FRIEZES AND A CONSTRUCTION OF THE EUCLIDEAN CLUSTER VARIABLES

I. ASSEM AND G. DUPONT

Abstract. Let Q be an euclidean quiver. Using friezes in the sense of Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical) cluster character associated to any object in the cluster category of Q. In particular, this algorithm allows to compute all the cluster variables in the cluster algebra associated to Q. It also allows to compute the sum of the Euler characteristics of the quiver grassmannians of any module M over the path algebra of Q.

Contents

1. Introduction 1 2. Cluster characters and friezes 5 3. Proofs of Theorems 1.3 and 1.4 6 4. Euler characteristics of complete quiver grassmannians 12 5. Explicit determination of B_{λ} and B′_{λ} 13 6. Algorithms 20 7. An example : Regular cluster variables in type 𝔼̃_{6} 22 8. Euler characteristics of complete grassmannians in type 𝔼̃ 24 Acknowledgements 25 References 25

1. Introduction

1.1. Cluster algebras. In [FZ02], Fomin and Zelevinsky introduced a class of commutative algebras, called cluster algebras, in order to design a combinatorial framework for studying total positivity in algebraic groups and canonical bases in quantum groups. Since then, cluster algebras have shown connections with various areas of mathematics like Lie theory, Teichmüller theory, Poisson geometry or the representation theory of algebras.

Let Q = (Q_{0}, Q_{1}) be an acyclic quiver, that is an oriented graph which contains no oriented cycle. Let Q_{0} denote the set of points in Q and Q_{1} the set of arrows in Q. We assume that these sets are finite and we fix a Q_{0}-tuple u = (u_{i}|i ∈ Q_{0}) of indeterminates. The cluster algebra A(Q) with initial seed (Q, u) is a subalgebra of ℤ[u^{±1}] = ℤ[u_{i}^{±1}|i ∈ Q_{0}], generated by a distinguished set of generators, called cluster variables defined recursively using a combinatorial process called mutation [FZ02].

Date: March 2, 2010.

The number of cluster variables in a cluster algebra can be finite or infinite. If the cluster algebra has finitely many cluster variables, it is of finite type. This is the case if and only if Q is a Dynkin quiver, that is a quiver of type 𝔸, 𝔻 or 𝔼 [FZ03]. In this case, it is possible to compute algorithmically the cluster variables in A(Q) by applying mutations recursively. If Q is not a Dynkin quiver, this recursive process does not end. Nevertheless, cluster variables can be parametrised using the representation theory of Q. In this paper, we exhibit an algorithm for constructing cluster variables with respect to this parametrisation when Q is a quiver of euclidean type, that is an acyclic orientation of an euclidean diagram of type 𝔸̃, 𝔻̃ or 𝔼̃ (these diagrams are also called extended Dynkin or affine in the literature).

More precisely, if C_{Q} denotes the cluster category introduced in [BMR^{+}06], Caldero, Chapoton and Keller introduced an explicit map

X_{?} : Ob(C_{Q})⟶ ℤ[u^{±1}]

called (canonical) cluster character (also called Caldero-Chapoton map in the literature) such that

{cluster variables in A(Q)} = {X_{M}|M indecomposable and rigid in C_{Q}} .

where an object is called rigid if it has no self-extension.

In general, in order to compute the character X_{M} of an object M, one has to compute the Euler characteristics of certain projective varieties called quiver grassmannians (see Section 2 for definitions). These Euler characteristics are hard to compute in practice. In some very simple cases, explicit computations were made [CZ06, Cer09, Cer10, Poe10] but in general there exists no simple way for computing these characteristics and thus for computing X_{M}.

1.2. Friezes. In this article, we provide a simple combinatorial method for computing these characters when Q is an euclidean quiver. Our methods are based on the concept of frieze on a stable translation quiver. This concept appeared in various independent works under different names : frises in [ARS09], frieze patterns in [Pro08, CC73a, CC73b], cluster-mesh algebras in [Dup09a].

1.2.1. Friezes on translation quivers. By definition, a stable translation quiver (Γ, τ) is a quiver Γ = (Γ_{0}, Γ_{1}) without loops together with a bijection τ : Γ_{0}⟶ Γ_{0}, called translation such that there is a bijection from the set of arrows j⟶ i in Γ_{1} to the set of arrows τi⟶ j in Γ_{1} for any two points i, j ∈ Γ_{0}. We always assume that a translation quiver is locally finite, that is, for any i ∈ Q_{0}, there exist only finitely many arrows starting or ending at i in Γ_{1}. Note that Γ_{0} and Γ_{1} may be infinite sets.

Throughout the paper, commutative rings are supposed unitary and, given a commutative ring R, we denote by 1 its identity.

Definition 1.1. Let (Γ, τ) be a stable translation quiver and R be a commutative ring. An R-frieze on (Γ, τ) is a map f : Γ_{0}⟶ R such that, for any i, j ∈ Γ_{0},

f(i)f(τ(i)) = ∏_{\begin{matrix}α∈Γ_{1} \\ α:j⟶ i \end{matrix}} f(j) + 1.

By a frieze on (Γ, τ), we mean an R-frieze for a certain commutative ring R.

It is clear from the definition that an R-frieze on a stable translation quiver (Γ, τ) is entirely determined by its restriction on each of the connected components of (Γ, τ).

1.2.2. Friezes on Auslander-Reiten quivers. If C is a Krull-Schmidt category with an Auslander-Reiten translation, we denote by Γ(C) its Auslander-Reiten quiver. It is naturally a translation quiver for the translation induced by the Auslander-Reiten translation in C.

If f is a frieze on Γ(C), we slightly abuse notations and view f as a function on objects in C. For any object M in C, we define f(M) as follows. If M is indecomposable, f(M) is the value of f on the point in Γ(C) corresponding to the isomorphism class of M. If M_{1}, …, M_{n} are indecomposable, we define f(M_{1} ⊕⋯⊕ M_{n}) as the product f(M_{1}) ⋯ f(M_{n}) and by convention, we set f(0) = 1.

1.3. Auslander-Reiten quivers of cluster categories. We fix an algebraically closed field k and an acyclic quiver Q = (Q_{0}, Q_{1}) such that Q_{0} and Q_{1} are finite. We denote by kQ the path algebra of Q and by mod-kQ the category of finitely generated right-kQ-modules. For any i ∈ Q_{0}, we denote by S_{i} the simple module associated to i, by P_{i} its projective cover and by I_{i} its injective hull. We denote by τ the Auslander-Reiten translation on mod-kQ. Note that a kQ-module M will be identified with a representation (M(i), M(α))_{i∈Q_{0},α∈Q_{1}} of Q.

Let D^{b}(mod−kQ) be the bounded derived category. We also denote by τ the Auslander-Reiten translation on D^{b}(mod−kQ) and we denote by [1] the suspension functor in D^{b}(mod−kQ). Following [BMR^{+}06], the cluster category of Q is the orbit category C_{Q} of the functor τ^{−1}[1] in D^{b}(mod−kQ). It is a triangulated category satisfying the 2-Calabi-Yau property, that is, there is a bifunctorial duality

Ext_{C_{Q}}^{1}(X, Y) ≃ DExt_{C_{Q}}^{1}(Y, X)

for any X, Y ∈ Ob(C_{Q}) where D = Hom_{k}(−, k) [Kel05, BMR^{+}06].

The set of indecomposable objects in C_{Q} can be identified with the disjoint union of the set of indecomposable kQ-modules and the set of shifts of the indecomposable projective kQ-modules [BMR^{+}06].

Under this identification, the Auslander-Reiten quiver Γ(C_{Q}) of the cluster category C_{Q} is a stable translation quiver for the translation induced by the shift [1] on C_{Q} which coincides with the Auslander-Reiten translation τ of induced by the Auslander-Reiten translation on D^{b}(mod−kQ).

All the projective kQ-modules belong to the same connected component P of Γ(C_{Q}). This component, called the transjective component, is isomorphic to ℤQ. If Q is a Dynkin quiver, P is the unique connected component in Γ(C_{Q}). Otherwise, Γ(C_{Q}) contains infinitely many other connected components, called regular. An indecomposable object in the transjective component is called transjective, it is of the form P_{i}[n + 1] for some n ∈ ℤ and i ∈ Q_{0}. If n < 0, then P_{i}[n + 1] is identified with the kQ-module τ^{−n}P_{i} and is called postprojective, if n > 0, then P_{i}[n + 1] is identified with the kQ-module τ^{n−1}I_{i} and is called preinjective. An indecomposable object in a regular component is identified with a regular kQ-module and is called regular. A decomposable object is transjective (or regular) if all its direct summands are transjective (or regular, respectively).

If Q is euclidean, regular components are parametrised by ℙ^{1}(k) and for any λ ∈ ℙ^{1}(k) the corresponding component T_{λ} is a tube of rank p_{λ} ≥ 1. If p_{λ} = 1, T_{λ} is called homogeneous, otherwise it is called exceptional. We set

ℙ^{E} = {λ ∈ ℙ^{1}(k)|p_{λ} > 1}

the index set of exceptional tubes and

ℙ^{H} = {λ ∈ ℙ^{1}(k)|p_{λ} = 1}

the index set of homogeneous tubes. There exists at most three exceptional tubes in Γ(C_{Q}), in other words |ℙ^{E}| ≤ 3. A module at the mouth of a tube is called quasi-simple.

If Q is euclidean, the Tits form of Q on ℤ^{Q_{0}} has corank one and there exists a unique positive generator of its radical which is denoted by δ. The vector δ is called the radical vector or positive minimal imaginary root of Q and it is known that, for any orientation of Q, there exists a point e ∈ Q_{0} which is either a sink or a source and such that δ_{e} = 1. Throughout the paper e denotes such a point.

1.4. Main results. Let Q be an euclidean quiver. For any λ ∈ ℙ^{1}(k), there exists a unique quasi-simple module M_{λ} in T_{λ} such that dim M_{λ}(e) = 1 and we set

N_{λ} = \{ \begin{matrix}M_{λ}[1] & if e is a source ; \\ M_{λ}[−1] & if e is a sink. \end{matrix}

We prove in Lemmas 3.10 and 3.11 the existence of transjective objects B_{λ} and B′_{λ} uniquely determined by the fact that there exist non-split triangles

N_{λ}⟶ B_{λ}⟶ S_{e}⟶ N_{λ}[1] and S_{e}⟶ B′_{λ}⟶ N_{λ}⟶ S_{e}[1]

if e is a source and non-split triangles

N_{λ}⟶ B′_{λ}⟶ S_{e}⟶ N_{λ}[1] and S_{e}⟶ B_{λ}⟶ N_{λ}⟶ S_{e}[1]

if e is a sink.

Definition 1.2. Let Q be an euclidean quiver. A cluster frieze on Γ(C_{Q}) is a frieze f on Γ(C_{Q}) with values in a commutative ring R such that for any λ ∈ ℙ^{E}

(1)  f(N_{λ}[k]) = \frac{f(B_{λ}[k]) + f(B′_{λ}[k])}{f(S_{e}[k])} for any k ∈ \{0, …, p_{λ} − 1\} .

A cluster frieze is called a strong cluster frieze if (1) holds for any λ ∈ ℙ^{1}(k).

Cluster friezes allow to realise cluster variables in the following sense :

Theorem 1.3. Let C_{Q} be the cluster category of an euclidean quiver Q and let f be a cluster frieze on Γ(C_{Q}) with values in ℚ(u_{i}, i ∈ Q_{0}) such that f(P_{i}[1]) = u_{i} for any i ∈ Q_{0}. Then f(M) = X_{M} for any indecomposable rigid object M in C_{Q}.

In particular, f induces a bijection between the set of indecomposable rigid objects in C_{Q} and the set of cluster variables in A(Q, u).

We actually prove a slightly more general result. Namely, we prove that if f is a cluster frieze on Γ(C_{Q}) such that f(P_{i}[1]) = u_{i} for any i ∈ Q_{0}, then f and X_{?} coincide on the transjective component and the exceptional tubes of Γ(C_{Q}).

Strong cluster friezes allow moreover to realise cluster characters associated to modules in homogeneous tubes :

Theorem 1.4. Let C_{Q} be the cluster category of an euclidean quiver Q and let f be a strong cluster frieze on Γ(C_{Q}) with values in ℚ(u_{i}, i ∈ Q_{0}) such that f(P_{i}[1]) = u_{i} for any i ∈ Q_{0}. Then f(M) = X_{M} for any indecomposable object M in C_{Q}.

Note that, if the objects B_{λ}, B′_{λ} can be explicitly constructed, Theorem 1.3 (and Theorem 1.4) provide a purely combinatorial method for computing every cluster variable (and every cluster character, respectively) in a cluster algebra of euclidean type. In Section 5, we provide such explicit constructions for any euclidean quiver.

1.5. Organisation of the paper. In Section 2, we prove that the cluster character X_{?} induces a frieze on the Auslander-Reiten quiver of any cluster category. Section 3 is dedicated to the proof of Theorems 1.3 and 1.4. As an application, we use friezes of numbers in Section 4 in order to compute the Euler characteristics of the socalled complete quiver grassmannians. Section 5 provides a case-by-case analysis in order to construct explicitly the objects B_{λ} and B′_{λ}. As an application, we provide an algorithm for computing all cluster variables in cluster algebras associated to euclidean quivers in Section 6. In Section 7 we use this algorithm to compute explicitly the regular cluster variables in type 𝔼̃_{6}. Finally, in Section 8, we use it to compute the Euler characteristics of the complete quiver grassmannians of quasi-simple regular modules in types 𝔼̃_{6}, 𝔼̃_{7} and 𝔼̃_{8}.

2. Cluster characters and friezes

We recall the definition of the cluster character associated to an acyclic quiver Q. For any kQ-module M, we denote by dim M ∈ ℤ_{≥0}^{Q_{0}} its dimension vector. For any e ∈ ℤ_{≥0}^{Q_{0}}, the quiver grassmannian of M of dimension e is

Gr_{e}(M) = {N ⊂ M|dim N = e} .

This is a projective variety and we denote by χ(Gr_{e}(M)) its Euler characteristic with respect to the simplicial (or étale) cohomology if k = ℂ (or if k is arbitrary, respectively).

We denote by ⟨−, −⟩ the Euler form on kQ-mod, which here coincides with the Tits form, given by

⟨M, N⟩ = dim Hom_{kQ}(M, N) − dim Ext_{kQ}^{1}(M, N).

It only depends on dim M and dim N (see [ASS05]).

Definition 2.1 ([CC06]). Let Q be an acyclic quiver. The cluster character associated to Q is the map

X_{?} : Ob(C_{Q})⟶ ℤ[u^{±1}]

defined as follows :

(ι) If M is an indecomposable kQ-module

(2)  X_{M} = ∑_{e∈ℤ_{≥0}^{Q_{0}}} χ(Gr_{e}(M)) ∏_{i∈Q_{0}} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim M−e⟩} ;

(ιι) If M ≃ P_{i}[1], then

X_{M} = X_{P_{i}[1]} ;

(ιιι) For any two objects M, N ∈ Ob(C_{Q}),

X_{M⊕N} = X_{M}X_{N}.

Note that (2) also holds for decomposable kQ-modules [CC06]. Moreover X_{M} = 1 if M = 0.

It was proved in [CK06] that X_{?} induces a 1-1 correspondence between the set of indecomposable rigid objects in C_{Q} and the set of cluster variables in A(Q). Thus, a cluster variable x ∈ A(Q) is called transjective (or regular) if x = X_{M} for some indecomposable rigid transjective (or indecomposable rigid regular, respectively) object M in C_{Q}.

Proposition 2.2. Let Q be an acyclic quiver. Then the cluster character associated to Q induces a ℤ[u^{±1}]-frieze on Γ(C_{Q}).

Proof. Let M be an indecomposable object in C_{Q}. Then either M is regular or M is transjective. If M is a regular kQ-module, it follows from [CC06, Proposition 3.10] that X_{M}X_{τM} = X_{B} + 1 where

0⟶ τM⟶ B⟶ M⟶ 0

is an almost split sequence of kQ-modules. It thus follows that X_{?} induces a frieze on the regular component T containing M. Explicitly, the frieze is given by f(i, n) = X_{R_{i}^{(n)}} for any i ∈ ℤ/pℤ and n ≥ 1 where the R_{i} for i ∈ ℤ/pℤ denote the quasi-simple modules in T ordered in such a way that τR_{i} ≃ R_{i−1} for any i ∈ ℤ/pℤ and R_{i}^{(n)} is the unique indecomposable regular kQ-module with quasi-socle R_{i} and quasi-length n.

If M is transjective,

Ext_{C_{Q}}^{1}(M[1], M) ≃ Hom_{C_{Q}}(M[1], M[1]) ≃ End_{C_{Q}}(M[1]) ≃ k.

It follows from [CK06, Theorem 2] that

X_{M}X_{M[1]} = X_{B} + X_{B′}

where B and B′ are the unique objects such that there exist non-split triangles

M[1]⟶ B⟶ M⟶ M[2] and M⟶ B′⟶ M[1]⟶ M[1].

Since End_{C_{Q}}(M[1]) ≃ k, B′ = 0 and thus X_{M}X_{M[1]} = X_{B} + 1. Moreover, since Ext_{C_{Q}}^{1}(M[1], M) ≃ k, the non-split triangle M[1]⟶ B⟶ M⟶ M[2] is almost split. Thus, X_{?} induces a frieze on the transjective component.

It follows that X_{?} induces a frieze on each connected component of Γ(C_{Q}) and thus on Γ(C_{Q}). By definition, X_{M} ∈ ℤ[u^{±1}] for any object M in C_{Q} so that the frieze induced by X_{?} takes its values in the commutative unitary ring ℤ[u^{±1}]. This completes the proof. □

3. Proofs of Theorems 1.3 and 1.4

The proofs consist of several steps. First, we prove that if Q is any acyclic quiver and M is a transjective object M, then X_{M} can be computed via a frieze on the transjective component. If Q is euclidean, we prove in Section 3.2 that a frieze on a tube in Γ(C_{Q}) is entirely determined by its values at the mouth of the tube. In Section 3.3, we give relations between cluster characters associated to quasi-simple modules in tubes and cluster characters associated to transjective objects. All these results are applied in Sections 3.4 and 3.5 in order to prove Theorems 1.3 and 1.4.

3.1. The transjective component. Given a locally finite quiver Q, we denote by ℤQ its repetition quiver, that is, the quiver whose points are labelled by ℤ × Q_{0} and arrows are given by (n, i)⟶ (n, j) and (n − 1, j)⟶ (n, i) if i⟶ j ∈ Q_{1}. It is a stable translation quiver for the translation τ(n, i) = (n − 1, i) for any n ∈ ℤ, i ∈ Q_{0}. A section in a repetition quiver ℤQ is a full connected subquiver Σ in ℤQ such that ℤΣ ≃ ℤQ (see [ASS05]).

Lemma 3.1. Let Q be an acyclic quiver and R be a commutative ring. Let f be a R-frieze on ℤQ. Then f is entirely determined by the image of a section in ℤQ.

Proof. Let f and g be R-friezes on ℤQ such that f and g coincide on a section Σ in ℤQ. Since ℤQ ≃ ℤΣ, we can label the points in ℤQ by ℤ × Σ_{0} such that points in Σ correspond to {0} × Σ_{0}. Since Σ is acyclic, we define a partial order on Σ_{0} by taking the closure ≺ of the order i ≺_{e} j if j⟶ i in Q_{0}.

We now prove that for any n ≥ 0 and any i ∈ Σ_{0}, f(n, i) = g(n, i). We put on ℤ_{≥0} × Σ_{0} the lexicographic order ≺ induced by ≤ and ≺. We prove the claim by induction on this lexicographic order. If n = 0, the result holds by hypothesis. Fix thus n ≥ 1 and i ∈ Σ_{0} and assume that f(m, j) = g(m, j) for any (j, m) ≺ (i, n). By definition of the frieze we have

f(n, i)f(n − 1, i) = ∏_{(m,j)⟶ (n,i)} f(m, j) + 1

and

g(n, i)g(n − 1, i) = ∏_{(m,j)⟶ (n,i)} g(m, j) + 1.

But if (m, j)⟶ (n, i), then it follows from the definition of ℤΣ that either m = n−1 and j⟶ i or m = n and i⟶ j. Hence, in any case, (m, j) ≺ (n, i). Thus,

f(n, i) = \frac{∏_{(m,j)⟶ (n,i)∈Σ_{1}} f(m, j) + 1}{f(n − 1, i)} = \frac{∏_{(m,j)⟶ (n,i)∈Σ_{1}} g(m, j) + 1}{g(n − 1, i)} = g(n, i).

If n ≤ 0, we use the same argument with the dual order on ℤ_{≤0} × Σ_{0}. □

Corollary 3.2. Let Q be an acyclic quiver. Let f be a frieze on Γ(C_{Q}) such that f(P_{i}[1]) = u_{i} for every i ∈ Q_{0}. Then for any indecomposable transjective object M,

f(M) = X_{M}.

In particular, f induces a bijection between the set of indecomposable transjective objects in C_{Q} and the set of transjective cluster variables in A(Q, u).

Proof. The transjective component P of Γ(C) is isomorphic to ℤQ. The full subquiver Σ whose points correspond to P_{i}[1], with i ∈ Q_{0}, is a section in ℤQ. By Proposition 2.2, the restrictions of X_{?} and f to P are friezes and it follows from Lemma 3.1 that they are entirely determined by their values on Σ. Since f and X_{?} coincide on Σ, they coincide on P.

The second statement follows from [CK06, Theorem 4]. □

Remark 3.3. For acyclic quivers, Corollary 3.2 provides a representation-theoretic interpretation of the combinatorial construction of cluster variables with friezes provided in [ARS09]. A similar interpretation will be provided for acyclic valued graphs in [DN].

3.2. Friezes on tubes. Let 𝔸_{∞} be the locally finite quiver with point set ℤ_{>0} and arrows i⟶ i + 1 for any i ∈ ℤ_{>0}. Let ℤ𝔸_{∞} be its repetition quiver. We recall that a (stable) tube T is a translation quiver isomorphic to ℤ𝔸_{∞}/(τ^{p}) for some integer p ≥ 1, called the rank of T . Points in T can be labelled by ℤ/pℤ × ℤ_{>0} and the translation on T induced by the translation on ℤ𝔸_{∞} is thus given by τ(i, n) = (i − 1, n) for any i ∈ ℤ/pℤ, n ∈ ℤ_{>0}. The mouth of a tube T is the set of points {(i, 1)|i ∈ ℤ/pℤ}.

Definition 3.4 ([Dup09a]). Let T_{i}, i ∈ ℤ_{≥0} be a family of indeterminates over ℤ_{≥0}. We define the n-th generalised Chebyshev polynomial P_{n} as follows. First, P_{−1} = 0, P_{0} = 1 and for any n ≥ 1, P_{n} is the polynomial in ℤ[T_{0}, …, T_{n−1}] given by

P_{n+1}(T_{0}, …, T_{n}) = T_{n}P_{n}(T_{0}, …, T_{n−1}) − P_{n−1}(T_{0}, …, T_{n−2}).

Note that, for any n ≥ 1,

P_{n}(T_{0}, …, T_{n−1}) = det [ \begin{matrix}T_{n−1} & 1 & & & (0) \\ 1 & ... & ... \\ & ... & ... & ... \\ & & ... & ... & 1 \\ (0) & & & 1 & T_{0} \end{matrix} ] ∈ ℤ[T_{0}, …, T_{n−1}].

If R is an arbitrary commutative unitary ring. The ring homomorphism ℤ⟶ R sending 1 to 1_{R} allows to view each generalised Chebyshev polynomial P_{n} as a polynomial function in n variables with coefficients in R. It was observed in [Dup09a] that friezes on tubes are entirely governed by generalised Chebyshev polynomials in the following sense.

Theorem 3.5 ([Dup09a]). Let T be a tube of rank p ≥ 1 and f be a frieze on T . Then for any i ∈ ℤ/pℤ and any n ≥ 1, we have

f(i, n) = P_{n}(f(i, 1), …, f(i + n − 1, 1)).
□

Corollary 3.6. A frieze on a tube is entirely determined by its values on the mouth of the tube. □

Remark 3.7. An interesting consequence is that if R is a ring and T is a tube with mouth M, every function M⟶ R can be completed into an R-frieze T ⟶ R. This contrasts with the situation of the repetition quivers ℤQ of an acyclic quiver Q where if Σ is a section in ℤQ, there may exist functions Σ⟶ R which do not extend to a R-frieze on ℤQ. For example, it is easily seen that if Q is a quiver of Dynkin type 𝔸_{2}, there exists no ℤ-frieze f on ℤQ such that f equals 2 on a section in ℤQ.

3.3. Characters associated to quasi-simple modules.

Lemma 3.8. Let Q be an acyclic quiver. Then the map σ : X_{M} ↦ X_{M[1]} induces a ℤ-algebra automorphism of A(Q).

Proof. It follows from [CK06] that A(Q) is the ℤ-algebra generated by X_{M} for M rigid in C_{Q} with the relations

X_{M}X_{M^{∗}} = X_{B} + X′_{B}

where (M, M^{∗}) is an exchange pair and B, B′ are the central terms of the corresponding exchange triangles (see [BMR^{+}06] for details). Since [1] is an autoequivalence of C_{Q}, (M[1], M^{∗}[1]) is also an exchange pair and B[1], B′[1] are the central terms of the corresponding exchange triangles. Thus,

X_{M[1]}X_{M^{∗}[1]} = X_{B[1]} + X_{B′[1]}

and X_{M} ↦ X_{M[1]} induces a ℤ-linear homomorphism A(Q)⟶ A(Q) whose inverse is induced by X_{M} ↦ X_{M[−1]}. □

From now on and until the end of Section 3, if Q denotes an euclidean quiver with radical vector δ then e ∈ Q_{0} denotes a point which is either a source or a sink such that δ_{e} = 1.

Lemma 3.9. Let Q be an euclidean quiver. Then for any λ ∈ ℙ^{1}(k), there exists a unique quasi-simple module M_{λ} in T_{λ} such that M_{λ}(e) ≃ k.

Proof. Let R_{0}, …, R_{p−1} be the quasi-simple modules in T . It is well-known (see for instance [CB92]) that

δ = ∑_{i=0}^{p−1} dim R_{i}

so that

1 = δ_{e} = ∑_{i=0}^{p−1} dim R_{i}(e).

Thus, there exists a unique i ∈ {0, …, p − 1} such that R_{i}(e) ≃ k. □

Lemma 3.10. Let Q be an euclidean quiver and λ ∈ ℙ^{1}(k). Assume that e is a source such that δ_{e} = 1. Set N_{λ} = M_{λ}[1]. Then the following hold :

(1) dim Ext_{C_{Q}}^{1}(N_{λ}, S_{e}) = 1 ;

(2) X_{N_{λ}}X_{S_{e}} = X_{B_{λ}} +X_{B′_{λ}} where B_{λ} is the (unique) kQ-module such that there exists a short exact sequence

0⟶ N_{λ}⟶ B_{λ}⟶ S_{e}⟶ 0

and B′_{λ} = Ker f⊕Coker f[−1] for any non-zero morphism f ∈ Hom_{C_{Q}}(N_{λ}, τS_{e}) ;

(3) B_{λ} and B′_{λ} are transjective.

Proof. Note that I_{e} ≃ S_{e} so that we have

dim Ext_{C_{Q}}^{1}(N_{λ}, S_{e}) = dim Ext_{C_{Q}}^{1}(N_{λ}, I_{e})
= dim Ext_{C_{Q}}^{1}(N_{λ}, P_{e}[2])
= dim Ext_{C_{Q}}^{1}(N_{λ}[−2], P_{e})
= dim Ext_{C_{Q}}^{1}(P_{e}, N_{λ}[−2])
= dim Hom_{C_{Q}}(P_{e}, N_{λ}[−1])
= dim Hom_{kQ}(P_{e}, τ^{−1}N_{λ})
= dim Hom_{kQ}(P_{e}, M_{λ})
= dim M_{λ}(e)
= 1.

Thus, according to [CK06],

X_{N_{λ}}X_{S_{e}} = X_{B_{λ}} + X_{B′_{λ}}

where B_{λ}, B′_{λ} are the (unique) objects such that there exist non-split triangles

N_{λ}⟶ B_{λ}⟶ S_{e}⟶ N_{λ}[1] and S_{e}⟶ B′_{λ}⟶ N_{λ}⟶ S_{e}[1]

in C_{Q}. Since Ext_{kQ}^{1}(S_{e}, N_{λ}) ≃ k, there exists a unique kQ-module M_{λ} such that there is a non-split short exact sequence

0⟶ N_{λ}⟶ B_{λ}⟶ S_{e}⟶ 0

inducing a non-split triangle N_{λ}⟶ B_{λ}⟶ S_{e}⟶ N_{λ}[1] in C_{Q}. Since B_{λ} is the middle term of a non-split exact sequence starting at a quasi-simple regular kQmodule and ending at a preinjective (and actually injective) module, it follows that B_{λ} is a preinjective kQ-module so that it is identified with a transjective object in C_{Q}.

Also, Hom_{C_{Q}}(N_{λ}, τS_{e}) ≃ Hom_{kQ}(N_{λ}, τS_{e}) ≃ k so that a non-zero morphism f ∈ Hom_{kQ}(N_{λ}, τS_{e}) induces a triangle

τS_{e}[−1]⟶ Ker f ⊕ Coker f[−1]⟶ N_{λ} ⟶^{f} τS_{e}

in D^{b}(mod−kQ). Since the projection functor D^{b}(mod−kQ)⟶ C_{Q} is triangulated, this gives a non-split triangle

S_{e}⟶ B′_{λ}⟶ N_{λ}⟶ S_{e}[1]

in C_{Q}. Since N_{λ} is quasi-simple and f is non-zero, then Ker f is a proper submodule of N_{λ} so that it is a postprojective kQ-module or zero. Also, Coker f is a quotient of the injective module τS_{e} so that it is a preinjective kQ-module or zero. Thus Coker f[−1] is a transjective object in C_{Q} and B′_{λ} is transjective. □

Lemma 3.11. Let Q be an euclidean quiver and λ ∈ ℙ^{1}(k). Assume that e is a sink such that δ_{e} = 1. Set N_{λ} = M_{λ}[−1]. Then the following hold :

(1) dim Ext_{C_{Q}}^{1}(N_{λ}, S_{e}) = 1 ;

(2) X_{N_{λ}}X_{P_{e}} = X_{B_{λ}} +X_{B′_{λ}} where B_{λ} is the (unique) kQ-module such that there exists a short exact sequence

0⟶ S_{e}⟶ B_{λ}⟶ N_{λ}⟶ 0

and B′_{λ} = Ker f⊕Coker f[−1] for any non-zero morphism f ∈ Hom_{C_{Q}}(S_{e}, τN_{λ}) ;

(3) B_{λ} and B′_{λ} are transjective.

Proof. Since e is a sink, we have S_{e} ≃ P_{e} so that

dim Ext_{C_{Q}}^{1}(N_{λ}, S_{e}) = dim Ext_{C_{Q}}^{1}(N_{λ}, P_{e})

= dim Ext_{kQ}^{1}(N_{λ}, P_{e})

= dim Hom_{kQ}(P_{e}, τN_{λ})

= dim Hom_{kQ}(P_{e}, M_{λ})

= dim M_{λ}(e)

= 1

The rest follows as for Lemma 3.10. □

3.4. Proof of Theorem 1.3. If order to prove Theorem 1.3, we need to prove that if f is a cluster frieze on Γ(C_{Q}) such that f(P_{i}[1]) = u_{i} for any i ∈ Q_{0}, then f(M) = X_{M} for any indecomposable rigid object M in C_{Q}. It is known that an indecomposable rigid object M is either transjective or belongs to an exceptional tube.

If M is transjective, the result follows from Corollary 3.2. We thus assume that M belongs to a tube T_{λ} for some λ ∈ ℙ^{E}. According to Corollary 3.6, since X_{?} and f induce friezes on T_{λ}, it is enough to prove that X_{?} and f coincide on the quasi-simple modules at the mouth of T_{λ}. In other words, it is enough to prove that X_{N_{λ}[k]} = f(N_{λ}[k]) for any k ∈ {0, …, p_{λ} − 1}. According to Lemmas 3.10 and 3.11, we have

X_{N_{λ}} = \frac{X_{B_{λ}} + X_{B′_{λ}}}{X_{S_{e}}}

so that applying Lemma 3.8, we get

X_{N_{λ}[k]} = \frac{X_{B_{λ}[k]} + X_{B′_{λ}[k]}}{X_{S_{e}[k]}}

for any k ∈ {0, …, p_{λ} − 1}.

Since S_{e}[k], B_{λ}[k] and B′_{λ}[k] are transjective, we have X_{S_{e}[k]} = f(S_{e}[k]), X_{B_{λ}[k]} = f(B_{λ}[k]) and X_{B′_{λ}[k]} = f(B′_{λ}[k]) for any k ∈ {0, …, p_{λ} − 1}. Thus, since f is a cluster frieze, then for any k ∈ {0, …, p_{λ} − 1}, we get

f(N_{λ}[k]) = \frac{f(B_{λ}[k]) + f(B′_{λ}[k])}{f(S_{e}[k])}

= \frac{X_{B_{λ}[k]} + X_{B′_{λ}[k]}}{X_{S_{e}[k]}}

= X_{N_{λ}[k]}

and Theorem 1.3 is proved. □

3.5. Proof of Theorem 1.4. If order to prove Theorem 1.3, we need to prove that if f is a strong cluster frieze on Γ(C_{Q}) such that f(P_{i}[1]) = u_{i} for any i ∈ Q_{0}, then f(M) = X_{M} for any indecomposable object M in C_{Q}.

Since f is a strong cluster frieze, it is in particular a cluster frieze and it thus follows that f(M) = X_{M} for any object M in the transjective component or in an exceptional tube of Γ(C_{Q}). It is thus sufficient to prove that f and X_{?} coincide on homogeneous tubes. According to Corollary 3.6, since X_{?} and f induce friezes on tubes, it is enough to prove that X_{?} and f coincide on the mouths of homogeneous tubes. Fix thus λ ∈ ℙ^{H}. It follows from Lemmas 3.10 and 3.11 that

X_{N_{λ}} = \frac{X_{B_{λ}} + X_{B′_{λ}}}{X_{S_{e}}} .

As before, since S_{e}, B_{λ} and B′_{λ} are transjective, we have X_{S_{e}} = f(S_{e}), X_{B_{λ}} = f(B_{λ}) and X_{B′_{λ}} = f(B′_{λ}). Moreover, since f is a strong cluster frieze, we have

f(N_{λ}) = \frac{f(B_{λ}) + f(B′_{λ})}{f(S_{e})}

= \frac{X_{B_{λ}} + X_{B′_{λ}}}{X_{S_{e}}}

= X_{N_{λ}} which proves Theorem 1.4. □

4. Euler characteristics of complete quiver grassmannians

Given an acyclic quiver Q, we construct ℤ-friezes on Γ(C_{Q}) and see that they allow to compute algorithmically the Euler characteristics of the following projective varieties.

Definition 4.1. Let Q be an acyclic quiver and M be a kQ-module. The complete quiver grassmannian is

Gr(M) = ⨆_{e∈ℤ_{≥0}^{Q_{0}}} Gr_{e}(M).

Since Gr_{e}(M) is empty for all but finitely many e ∈ ℤ_{≥0}^{Q_{0}}, then Gr(M) is a finite union of projective varieties and is thus a projective variety. We denote by χ(Gr(M)) its Euler characteristic.

Let ev be the ring homomorphism

ev : \{ \begin{matrix}ℤ[u^{±1}] & ⟶ & ℤ, \\ u_{i} & ↦ & 1 & for any i ∈ Q_{0}. \end{matrix}

Lemma 4.2. Let Q be an acyclic quiver, then for any kQ-module M,

ev(X_{M}) = χ(Gr(M)).

Proof. Let M be a kQ-module. Then

ev(X_{M}) = ∑_{e∈ℤ_{≥0}^{Q_{0}}} χ(Gr_{e}(M))ev(u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim M−e⟩})

= ∑_{e∈ℤ_{≥0}^{Q_{0}}} χ(Gr_{e}(M))

= χ( ⨆_{e∈ℤ_{≥0}^{Q_{0}}} Gr_{e}(M))

= χ(Gr(M)).

□

Corollary 4.3. Let Q be an acyclic quiver and f be a ℤ-frieze on Γ(C_{Q}) such that f(P_{i}[1]) = 1 for any i ∈ Q_{0}. Then f(M) = χ(Gr(M)) for any transjective object M.

In particular, χ(Gr(M)) > 0 for any postprojective or preinjective kQ-module M.

Proof. It follows from Proposition 2.2 that X_{?} induces a frieze on Γ(C_{Q}). Since ev : ℤ[u^{±1}]⟶ ℤ is a ring homomorphism, it follows that ev ∘ X_{?} is a ℤ-frieze on Γ(C_{Q}) such that (ev ∘ X_{?})(P_{i}[1]) = 1 for any i ∈ Q_{0}. Thus, f and ev ∘ X_{?} coincide on a section in the transjective component P. By Lemma 3.1 the friezes coincide on P. It thus follows from Lemma 4.2 that f(M) = χ(Gr(M)) for any transjective object M. An easy induction on the transjective component proves that f(M) > 0 for any transjective object M. □

Corollary 4.4. Let C_{Q} be the cluster category of an euclidean quiver Q and let f be a cluster ℤ-frieze on Γ(C_{Q}) such that f(P_{i}[1]) = 1 for any i ∈ Q_{0}. Then f(M) = χ(Gr(M)) for any rigid kQ-module M.

Proof. As above, f and M ↦ χ(Gr(M)) coincide on the transjective component. Now, as for Theorem 1.3, it is easy to check that if f is a ℤ-frieze on Γ(C_{Q}) such that f(P_{i}[1]) = 1 for any i ∈ Q_{0}, then f coincides with ev ∘ X_{?} on every object contained either in the transjective component or in an exceptional tube. In particular, f(M) = ev(X_{M}) = χ(Gr(M)) for any rigid kQ-module M. □

Similarly, we obtain :

Corollary 4.5. Let C_{Q} be the cluster category of an euclidean quiver Q. Let f be a strong cluster ℤ-frieze on Γ(C_{Q}) such that f(P_{i}[1]) = 1 for any i ∈ Q_{0}. Then f(M) = χ(Gr(M)) for any kQ-module M. □

5. Explicit determination of B_{λ} and B′_{λ}

Let Q be an euclidean quiver. Once defined, the cluster frieze provides an efficient tool for computations in the cluster algebra A(Q). Nevertheless, the definition of a cluster frieze on Γ(C_{Q}) requires the determination of the pair (B_{λ}, B′_{λ}) of objects arising as middle terms of triangles with extreme terms N_{λ} and S_{e}. This determination is the only obstruction for getting a complete algorithm for computing cluster variables. The aim of this section is to explicit these objects for arbitrary euclidean quivers. This is based on a case-by-case analysis.

5.1. Homogeneous tubes. We first prove that for any euclidean quiver Q and any parameter λ ∈ ℙ^{H} of an homogeneous tube, the transjective objects B_{λ} and B′_{λ} do not depend on the choice of λ.

Lemma 5.1. Let Q be an euclidean quiver. Assume that e is a sink. Then, for any λ ∈ ℙ^{H}, the following hold :

(1) B_{λ} is the unique (up to isomorphism) indecomposable module of dimension vector δ + dim S_{e} ;

(2) B′_{λ} ≃ C[−1] where C is the unique (up to isomorphism) indecomposable module of dimension vector δ − dim S_{e}.

Proof. According to Lemma 3.11, B_{λ} is the only module such that there exists a non-split exact sequence

0⟶ S_{e}⟶ B_{λ}⟶ N_{λ}⟶ 0

and B_{λ} is postprojective.

We denote by ∂_{X} = ⟨δ, dim X⟩ the defect of a representation X of Q (see e.g.

[DR76] for details). Since δ_{e} = 1, we have ∂_{S_{e}} = ∂_{P_{e}} = −1. Applying this form to the short exact sequence above, we get ∂_{B_{λ}} = ∂_{S_{e}} = −1 so that B_{λ} is an indecomposable postprojective module. Up to isomorphism, it is thus determined by its dimension vector which is δ + dim S_{e}.

Now, according to Lemma 3.11, B′_{λ} is isomorphic to Ker f ⊕ Coker f[−1] for any non-zero morphism f ∈ Hom_{kQ}(S_{e}, τN_{λ}). Since S_{e} is simple, then Ker f is zero and thus B′_{λ} ≃ C[−1] where C = Coker f. Applying the defect form to the short exact sequence

0⟶ S_{e} ⟶^{f} τN_{λ}⟶ C⟶ 0

we get ∂_{C} = −∂_{S_{e}} = 1. Since C is preinjective, it is indecomposable and thus entirely determined by its dimension vector which is δ − dim S_{e}. □

We put a partial order ≤ on ℤ^{Q_{0}} by setting (e_{i})_{i∈Q_{0}} ≤ (f_{i})_{i∈Q_{0}} if e_{i} ≤ f_{i} for every i ∈ Q_{0}.

Lemma 5.2. Let Q be an euclidean quiver. Assume that e is a source. Then, for any λ ∈ ℙ^{H}, the following hold :

(1) B_{λ} is the unique (up to isomorphism) indecomposable module of dimension vector δ + dim S_{e} ;

(2) if dim (τS_{e}) ≤ δ, then B′_{λ} is the unique (up to isomorphism) indecomposable module of dimension vector δ − dim (τS_{e}) ;

(3) if δ ≤ dim (τS_{e}), then B′_{λ} ≃ C[−1] where C is the unique (up to isomorphism) indecomposable module of dimension vector dim (τS_{e}) − δ.

Proof. According to Lemma 3.10, B_{λ} is the only module such that there exists a non-split exact sequence

0⟶ N_{λ}⟶ B_{λ}⟶ S_{e}⟶ 0.

Since N_{λ} is quasi-simple and S_{e} is injective, then B_{λ} is preinjective. Applying the defect form, we get ∂_{B_{λ}} = ∂_{S_{e}} = 1 so that B_{λ} is indecomposable preinjective. Up to isomorphism, it is thus determined by its dimension vector which is δ + dim S_{e}.

Now, according to Lemma 3.10, B′_{λ} is isomorphic to Ker f ⊕ Coker f[−1] for any non-zero morphism f ∈ Hom_{kQ}(N_{λ}, τS_{e}). We set K = Ker f and C = Coker f. Since N_{λ} is quasi-simple, then K is necessarily postprojective and C is necessarily preinjective. Applying the defect form to the exact sequence

0⟶ K⟶ N_{λ} ⟶^{f} τS_{e}⟶ C⟶ 0

we get ∂_{K} = 1 + ∂_{C}. Since ∂_{K} ≤ 0 and ∂_{C} ≥ 0, we get that either K = 0 and C is indecomposable or C = 0 and K is indecomposable.

If C = 0, then f is an epimorphism and thus dim (τS_{e}) ≤ δ. In this case, B′_{λ} = K is entirely determined by its dimension vector which is δ − dim (τS_{e}).

If K = 0, then f is a monomorphism so that δ ≤ dim (τS_{e}). In this case C is entirely determined by its dimension vector which is dim (τS_{e}) − δ and B′_{λ} = C[−1]. □

Corollary 5.3. Let Q be an euclidean quiver. Then for any λ, μ ∈ ℙ^{H}, we have

B_{λ} ≃ B_{μ} and B′_{λ} ≃ B′_{μ}.

□

5.2. Exceptional tubes in type 𝔸̃. Let Q be an euclidean quiver of type 𝔸̃_{r,s}, that is, with r arrows going clockwise and s arrows going counter-clockwise. The Auslander-Reiten-quiver Γ(C_{Q}) contains two exceptional tubes T_{0} and T_{1} of respective ranks r and s.

Since Q is acyclic it contains at least one sink and at least one source. Moreover, since δ_{i} = 1, for any i ∈ Q_{0}, in this case, the point e can be any of these sinks or sources. In order to fix notations, we consider the case where e is a sink.

There exists thus a unique clockwise (or counter-clockwise, respectively) arrow in Q_{1} with target e. We denote by i_{+} (or i_{−}, respectively) the source of this unique clockwise (or counter-clockwise, respectively) arrow. Locally, Q can be depicted as follows :

[figure: ⋯; ⋯; i_{+}; e.; ⋯; ⋯; i_{−}]

We denote by M_{0} (or M_{1}) the quasi-simple module in T_{0} (or T_{1}, respectively) given by Lemma 3.9.

It is not hard to prove (see for instance [Dup08, §9.6]) that M_{0} and M_{1} are given by :

\xymatrix{ & ⋯ & 0 \ar[l] \ar[r] & ⋯ \ar[r] & 0 \ar[dr] \\ M_{0} : &  &  &  &  & k \\  & ⋯ 0 & k \ar[l]^{0} \ar[r]_{1} & ⋯ \ar[r]_{1} & k \ar[ur]_{1}}

\xymatrix{ & ⋯ 0 & k \ar[l]_{0} \ar[r]^{1} & ⋯ \ar[r]^{1} & k \ar[dr]^{1} \\ M_{1} : &  &  &  &  & k. \\  & ⋯ & 0 \ar[l] \ar[r] & ⋯ \ar[r] & 0 \ar[ur]}

The set {τ^{k}M_{0}|0 ≤ k ≤ r − 1} is a complete list of pairwise non-isomorphic quasi-simple modules in T_{0} and the set {τ^{k}M_{1}|0 ≤ k ≤ s − 1} is a complete list of pairwise non-isomorphic quasi-simple modules in T_{1}. Since we considered a sink e, we set N_{0} = M_{0}[−1] and N_{1} = M_{1}[−1]. We easily check that N_{0} ≃ P_{i+}/S_{e}. We have a non-split extension of kQ-modules

0⟶ S_{e}⟶ P_{i_{+}}⟶ N_{0}⟶ 0.

Moreover, if f is a non-zero morphism S_{e}⟶ τN_{0} ≃ M_{0}, then Ker f = 0 since S_{e} is simple and Coker f ≃ M_{0}/S_{e} = I_{i_{−}}. Thus, we get

B_{0} ≃ P_{i_{+}} and B′_{0} ≃ I_{i_{−}}[−1] ≃ P_{i_{−}}[1].

Similarly,

B_{1} ≃ P_{i_{−}} and B′_{1} ≃ I_{i_{+}}[−1] ≃ P_{i_{+}}[1].

For λ ∈ ℙ^{H}, it follows from Lemma 5.1 that B_{λ} is the unique indecomposable postprojective module E with dimension vector δ+dim S_{e} and B′_{λ} ≃ C[−1] where C is the unique indecomposable preinjective module with dimension vector δ−dim S_{e}.

Summing up in a table, we get :

\begin{matrix}λ ∈ ℙ^{1}(k) & p_{λ} & B_{λ} & B′_{λ} \\ 0 & r & P_{i_{+}} & P_{i_{−}}[1] \\ 1 & s & P_{i_{−}} & P_{i_{+}}[1] \\ λ ∈ ℙ^{H} & 1 & E & C[−1] \end{matrix}

Table 1. B_{λ} and B′_{λ} in type 𝔸̃_{r,s}

5.3. Exceptional tubes in type 𝔻̃. We now assume that Q is an euclidean quiver of type 𝔻̃_{n+3} with n ≥ 1. We moreover assume that Q is equipped with the orientation in the tables of [DR76]. That is

\xymatrix{ & a_{1} \ar[dr] &  &  &  & b_{1} \\ Q : &  & c_{1} \ar[r] & ⋯ \ar[r] & c_{n} \ar[dr] \ar[ur] \\  & a_{2} \ar[ur] &  &  &  & b_{2}}

We set e = b_{1} which is a sink.

There are three exceptional tubes T_{1}, T_{∞} and T_{0} of respective ranks n + 1, 2 and 2. We denote by M_{1}, M_{∞} and M_{0} the quasi-simple modules given by Lemma 3.9.

5.3.1. The tube of rank n + 1. The quasi-simple modules in T_{1} are the S_{c_{i}}, i = 1, …, n and the sincere module M_{1} such that M_{1}(i) ≃ k for every i ∈ Q_{0} and M_{1}(α) = 1_{k} for every α ∈ Q_{1}. We have τS_{c_{i}} ≃ S_{c_{i+1}} if i ≠ n and τS_{c_{n}} ≃ M_{1}.

We set N_{1} = M_{1}[−1] ≃ S_{c_{n}}. With the notations of Lemma 3.11, we get

B_{1} ≃ P_{c_{n}}/P_{b_{2}}

and B′_{1} = Ker f ⊕ Coker f[−1] for 0 ≠ f ∈ Hom_{kQ}(S_{b_{1}}, M_{1}). But Ker f = 0 since P_{b_{1}} ≃ is simple to that

B′_{1} = Coker f[−1] ≃ I_{b_{2}}[−1] ≃ P_{b_{2}}[1].

5.3.2. Tubes of rank 2. Let T_{0} be the tube whose quasi simples are

\xymatrix{ & 0 \ar[dr] &  &  &  & k \\ M_{0} = &  & k \ar[r] & ⋯ \ar[r] & k \ar[dr] \ar[ur] \\  & k \ar[ur] &  &  &  & 0}

\xymatrix{ & k \ar[dr] &  &  &  & 0 \\ N_{0} = &  & k \ar[r] & ⋯ \ar[r] & k \ar[dr] \ar[ur] \\  & 0 \ar[ur] &  &  &  & k}

With the notations of Lemma 3.11, we get

B_{0} ≃ P_{a_{1}}

and B′_{0} = Ker f ⊕ Coker f[−1] for 0 ≠ f ∈ Hom_{kQ}(S_{b_{1}}, M_{0}). Since S_{b_{1}} is simple, Ker f = 0 and thus Coker f is the representation

\xymatrix{0 \ar[dr] &  &  &  & 0 \\  & k \ar[r] & ⋯ \ar[r] & k \ar[dr] \ar[ur] \\ k \ar[ur] &  &  &  & 0.}

In particular, Coker f is indecomposable preinjective so that it is determined by its dimension vector. An easy induction on n shows that dim Coker f = dim τ^{n}I_{a_{2}} so that Coker f ≃ τ^{n}I_{a_{2}} ≃ I_{a_{2}}[n] and thus

B′_{0} ≃ I_{a_{2}}[n − 1].

For T_{∞}, we notice that the group ℤ/2ℤ acts by automorphisms on C_{Q} by exchanging points a_{1} and a_{2}. This action preserves the point e = b_{1} so that

B_{∞} ≃ P_{a_{2}} and B′_{∞} ≃ I_{a_{1}}[n − 1].

For λ ∈ ℙ^{H}, it follows from Lemma 5.1 that B_{λ} is the unique indecomposable postprojective module E with dimension vector δ+dim S_{e} and B′_{λ} ≃ C[−1] where C is the unique indecomposable preinjective module with dimension vector δ−dim S_{e}.

Summing up in a table, we get :

\begin{matrix}λ ∈ ℙ^{1}(k) & p_{λ} & B_{λ} & B′_{λ} & e \\ 1 & n + 1 & P_{c_{n}}/P_{b_{2}} & I_{b_{2}}[−1] & e = b_{1} \\ 0 & 2 & P_{a_{1}} & I_{a_{2}}[n − 1] & e = b_{1} \\ ∞ & 2 & P_{a_{2}} & I_{a_{1}}[n − 1] & e = b_{1} \\ λ ∈ ℙ^{H} & 1 & E & C[−1] & e = b_{1} \end{matrix}

Table 2. B_{λ} and B′_{λ} in type 𝔻̃_{n+3}, n ≥ 1

5.4. Exceptional tubes in type 𝔼̃_{6}. Let Q be the quiver of euclidean type 𝔼̃_{6} with the orientation considered in the tables of [DR76].

\xymatrix{ &  &  & 5 \ar[d] \\  &  &  & 4 \ar[d] \\ Q : & 3 \ar[r] & 2 \ar[r] & 1 & 6 \ar[l] & 7 \ar[l]}

There are three exceptional tubes T_{∞}, T_{0}, T_{1} of respective ranks 2, 3 and 3. We set e = 7 which is a source. Note that I_{e} ≃ S_{e} = S_{7} and that τS_{e} ≃ S_{6}.

For any λ ∈ ℙ^{E}, we denote by M_{λ} the quasi-simple module given in Lemma 3.9 and set N_{λ} ≃ M_{λ}[1]. We denote by B_{λ} and B′_{λ} the unique objects in C_{Q}, given by Lemma 3.10 such that there is a non-split short exact sequence

0⟶ N_{λ}⟶ B_{λ}⟶ S_{e}⟶ 0

and B′_{λ} = Ker f ⊕ Coker f[−1] for some non-zero morphism f ∈ Hom_{kQ}(N_{λ}, τS_{e}). Since τS_{e} is simple, B′_{λ} = Ker f. As in the proof of Lemma 5.2, it is easy to see that B_{λ} and B′_{λ} are determined by their dimension vectors.

The following table sums up the dimension vectors for T_{0} and T_{∞} :

λ ∈ ℙ^{1}(k) dim N_{λ} dim B_{λ} dim B′_{λ}
∞  [1101010] [1101011] [1101000]
0  [1110010] [1101011] [1110000]

Now it is possible, using for example the algorithms provided in Section 6.1, to compute explicitly the indecomposable modules having these dimension vectors. We get

B_{∞} ≃ τ^{3}I_{7} and B′_{∞} ≃ τ^{−1}P_{7}

and

B_{0} ≃ τ^{2}I_{5} and B′_{0} ≃ P_{3}.

For T_{1}, we deduce it from T_{0} using the fact that the action of the product of transpositions (2, 6)(3, 7) on Q_{0} induces an action on C_{Q} by automorphisms. Since this action exchanges 7 and 3, in this case, we have to take e = 3 and we get

B_{1} ≃ τ^{2}I_{5} and B′_{1} ≃ P_{7}.

For λ ∈ ℙ^{H} and e = 7, we can apply Lemma 5.2 and see that B_{λ} is the unique indecomposable (preinjective) module with dimension vector [3212122] and B′_{λ} is the unique indecomposable (preprojective) module with dimension vector [3212111]. Applying the recognition algorithm given in Section 6.1, we get

B_{λ} ≃ τ^{6}I_{7} and B′_{λ} ≃ τ^{−4}P_{7}.

Summing up in a table, we get

\begin{matrix}λ ∈ ℙ^{1}(k) & p_{λ} & B_{λ} & B′_{λ} & e \\ 0 & 2 & τ^{3}I_{7} & τ^{−1}P_{7} & e = 7 \\ 1 & 3 & τ^{2}I_{5} & P_{3} & e = 7 \\ ∞ & 3 & τ^{2}I_{5} & P_{7} & e = 3 \\ λ ∈ ℙ^{H} & 1 & τ^{7}I_{7} & τ^{−4}P_{7} & e = 7 \end{matrix}

Table 3. B_{λ} and B′_{λ} in type 𝔼̃_{6}

Note that an explicit computations of the corresponding variables is given in Section 7.

5.5. Exceptional tubes in type 𝔼̃_{7}. Let Q be the quiver of euclidean type 𝔼̃_{7} with the orientation considered in [DR76].

\xymatrix{ &  &  &  & 5 \ar[d] \\ Q : & 4 \ar[r] & 3 \ar[r] & 2 \ar[r] & 1 & 6 \ar[l] & 7 \ar[l] & 8 \ar[l]}

There are three exceptional tubes T_{∞}, T_{0}, T_{1} of respective ranks 2,3 and 4. We consider e = 8 which is a source. We have I_{e} ≃ S_{e} = S_{8} and τS_{e} ≃ S_{7}. As for type 𝔼̃_{6}, we get the following table :

\begin{matrix}λ ∈ ℙ^{1}(k) & p_{λ} & B_{λ} & B′_{λ} & e \\ ∞ & 2 & τ^{6}I_{4} & τ^{−4}P_{4} & e = 8 \\ 0 & 3 & τ^{4}I_{8} & τ^{−2}P_{8} & e = 8 \\ 1 & 4 & τ^{3}I_{4} & τ^{−1}P_{4} & e = 8 \\ λ ∈ ℙ^{H} & 1 & τ^{12}I_{8} & τ^{−10}P_{8} & e = 8 \end{matrix}

Table 4. B_{λ} and B′_{λ} in type 𝔼̃_{7}

5.6. Exceptional tubes in type 𝔼̃_{8}. Let Q be the quiver of euclidean type 𝔼̃_{8} with the canonical orientation

\xymatrix{ &  &  & 4 \ar[d] \\ Q : & 3 \ar[r] & 2 \ar[r] & 1 & 5 \ar[l] & 6 \ar[l] & 7 \ar[l] & 8 \ar[l] & 9 \ar[l]}

There are three exceptional tubes T_{∞}, T_{0}, T_{1} of respective ranks 2,3 and 5. We take e = 9 which is a source. We have I_{e} ≃ S_{e} = S_{9} and τS_{e} ≃ S_{8}. As before, we get :

\begin{matrix}λ ∈ ℙ^{1}(k) & p_{λ} & B_{λ} & B′_{λ} \\ ∞ & 2 & τ^{15}I_{9} & τ^{−13}P_{9} \\ 0 & 3 & τ^{10}I_{9} & τ^{−8}P_{9} \\ 1 & 5 & τ^{6}I_{9} & τ^{−4}P_{9} \\ λ ∈ ℙ^{H} & 1 & τ^{30}I_{9} & τ^{−28}P_{9} \end{matrix}

Table 5. B_{λ} and B′_{λ} in type 𝔼̃_{8}

5.7. Changing orientations. For simplicity, we worked with prescribed orientations, except for 𝔸̃. Nevertheless, our techniques can actually be adapted to any orientation.

A first method would consist in doing again the computations made earlier. Indeed, we worked with a fixed orientation in order to use the tables provided in [DR76] but there is no theoretical obstruction for doing all the computations with a different orientation.

A second method consists in observing that the strong isomorphism of cluster algebras (in the sense of [FZ03]) corresponding to the orientation change can be realised combinatorially. We detail this method here.

Let Q′ be an euclidean quiver equipped with an arbitrary orientation. The cluster algebra A(Q′, x) contains a seed (Q, u) such that Q is an euclidean quiver of the same euclidean type as Q′ equipped with one of the orientations considered above. The cluster categories C_{Q} and C_{Q′} are equivalent and in particular, have the same Auslander-Reiten quiver Γ.

We denote by P′_{i}, with i ∈ Q′_{0}, the indecomposable projective kQ′-modules and by P_{i}, with i ∈ Q_{0}, the indecomposable projective kQ-modules. We denote by X_{?} : Ob(C_{Q})⟶ ℤ[u^{±1}] (or X′_{?} : Ob(C_{Q′})⟶ ℤ[x^{±1}] the cluster character associated to Q (or Q′, respectively) and we set f : Γ⟶ ℤ[u^{±1}] the cluster frieze on Γ sending P_{i}[1] to u_{i} for every i ∈ Q_{0}.

Let ϕ : A(Q, u)⟶ A(Q′, x) be the strong isomorphism of cluster algebras. According to Theorem 1.3, the cluster variables in A(Q, u) are the f(M) where M runs over the points of Γ corresponding to indecomposable rigid objects in C_{Q}. Let v be such a point in Γ. Then v corresponds to some indecomposable object M in C_{Q} and to Φ(M) in C_{Q′}. It follows from [Pal08, §5] that

X′_{M} = ϕ(X_{Φ(M)}) = ϕ(f(Φ(M))).

Now, in order to compute cluster variables in A(Q′, x), it is enough to compute first the cluster frieze f on Γ and then to replace each u_{i} by its Laurent expansion in the cluster x. The first part can be done algorithmically using Theorem 1.3 and in order to find the expansion of u_{i} in x, one simply constructs the frieze f′ on P such that f′(P′_{i}[1]) = x_{i} for any i ∈ Q_{0} and then replace u_{i} by f′(P_{i}[1]) in the values of the frieze f.

6. Algorithms

6.1. Recognising indecomposable transjective modules. We now detail the algorithms used in Section 5 in order to recognise indecomposable modules in the transjective component from their dimension vectors. The algorithm is elementary and actually works for any acyclic quiver.

Let Q be an acyclic quiver. We assume that we know explicitly the dimension vectors of the indecomposable projective kQ-modules and of the indecomposable injective kQ-modules (this can be achieved with an elementary computer program).

Let C be the Cartan matrix of kQ, that is the matrix whose i-th column is dim P_{i} or, equivalently whose i-th line is dim I_{i} (see for instance [ASS05, §III]). Let Φ = −C^{t}C^{−1} be the Coxeter matrix of kQ. Note that if Q is an euclidean quiver equipped with one of the orientations provided in Section 5, the explicit Coxeter matrix can be found in [SS07].

Then for any i ∈ Q_{0}, n ≥ 0, we have

dim τ^{−n}P_{i} = Φ^{−n}dim P_{i} ;

dim τ^{n}I_{i} = Φ^{n}dim P_{i}.

Thus, if M is an indecomposable postprojective module whose dimension vector is known, one can apply the following procedure.

Algorithm 1 An algorithm for recognising indecomposable postprojective modules Require: M is indecomposable postprojective 1: n = 0 2: while f = 0 do 3: for i ∈ Q_{0} do 4: if dim M = Φ^{−n}dim P_{i} then 5: R := τ^{−n}P_{i} 6: f := 1 7: end if 8: end for 9: n := n + 1 10: end while 11: return R // The module M is thus isomorphic to R.

The algorithm is dual for indecomposable preinjective modules.

6.2. Computing cluster characters in the euclidean case. Let Q be an euclidean quiver. In this section, we give algorithms for computing cluster characters associated to arbitrary indecomposable kQ-modules. As usual, for any λ ∈ ℙ^{1}(k), M_{λ} is the quasi-simple module in the tube T_{λ} given by Lemma 3.9 and p_{λ} is the rank of T_{λ}.

First, we see that cluster characters associated to indecomposable postprojective modules can be computed recursively :

Algorithm 2 Recursive computation for postprojective modules Require: n ≥ 0, i ∈ Q_{0} 1: for j ∈ Q_{0} do 2: X_{P_{j}[1]} := u_{j} 3: end for

4:

X_{P_{i}[−n]} := \frac{∏_{(j,m)⟶ (i,−n)} X_{P_{j}[m]}}{X_{P_{i}[−n+1]}}

5: return X_{P_{i}[−n]}

The algorithm is dual for indecomposable preinjective modules. Now that we are able to compute algorithmically characters associated to post-

projective and preinjective modules, we give an algorithm to compute characters associated to indecomposable regular modules.

Algorithm 3 Computation of X_{M_{λ}[k]^{(l)}} for l ≥ 1, k ∈ [0, p_{λ} − 1] and λ ∈ ℙ^{1}(k)

Require: λ ∈ ℙ^{1}(k), M ≃ M_{λ}[k]^{(l)} with l ≥ 1 and k ∈ [0, p_{λ} − 1] 1: for k ∈ [0, p_{λ} − 1] do

2:

X_{M_{λ}[k]} := \frac{X_{B_{λ}[k]} + X_{B′_{λ}[k]}}{X_{S_{e}[k]}}

// Characters arising in the right-hand side can be computed using Algorithm 2 and its dual version for preinjective modules

3: end for 4: X_{M} := P_{l}(X_{M_{λ}[(k mod p_{λ})]}, …, X_{M_{λ}[((k+l−1) mod p_{λ})]}) 5: return X_{M}

7. An example : Regular cluster variables in type 𝔼̃_{6}

Let Q be the euclidean quiver of type 𝔼̃_{6} considered in Section 5.4. For any λ ∈ ℙ^{1}(k), we provide an explicit expansion formula of the cluster variable corresponding to regular modules at the mouths of the exceptional tubes.

In the tube T_{0} of rank two, we have :

X_{N_{0}} = \frac{1}{u_{1}u_{2}u_{4}u_{6}} (u_{2}u_{3}u_{4}u_{5}u_{6}u_{7} + u_{1}^{3} + u_{1}^{2}u_{3} + u_{1}^{2}u_{5}
+ u_{1}u_{3}u_{5} + u_{1}^{2}u_{7} + u_{1}u_{3}u_{7} + u_{1}u_{5}u_{7} + u_{3}u_{5}u_{7})

and

X_{N_{0}[1]} = \frac{1}{u_{1}^{2}u_{2}u_{3}u_{4}u_{5}u_{6}u_{7}} (u_{2}^{2}u_{3}u_{4}^{2}u_{5}u_{6}^{2}u_{7}

+ u_{1}u_{2}u_{3}u_{4}u_{5}u_{6}^{2} + u_{1}u_{2}u_{3}u_{4}^{2}u_{6}u_{7} + u_{1}u_{2}^{2}u_{4}u_{5}u_{6}u_{7}

+ u_{1}^{3}u_{2}u_{4}u_{6} + u_{1}u_{2}u_{3}u_{4}u_{5}u_{6} + u_{1}u_{2}u_{3}u_{4}u_{6}u_{7} + u_{1}u_{2}u_{4}u_{5}u_{6}u_{7}

+ 2u_{2}u_{3}u_{4}u_{5}u_{6}u_{7} + u_{1}^{3}u_{2}u_{4} + u_{1}^{3}u_{2}u_{6} + u_{1}^{3}u_{4}u_{6} + u_{1}^{2}u_{3}u_{4}u_{6}

+ u_{1}^{2}u_{2}u_{5}u_{6} + u_{1}^{2}u_{2}u_{4}u_{7} + u_{1}^{3}u_{2} + u_{1}^{3}u_{4} + u_{1}^{2}u_{3}u_{4} + u_{1}^{2}u_{2}u_{5}

+ u_{1}^{3}u_{6} + u_{1}^{2}u_{3}u_{6} + u_{1}^{2}u_{5}u_{6} + u_{1}u_{3}u_{5}u_{6} + u_{1}^{2}u_{2}u_{7} + u_{1}^{2}u_{4}u_{7}

+ u_{1}u_{3}u_{4}u_{7} + u_{1}u_{2}u_{5}u_{7} + u_{1}^{3} + u_{1}^{2}u_{3} + u_{1}^{2}u_{5} + u_{1}u_{3}u_{5}

+u_{1}^{2}u_{7} + u_{1}u_{3}u_{7} + u_{1}u_{5}u_{7} + u_{3}u_{5}u_{7})

In the tube T_{1} of rank three, we obtain :

X_{N_{1}} = \frac{u_{2}u_{3}u_{4}u_{6}u_{7} + u_{1}^{2}u_{2} + u_{1}u_{2}u_{7} + u_{1}^{2} + u_{1}u_{3} + u_{1}u_{7} + u_{3}u_{7}}{u_{1}u_{2}u_{3}u_{6}}

X_{N_{1}[1]} = \frac{u_{2}u_{3}u_{4}u_{5}u_{6} + u_{1}^{2}u_{4} + u_{1}u_{3}u_{4} + u_{1}^{2} + u_{1}u_{3} + u_{1}u_{5} + u_{3}u_{5}}{u_{1}u_{2}u_{4}u_{5}}

X_{N_{1}[2]} = \frac{u_{2}u_{4}u_{5}u_{6}u_{7} + u_{1}^{2}u_{6} + u_{1}u_{5}u_{6} + u_{1}^{2} + u_{1}u_{5} + u_{1}u_{7} + u_{5}u_{7}}{u_{1}u_{4}u_{6}u_{7}}

In the tube T_{∞} of rank three, we get :

X_{N_{∞}} = \frac{u_{2}u_{3}u_{4}u_{6}u_{7} + u_{1}^{2}u_{6} + u_{1}u_{3}u_{6} + u_{1}^{2} + u_{1}u_{3} + u_{1}u_{7} + u_{3}u_{7}}{u_{1}u_{2}u_{6}u_{7}}

X_{N_{∞}[1]} = \frac{u_{2}u_{4}u_{5}u_{6}u_{7} + u_{1}^{2}u_{4} + u_{1}u_{4}u_{7} + u_{1}^{2} + u_{1}u_{5} + u_{1}u_{7} + u_{5}u_{7}}{u_{1}u_{4}u_{5}u_{6}}

X_{N_{∞}[2]} = \frac{u_{2}u_{3}u_{4}u_{5}u_{6} + u_{1}^{2}u_{2} + u_{1}u_{2}u_{5} + u_{1}^{2} + u_{1}u_{3} + u_{1}u_{5} + u_{3}u_{5}}{u_{1}u_{2}u_{3}u_{4}}

This provides an interesting result in view of Fomin-Zelevinsky’s positivity conjecture :

Proposition 7.1. Let Q be an euclidean quiver of type 𝔼̃_{6} with the orientation considered in Section 5. Then the regular cluster variables of A(Q) belong to ℤ_{≥0}[u^{±1}].

Proof. A regular cluster variable in A(Q) is by definition of the form X_{M} for some indecomposable rigid regular kQ-module M. Such an indecomposable regular rigid is of the form R^{(l)} where R is quasi-simple in an exceptional tube T_{λ} and 0 < l < p_{λ}. If M is quasi-simple, the result holds according to the above expressions. Thus, the only case to consider is when λ ∈ {1, ∞} and l = 2. In this case, the proof is inspired by [Dup09b]. According to the previous expressions, for any quasi-simple module R in T_{0} or T_{∞}, the character X_{R} can be written as

X_{R} = ∑_{i=1}^{n_{R}} L_{R,i}

where L_{R,i} is a (monic) Laurent monomial in the cluster u. Now, we observe that for any R, there exist integers 1 ≤ i_{0}^{R} ≤ n_{R} and 1 ≤ j_{0}^{R} ≤ n_{R[−1]} such that

X_{R[−1]} = \frac{1}{L_{R,i_{0}}} + ∑_{i≠j_{0}^{R}; i=1}^{n_{R[−1]}} L_{R[−1],i}.

These monomials are explicitly listed in the following table.

\begin{matrix}R & L_{R,i_{0}^{R}} \\ N_{1} & \frac{u_{4}u_{7}}{u_{1}} \\ N_{1}[1] & \frac{u_{2}u_{5}}{u_{1}} \\ N_{1}[2] & \frac{u_{3}u_{6}}{u_{1}} \\ N_{∞} & \frac{u_{3}u_{4}}{u_{1}} \\ N_{∞}[1] & \frac{u_{2}u_{7}}{u_{1}} \\ N_{∞}[2] & \frac{u_{5}u_{6}}{u_{1}} \end{matrix}

Thus, we get

X_{R^{(2)}} = P_{2}(X_{R}, X_{R[−1]})

= X_{R}X_{R[−1]} − 1

= (L_{R,i_{0}^{R}} + ∑_{\begin{matrix}i=1 \\ i≠i_{0}^{R} \end{matrix}}^{n_{R}} L_{R,i}) (\frac{1}{L_{R,i_{0}^{R}}} + ∑_{\begin{matrix}i=1 \\ i≠j_{0}^{R} \end{matrix}}^{n_{R[−1]}} L_{R[−1],i}) − 1 ∈ ℤ_{≥0}[u^{±1}].

□

We can similarly compute the character X_{M_{λ}} associated to a quasi-simple module at the mouth of an homogeneous tube but the result is far too long to be presented here. It is a sum of 322 Laurent monomials so that it is in particular an element of ℤ_{≥0}[u^{±1}]. Written as an irreducible Laurent polynomial, its denominator is ∏_{i} u_{i}^{δ_{i}}, accordingly to [CK06, Theorem 3].

8. Euler characteristics of complete grassmannians in type 𝔼̃

In type 𝔼̃_{n} with n = 6, 7, 8, our algorithm allows to compute cluster characters associated to any object in the cluster category. The explicit Laurent expansions of these characters are so long that it would neither be reasonable nor useful to present them here. Nevertheless, our algorithms may be used to compute Euler characteristics of complete quiver grassmannians. In order to so so, it is enough to replace the lines “X_{P_{j}[1]} := u_{j}” by X_{P_{j}[1]} := 1 in Algorithm 2 and its dual.

In this section, we give the complete list of these Euler characteristics for quasisimple modules in tubes of Γ(C_{Q}) when Q is an euclidean quiver of type 𝔼̃_{n} with n = 6, 7, 8 equipped with one of the orientations considered in Section 5.

In the following tables, we use the notations of Section 5. The omitted values correspond to τ-periodic modules for which the value of the Euler characteristic of the complete quiver grassmannian can already be found in the table.

λ 0 1 ∞ λ ∈ ℙ^{H}
p_{λ} 2 3 3  1
N_{λ} 9 7 7  322
τN_{λ} 36 7 7
τ^{2}N_{λ}  7 7

Table 6. Characteristics of complete quiver grassmannians in type 𝔼̃_{6}

λ 0 1 ∞ λ ∈ ℙ^{H}
p_{λ} 3 4 2  1
N_{λ} 10 7 61 3719
τN_{λ} 9 9 61
τ^{2}N_{λ} 42 7
τ^{3}N_{λ}  9

Table 7. Characteristics of complete quiver grassmannians in type 𝔼̃_{7}

λ 0 1 ∞ λ ∈ ℙ^{H}
p_{λ} 3 5 2  1
N_{λ} 88 47 779 403520
τN_{λ} 74 11 518
τ^{2}N_{λ} 62 9
τ^{3}N_{λ}  10
τ^{4}N_{λ}  9

Table 8. Characteristics of complete quiver grassmannians in type 𝔼̃_{8}

Note that the Euler characteristic of the complete quiver grassmannian of a kQ-module M is also equal to the number of Laurent monomials occurring in the Laurent expansion of X_{M} in the initial cluster u so that these numbers give an idea of the complexity of the associated variables.

Acknowledgements

This paper was written while the second author was at the university of Sherbrooke as a CRM-ISM postdoctoral fellow under the supervision of the first author, Thomas Brüstle and Virginie Charette.

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