projective indecomposable objects of C_{T} are diagonals in r^{+}(T).

(v) The injective indecomposable objects of C_{T} are diagonals in r^{−}(T).

Proof. (i) and then (ii) are clear by construction of the category C_{T}. By (i) and (ii), extremal terms of an almost split sequence are given by the diagonals α and α′ of Figure 2. This proves (iii). (iv) and (v) follow from (iii). □

The assertions (iv) and (v) of Theorem 5.1 suggest an interpretation of the diagonals of the triangulation T in terms of a category. Indeed, we will consider those diagonals, at least in the hereditary case, as shifts of the projectives in the derived category DC_{T}.

In order to simplify the construction, suppose that T is a triangulation corresponding to the unioriented quiver A_{n} with simple projective P_{1}. The category C_{T} ≃ Mod Q_{T} is hereditary, so, the indecomposable objects of the derived category D Mod Q_{T} are the shifts M[m], m ∈ ℤ, of the indecomposables M of Mod Q_{T}. Let F be the functor of D Mod Q_{T} given by M ↦ τ^{−1}M[1], where τ is the Auslander-Reiten translate in the derived category. We define the orbit category \overline{D Mod Q_{T}} whose objects are objects of D Mod Q_{T} and morphisms are given by Hom\overline{D Mod Q_{T}} (M, N) := ⊕_{i∈ℤ}Hom_{D Mod Q_{T}} (M, F^{i}(N)), M, N ∈ D Mod Q_{T}. The set Ind(Q_{T}) ∪ \{P_{i}[1], 1 ≤ i ≤ n\} is the set of indecomposable objects of \overline{D Mod Q_{T}} up to isomorphism. Note that the category \overline{D Mod Q_{T}} is not abelian in general. By a result of Bernhard Keller, [9], it is a triangulated category.

We can also construct the total category C generated by all the diagonals of the (n + 3) polygon. The construction is analogue to the construction of C_{T}: indecomposable objects are positive roots and simple negative ones. The homomorphisms

Figure 6. Example of Auslander-Reiten Quiver

and the mesh relations are defined as in §2.2 without the point (ii) in the convention made there.

Theorem 5.2. Let T be a triangulation corresponding to an orientation of the quiver A_{n}. Then, the categories C and \overline{D Mod Q_{T}} are equivalent.

We give here a sketch of the proof. The derived category of representations of the an oriented quiver of type A is well-known. In particular, it does not depend on the orientation of the quiver. Hence, we can suppose that T correponds to the unioriented quiver as above. The indecomposable objects of D Mod Q_{T} can be indexed by κ: ℤ×{1, …, n} → Ind D Mod Q_{T}, by the rule κ(1, i) = P_{i}, κ(i+1, j) = A^{−1}(κ(i, j)), κ(i + j, n + 1 − j) = κ(i, j)[1]. This implies that the indecomposable objects of \overline{D Mod Q_{T}} can be indexed by ℤ × \{1, …, n\}/(i, j) ≡ (i + j + 1, n − j + 1).

For all (i, j) in ℤ × {1, …, n}, we define the quadrilateral R_{(i,j)} by its vertices (i, j), (i, n), (i + j − 1, 1), (i + j − 1, n − j + 1). Let M be the indecomposable object in \overline{D Mod Q_{T}} indexed by (i, j), and let N be any indecomposable object. Then, Hom\overline{D Mod Q_{T}} (M, N) ≠ 0 if and only if N is indexed by a point inside R_{(i,j)}. In this case, it is ℂ as a space and the composition of morphisms is given by the multiplication. Now, let us index the vertices of the (n + 3) polygon by the group ℤ/(n + 3)ℤ and let [i, j] be the diagonal from i to j, j − i ≠ 1, 0, −1. By the description above, the additive functor defined by [i, j] ↦ κ(i, j − i − 1) gives an equivalence of categories.

Remark 5.3. Using the equivalence above, it is easy to see that given two diagonals α and α′, the group Ext^{1}(α, α′) is non zero if and only if α and α′ cross. Hence, a triangulation of the polygon correspond to a maximal set of pairwise extension free diagonals.

Remark 5.4. The orbit category \overline{D Mod Q_{T}} was introduced in [3] for all simplylaced root systems. Its construction was given to us by Bernhard Keller.

An example of the Auslander-Reiten quiver is provided in Figure 6, for the Quiver with relation shown in the right part of Figure 3.

6. Denominators of Laurent polynomials

In this section, we will prove Conjecture 1.1 for cluster algebras of type A. The existence of the bijection will be an easy consequence of section 4 and older results of Fomin and Zelevinsky. Most of this section is concerned with the calculation of the exponents in the denominators of the Laurent polynomials, in order to prove the equation of the Conjecture.

Throughout this section we will use the following setup. Let Φ_{≥−1} be the set of almost positive roots and τ_{+} and τ_{−} the involutions on Φ_{≥−1} defined by Fomin and Zelevinsky in [7]. For any α ∈ Φ_{≥−1}, let w_{α} be the cluster variable corresponding to α by the bijection of Fomin and Zelevinsky [6, Theorem 1.9]. Let C = {u_{1}, …, u_{n}} be a cluster and let β_{1}, …, β_{n} be the almost positive roots such that w_{β_{i}} = u_{i}. Recall the following properties of τ_{±}:

Proposition 6.1. 1. Every ⟨τ_{−}, τ_{+}⟩–orbit in Φ_{≥−1} contains a negative simple

root.

2. There is a unique function ( ||) : Φ_{≥−1} × Φ_{≥−1} → ℤ_{≥0} such that (i) (−α_{i} || β) = max(n_{i}(β), 0), (ii) (τ_{±}α || τ_{±}β) = (α || β). Furthermore ( ||) is symmetric for simply-laced root systems.

3. The set τ_{±}(C) = {w_{τ_{±}(β_{i})} | i = 1, …, n} is a cluster.

Proof. 1. is shown in [7, Theorem 2.6], 2. in [7, section 3.1] and 3. in [7, Proposition 3.5]. □

By the Laurent phenomenon [5], we can write, for any almost positive root α,

(2)  w_{α} = \frac{R_{α,C}}{∏_{i=1}^{n} u_{i}^{[α,β_{i},C]}} ,

where R_{α,C} is a polynomial in the variables u_{1}, …, u_{n} such that none of the u_{i} divides R_{α,C}, and [α, β_{i}, C] ∈ ℤ. The following Lemma is crucial.

Lemma 6.2. For any pair of almost positive roots α, β_{i} and any pair of clusters C, C′ such that u_{i} = w_{β_{i}} ∈ C ∩ C′, we have

[α, β_{i}, C] = [α, β_{i}, C′].

Proof. It is sufficient to prove assertions (a) and (b) below.

(a) All clusters containing the given cluster variable u_{i} are connected in the mutation graph.

(b) In mutations which do not exchange u_{i}, the exponent of u_{i} in the denominator of w_{α} is unchanged.

Assertion (a) can either be seen as a classical statement on the link of a simplex in a simplicial sphere, or can be checked directly using the recursive properties of clusters. First the adjacency graph of clusters containing a fixed cluster variable is mapped to an isomorphic graph by the action of τ_{+} and τ_{−} because this action respects the compatibility function. Thus one can suppose that the fixed cluster variable corresponds to a negative simple root. But then the graph of clusters containing this negative simple root is isomorphic to a product of whole cluster adjacency graphs for smaller sub-root systems. Therefore it is connected as a product of connected graphs.

Let us show (b) now. Consider the mutation C → C′ that exchanges the cluster variables u_{j} ∈ C and u′_{j} ∈ C′. The exchange relation gives u_{j} = \frac{M_{1}+M_{2}}{u′_{j}}, where M_{1} and M_{2} are monomials without common divisors in the variables C \ {u_{j}}. From equation (2), we obtain by substitution

w_{α} = \frac{R_{α,C}(u_{1}, …, u_{j−1}, \frac{M_{1}+M_{2}}{u′_{j}} , u_{j+1}, …, u_{n})}{∏_{l≠j} u_{l}^{[α,β_{l},C]}(\frac{M_{1}+M_{2}}{u′_{j}})^{[α,β_{j},C]}}.

By the Laurent phenomenon, we know that w_{α} is a Laurent polynomial in the cluster variables u_{1}, …, u′_{j}, …, u_{n}. We want to prove that the exponent of u_{i} in the denominator is still [α, β_{i}, C]. Clearly, by properties of M_{1} and M_{2}, this is true if and only if the Laurent polynomial R_{α,C}(u_{1}, …, \frac{M_{1}+M_{2}}{u′_{j}} , …, u_{n}) is not zero after evaluation at u_{i} = 0. To conclude, remark that, by properties of the monomials M_{1} and M_{2} stated above, the value of this Laurent polynomial at u_{i} = 0 is obtained by an invertible substitution from the value of R_{α,C} at u_{i} = 0, which is known not to be zero. □

Thus [α, β, C] will be denoted simply [α, β] from now on.

The following Lemma is proved for all simply-laced root-systems.

Lemma 6.3. Let α,β be two almost positive roots. Then

(3)  [α, β] = [τ_{±}α, τ_{±}β].

Proof. Let us consider a sequence of adjacent clusters

(4)  C_{0} ↔ C_{1} ↔ ⋯ ↔ C_{N},

where α ∈ C_{0} and β ∈ C_{N}. The exchange relations depend only on the matrices B(C_{0}), …, B(C_{N}) associated to these clusters. As the action of τ respects the compatibility relation, one gets another chain of adjacent clusters

(5)  τ(C_{0}) ↔ τ(C_{1}) ↔ ⋯ ↔ τ(C_{N}),

where τ(α) ∈ τ(C_{0}) and τ(β) ∈ τ(C_{N}). By Lemma 4.8 in [6], one has

(6)  B_{τγ′,τγ′′}(τ(C)) = −B_{γ′,γ′′}(C),

for any cluster C and roots γ′, γ′′ in it. This minus sign does not change the exchange relations. From this one deduces that the expression of the cluster variable u_{α} in the variables of the cluster C_{N} is the same as the expression of the cluster variable u_{τ(α)} in the variables of the cluster τ(C_{N}). This proves the Lemma. □

Lemma 6.4. Let −α_{i} be a simple negative root and α an almost positive root. Then

[α, −α_{i}] = n_{i}(α).

Proof. The quantity [α, −α_{i}] is computed using the expression of the cluster variables α in the cluster made of negative roots. Then the bijection of Fomin and Zelevinsky between cluster variables and roots ([6, Theorem 1.9]) implies that this is (α || −α_{i}). By Proposition 6.1.2.(i) and symmetry of ( ||) in the simply-laced cases, the conclusion follows. □

Proposition 6.5. Let α,β be two distinct almost positive roots. Then

[α, β] = (α || β).

Proof. Define a function b : Φ_{≥−1} × Φ_{≥−1} → ℤ_{≥0} by

b(α, β) = \{ \begin{matrix}[α, β] & if α ≠ β, \\ 0 & if α = β. \end{matrix}

This function is well defined by Lemma 6.2. Moreover (1) b(−α_{i}, β) = max(n_{i}(β), 0) (by Lemma 6.4)

(2) b(τ_{±}α, τ_{±}β) = b(α, β) (by Lemma 6.3).

By Proposition 6.1(2), the function ( || ) : Φ_{≥−1} × Φ_{≥−1} → ℤ_{≥0} is the unique function having the properties (1) and (2), thus b(α, β) = (α || β). Therefore [α, β] = (α || β) if α ≠ β. □

The following Theorem establishes the Conjecture 1.1 for the type A_{n}.

Theorem 6.6. Let C = {u_{1}, …, u_{n}} be a cluster of a cluster algebra of type A_{n} and let V be the set of all cluster variables of the algebra. Let Q_{C} be the quiver with relations associated to C and Ind(Q_{C}) the set of isoclasses of indecomposable modules. Then there is a bijection

Ind(Q_{C}) → V \ C, α ↦ w_{α},

such that

w_{α} = \frac{P (u_{1}, …, u_{n})}{∏_{i=1}^{n} u_{i}^{n_{i}(α)}},

where P is a polynomial such that none of the u_{i} divides P (i = 1, …, n) and n_{i}(α) is the multiplicity of the simple module α_{i} in the module α.

Proof. Let T_{C} = {β_{1}, …, β_{n}} be the triangulation of the (n + 3) polygon corresponding to the cluster C and let D be the set of diagonals of the polygon; thus T_{C} ⊂ D. Let T_{0} = {−α_{1}, …, −α_{n}} be the “snake triangulation” [6, 12.2]. Q_{T_{0}} is the alternating quiver of type A_{n} and the diagonals −α_{i} ∈ T_{0} are the negative simple roots. Fomin and Zelevinsky have shown that there is a bijection α ↦ w_{α} between the set of almost positive roots Φ_{≥−1} and the set of cluster variables V. In type A, they identified Φ_{≥−1} with D and proved that for any cluster C there is a bijection α ↦ w_{α} between D \ T_{C} and V \ C. In section 4, we have shown the bijection M_{α} ↦ α between Ind(Q_{C}) and D \ T_{C}. This establishes a bijection Ind(Q_{C}) → V \C. This bijection sends the simple module in Ind(Q_{C}) at the vertex i ∈ Q_{C} to the variable w_{β} where β is the unique diagonal in D \ T_{C} that crosses β_{i} and does not cross any diagonal in T_{C} \ {β_{i}}

Let M_{α} be an element of Ind(Q_{C}) with α the corresponding diagonal in D \ T_{C}. By Lemma 6.2, we have w_{α} = _{\frac{P (u_{1},...,u_{n})}{∏_{i=1}^{n} u_{i}^{[α,β_{i}]}}} . We have to show that [α, β_{i}] = n_{i}(α)

for all i = 1, …, n. Note that α ≠ β_{i} since α ∉ T_{C}, hence using Proposition 6.5, we get [α, β_{i}] = (α || β_{i}) and by [6, 12.2] this is equal to 1 if the diagonals α and β_{i} are crossing, and zero otherwise. Thus

[α, β_{i}] = \{ \begin{matrix}1 & if i ∈ Supp α in the sense of section 4 \\ 0 & otherwise. \end{matrix} \} = n_{i}(α).
□

References

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