GENERIC CLUSTER CHARACTERS

G. DUPONT

Abstract. Let C be a Hom-finite triangulated 2-Calabi-Yau category with constructible cones and let T be a cluster-tilting object in C. We introduce a set G^{T} (C) of generic values of the cluster character associated to T parameterized by the Grothendieck group K_{0}(add T). We prove that the set G^{T} (C) naturally contains the cluster monomials of the cluster algebra associated to the Gabriel quiver of the cluster-tilted algebra End_{C}(T)^{op} .

When C = C_{Q} is the cluster category of an acyclic quiver Q and T = kQ is the canonical cluster-tilting object, we prove that G^{T} (C) satisfies multiplicative properties compatible with the virtual generic decomposition introduced by Igusa-Orr-Todorov-Weyman. As a consequence, we prove that this set coincides with the set of generic variables previously introduced by the author in the context of acyclic cluster algebras.

Contents

1. Introduction and main results 1 2. Preliminaries 3 3. Generic cluster characters for 2-Calabi-Yau triangulated categories 5 4. Cluster monomials and generic characters 8 5. Indices and dimension vectors in cluster categories 9 6. Generic cluster characters and virtual generic decomposition 12 7. Generic cluster characters and classical generic variables 13 Acknowledgments 17 References 17

1. Introduction and main results

1.1. Cluster algebras. Cluster algebras were introduced in [FZ02] in order to design a combinatorial framework for studying total positivity in algebraic groups and canonical bases in quantum groups. It is expected, and proved in some cases, that a cluster algebra has distinguished linear bases providing combinatorial models for canonical or semicanonical bases in quantum groups [FZ02, GLS10, Nak09]. Besides this question, cluster algebras have shown interactions with various areas of mathematics like Lie theory, combinatorics, Teichmüller theory, Poisson geometry or quiver representations. The connection with representation theory was particularly fruitful for the construction of linear bases in a large class of cluster algebras [CZ06, CK08, Cer09, Dup08, DXX09, GLS10].

In full generality, a (coefficient-free) cluster algebra can be associated to any pair (Q, x) where Q = (Q_{0}, Q_{1}) is a quiver and x = (x_{i}|i ∈ Q_{0}) is a Q_{0}-tuple of indeterminates over ℤ. By a quiver Q = (Q_{0}, Q_{1}), we always mean an oriented graph such that Q_{0} is a finite set of vertices and Q_{1} is a finite set of arrows. Moreover, we always assume that a quiver Q does not contain any loops or 2cycles. Given such a pair (Q, x), we denote by A(Q, x) the corresponding cluster algebra.

A(Q, x) is a subalgebra of the algebra ℤ[x^{±1}] of Laurent polynomials in the x_{i} with i ∈ Q_{0}. It is equipped with a distinguished set of generators called cluster variables, gathered into possibly overlapping sets of fixed cardinality called clusters, generated by a recursive process called mutation. We denote by Cl(Q, x) the set of cluster variables in A(Q, x). Monomials in variables belonging to a same cluster are called cluster monomials and we denote by M(Q, x) the set of cluster monomials in A(Q, x).

A ℤ-basis in the cluster algebra A(Q, x) is a free generating set of A(Q, x) viewed as a ℤ-module. If A(Q, x) has finitely many cluster variables, the set M(Q, x) of cluster monomials is a ℤ-basis in A(Q, x) [CK08]. It is conjectured, and proved in most cases, that cluster monomials are always linearly independent over ℤ. Nevertheless, it was observed in [SZ04] that they are not sufficient to generate A(Q, x) as a ℤ-module if A(Q, x) has infinitely many cluster variables. The aim of this article is to provide, for a wide class of cluster algebras, a general construction of a distinguished family of elements in ℤ[x^{±1}] containing naturally the cluster monomials in A(Q, x) and which is expected to constitute a ℤ-basis in the cluster algebra A(Q, x).

1.2. Triangulated 2-Calabi-Yau realizations. Given a triangulated 2-Calabi-Yau category C over an algebraically closed field k with a cluster-tilting object T, the set of cluster-tilting objects in C reachable from T is in bijection with the set of cluster variables in A(Q_{T} , x) where Q_{T} is the Gabriel quiver of the algebra End_{C}(T)^{op} [BIRS09] (see Section 2 for details). This bijection can be made explicit using the so-called cluster character

X_{?}^{T} : Ob(C)⟶ ℤ[x^{±1}]

introduced in [Pal08] whose definition is recalled in Section 2.

When C = C_{Q} is the cluster category of an acyclic quiver and T = kQ is the canonical cluster-tilting object, the cluster character X^{T} coincides with the Caldero-Chapoton map CC introduced in [CC06, CK06]. In [Dup08], the author introduced and studied generic values of the Caldero-Chapoton map under the name of generic variables. It is known that these generic variables form a ℤ-basis in the cluster algebra A(Q, x) if Q is a tame quiver [Dup08, DXX09] and a ℂ-basis of the ℂ-vector space ℂ ⊗_{ℤ} A(Q, x) if Q is acyclic [GLS10, Sch09]. In this article, we generalize this construction to a much wider context and conjecture that it still provides a ℤ-linear basis of the corresponding cluster algebra.

1.3. Main results. If k is the field of complex numbers and C has constructible cones in the sense of [Pal09], we associate to any element γ ∈ K_{0}(add T) a Laurent polynomial X(γ) by taking the character of the cone of a generic morphism in End_{C}(T)^{op} -mod with δ-vector γ in the sense of [DF09] (see Section 3 for details). The set

G^{T} (C) = {X(γ)|γ ∈ K_{0}(add T)}

is called the set of generic characters.

Theorem 4.1 states that the set G^{T} (C) of generic characters contains naturally the set of cluster monomials (and thus of cluster variables) in the cluster algebra M(Q_{T} , x).

When C = C_{Q} is the cluster category of an acyclic quiver Q and T = kQ is the canonical cluster-tilting object, we prove that G^{T} (C) satisfies multiplicative properties compatible with the virtual generic decomposition of [IOTW09] (Theorem 6.1).

In this case, we also prove that the set of G^{T} (C) coincides with the set G(Q) of generic variables introduced in [Dup08]. In particular, it provides a linear basis in the acyclic cluster algebra A(Q, x) (Theorem 7.3 and Corollary 7.5).

1.4. Organization of the paper. In Section 2, we recall necessary background concerning 2-Calabi-Yau triangulated realizations of cluster algebras. In Section 3, we define generic cluster characters in full generality and connect this construction to general presentations of modules introduces in [DF09]. In Section 4 we prove that generic cluster characters naturally contain cluster monomials of the corresponding cluster algebra. In the sequel, we focus on the case of cluster categories associated to acyclic quivers. In Section 5, we relate indices and dimension vectors in cluster categories in order to obtain a parameterization of generic characters by their gvectors. In Section 6, we prove the compatibility of generic characters with the virtual generic decomposition of [IOTW09]. Finally, in Section 7, we prove that this construction indeed generalizes the construction of generic variables provided in [Dup08].

2. Preliminaries

2.1. Triangulated 2-Calabi-Yau realizations. Let k be the field of complex numbers. Throughout the paper, C will be a Hom-finite triangulated Krull-Schmidt k-category satisfying the 2-Calabi-Yau property. It means that there is a bifunctorial isomorphism

Hom_{C}(M, N) ≃ DHom_{C}(N, M[2])

for any two objects M, N in C where D = Hom_{k}(−, k) is the standard duality and [1] is the suspension functor in C.

A cluster-tilting object in C is a basic rigid (that is, without self-extension) object T such that Ext_{C}^{1}(T, X) = 0 implies X ∈ add T. We always assume that C contains a cluster-tilting object T = T_{1} ⊕ ⋯ ⊕ T_{n}. We denote by Q_{T} the Gabriel quiver of the algebra B_{T} = End_{C}(T)^{op} and we fix a n-tuple x of indeterminates over ℤ. We denote by T the additive category add T. We recall that a cluster tilting-object T′ in C is called reachable from T if T and T′ are in the same connected component of the cluster-tilting graph of C.

A fundamental example of triangulated 2-Calabi-Yau category is the so-called cluster category C_{Q} of an acyclic quiver Q introduced in [BMR^{+}06] (see also [CCS06] for Dynkin type 𝔸). The path algebra kQ of Q can be identified with a clustertilting object in C_{Q}, called canonical cluster-tilting object in C_{Q} and in this case, every rigid object in C is reachable from kQ.

Given a morphism f ∈ Hom_{C}(M, N) with M, N in C, the cone of f is the unique (up to isomorphism) object Cone (f) in C such that there exists a triangle

M ⟶^{f} N⟶ Cone (f)⟶ M[1].

Throughout the paper, we will assume that the category C has constructible cones in the sense of [Pal09, §1.3]. This situation includes in particular cluster categories, stable categories of Frobenius categories and Amiot generalized cluster categories [Ami10] as shown in [Pal09].

Definition 2.1. A T -morphism in C is a morphism f ∈ Hom_{C}(T_{1}, T_{0}) for some objects T_{0}, T_{1} in T .

We denote by F_{T} the functor

F_{T} = Hom_{C}(T, −) : C⟶ B_{T} -mod .

inducing an equivalence of categories

C/(T [1]) ⟶^{∼} B_{T} -mod

where B_{T} -mod denotes the category of finitely generated left B_{T} -modules. Note that projective B_{T} -modules are given by the F_{T} M where M runs over T .

As shown in [KR08], for any object M in C, there exist (non-unique) triangles

T_{1}^{M}⟶ T_{0}^{M}⟶ M⟶ T_{1}^{M}[1],
M⟶ T_{M}^{0}[2]⟶ T_{M}^{1}[2]⟶ M[1]

with T_{i}^{M}, T_{M}^{i} in T for any i ∈ {1, 2}.

Following [Pal08], the index of M (with respect to T )

ind_{T} (M) = [T_{0}^{M}] − [T_{1}^{M}]

and the coindex of M (with respect to T )

coind_{T} (M) = [T_{M}^{0}] − [T_{M}^{1}]

are well-defined elements in the Grothendieck group K_{0}(T ).

2.2. Cluster characters. Given a B_{T} -module M and e ∈ K_{0}(B_{T} -mod ), we denote by Gr_{e}(M) the variety of sub-B_{T} -modules of M whose class in K_{0}(B_{T} -mod ) is e. This is a projective variety and we denote by χ(Gr_{e}(M)) its Euler characteristic with respect to the simplicial cohomology.

Let ⟨−, −⟩ be the bilinear form on the split Grothendieck group K_{0}(B_{T} -mod )^{split} given by

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

It is well-defined on the Grothendieck group K_{0}(B_{T} -mod ) if B_{T} is hereditary but not in general.

For any i ∈ {1, …, n}, let S_{i} be the simple B_{T} -module associated to i. Then the linear form

⟨S_{i}, −⟩_{a} : M ↦ ⟨S_{i}, M⟩ − ⟨M, S_{i}⟩

is well-defined on K_{0}(B_{T} -mod ) [Pal08, Lemma 1.3].

Definition 2.2 ([Pal08]). The cluster character associated to T is the map

X_{?}^{T} : Ob(C)⟶ ℤ[x^{±1}]

given by

X_{M}^{T} = \{ \begin{matrix}x_{i} if M ≃ T_{i}[1] ; \\ ∑_{e∈K_{0}(B_{T} -mod )} χ(Gr_{e}(F_{T} M)) ∏_{i=1}^{n} x_{i}^{⟨S_{i},e⟩_{a}−⟨S_{i},F_{T} M⟩} otherwise. \end{matrix}

Then, it follows from [Pal08, Corollary 5.4] that

{X_{M}^{T}|M is indecomposable rigid in C and reachable from T } = Cl(Q_{T} , x)

and

{X_{M}^{T}|M is rigid in C and reachable from T } = M(Q_{T} , x).

When C = C_{Q} is the cluster category of an acyclic quiver and T = kQ, the cluster character X_{?}^{T} coincides with the so-called Caldero-Chapoton map introduced in [CC06, CK06].

3. Generic cluster characters for 2-Calabi-Yau triangulated categories

3.1. Generic characters. In [Dup08], we observed that the Caldero-Chapoton map takes generic values on the irreducible components of the set of objects in the corresponding cluster category. For an arbitrary 2-Calabi-Yau category C, there is in general no obvious geometry on Ob(C). However, since C is k-linear, Hom-spaces in C are k-vector spaces and in particular, they are irreducible affine varieties. Thus, for geometric statements, it is more convenient to consider morphisms instead of objects. This philosophy is also clear from the point of view of representation theory of algebras, as shown in [IOTW09, DF09].

Definition 3.1. For any objects T_{0}, T_{1} in T and f ∈ Hom_{C}(T_{1}, T_{0}), we set

X : \{ \begin{matrix}Hom_{C}(T_{1}, T_{0}) & ⟶ & ℤ[x^{±1}] \\ f & ↦ & X_{Cone (f)}^{T} \end{matrix}

The group Aut_{C}(T_{0}) × Aut_{C}(T_{1})^{op} acts on Hom_{C}(T_{1}, T_{0}) by

(g_{0}, g_{1}).f = g_{0}fg_{1}

and the map X is invariant under this action.

If M is a rigid object in C, consider a triangle

T_{1}^{M} ⟶^{f} T_{0}^{M}⟶ M⟶ T_{1}^{M}[1]

with T_{0}^{M}, T_{1}^{M} in T . Then, it follows from [DK08, §2.1] that the orbit of f under this action is a dense open subset in Hom_{C}(T_{1}^{M}, T_{0}^{M}) so that X is constant over a dense open subset of Hom_{C}(T_{1}^{M}, T_{0}^{M}). The following lemma proves that such a dense open subset actually exists for any two objects T_{0}, T_{1} in T .

Lemma 3.2. Let T_{0}, T_{1} be objects in T . Then there exists a Zariski dense open subset U_{(T_{1},T_{0})} ⊂ Hom_{C}(T_{1}, T_{0}) such that X is constant on U_{(T_{1},T_{0})}.

Moreover, if U′_{(T_{1},T_{0})} is another such open dense subset, then the values of X on U_{(T_{1},T_{0})} and U′_{(T_{1},T_{0})} coincide. We denote by X(T_{1}, T_{0}) this common value.

Proof. With the assumptions on the category C, it follows from [Pal09] that the map X : f ↦ X_{Cone (f)}^{T} is constructible on Hom_{C}(T_{1}, T_{0}). Thus, there exists a nonempty open (for the Zariski topology) subset U(T_{0}, T_{1}) ⊂ Hom_{C}(T_{1}, T_{0}) such that X is constant over U(T_{0}, T_{1}). Since Hom_{C}(T_{1}, T_{0}) is an irreducible affine variety, U(T_{0}, T_{1}) is dense and the lemma follows. □

3.2. Stabilization maps and cluster characters. Let T_{0}, T_{1}, T′_{0}, T′_{1} be objects in T . Then [T_{0}]−[T_{1}] = [T′_{0}]−[T′_{1}] in K_{0}(T ) if and only if there exists T, T′ ∈ T such that T_{0}⊕T = T′_{0}⊕T′ and T_{1}⊕T = T′_{1}⊕T′. We define on T a structure of right-filter by setting T′ ≤ T′′ iff there exists T^{(3)} ∈ T such that T′′ = T′⊕T^{(3)}. For any T_{0}, T_{1} in T , if T′ ≤ T′′ in T as above, the stabilization map from Hom_{C}(T_{1} ⊕ T′, T_{0} ⊕ T′) to Hom_{C}(T_{1} ⊕ T′′, T_{0} ⊕ T′′) is

St_{T′,T′′} : \{ \begin{matrix}Hom_{C}(T_{1} ⊕ T′, T_{0} ⊕ T′) & ⟶ & Hom_{C}(T_{1} ⊕ T′′, T_{0} ⊕ T′′) \\ f & ↦ & f ⊕ id_{T^{(3)}}. \end{matrix}

We set St_{T′} = St_{0,T′} the stabilization map Hom_{C}(T_{1}, T_{0})⟶ Hom_{C}(T_{1} ⊕T′, T_{0} ⊕T′) sending f to f ⊕ id_{T′}.

Given an element γ ∈ K_{0}(T ), there exists a unique pair (T_{0}^{min} (γ), T_{1}^{min} (γ)) of objects in T such that

γ = [T_{0}^{min} (γ)] − [T_{1}^{min} (γ)]

and such that T_{0}^{min} (γ) and T_{1}^{min} (γ) have no common direct factor.

Remark 3.3. Let T′, T_{0}, T_{1} be objects in T and f ∈ Hom_{C}(T_{1}, T_{0}). The triangles

T_{1} ⟶^{f} T_{0}⟶ Cone (f)⟶ T_{1}[1] and T′ ⟶^{id_{T′}} T′⟶ 0⟶ T [1]

give rise to

T_{1} ⊕ T′ ⟶^{f⊕id_{T′}} T_{0} ⊕ T′⟶ Cone (f)⟶ T_{1}[1] ⊕ T′[1]

so that Cone (f ⊕ id_{T′}) ≃ Cone (f) and thus X(f ⊕ id_{T′}) = X(f). Thus, X is invariant under stabilization.

Note that if f is generic in Hom_{C}(T_{1}, T_{0}), the image of its Aut(P_{0}) × Aut(P_{1})^{op} - orbit under a stabilization map is a Zariski dense open subset in the image of Hom_{C}(T_{1}, T_{0}) under this stabilization map. Theorem 3.4 proves that the image of f under stabilization is actually generic in Hom_{C}(T_{1} ⊕ T, T_{0} ⊕ T).

Theorem 3.4 (Stability Theorem). Let T_{0}, T_{1} be objects in T and γ = [T_{0}]−[T_{1}] ∈ K_{0}(T ). Then, the generic element in Hom_{C}(T_{1}, T_{0}) is isomorphic to an element in the image of the stabilization map Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ))⟶ Hom_{C}(T_{1}, T_{0}).

Proof. The proof is the same as [IOTW09, Theorem 5.2.2]. We recall it for completeness. Let T_{0}, T_{1} be objects in T and γ = [T_{0}] − [T_{1}] ∈ K_{0}(T ). Then, there exists some T ∈ T such that T_{0} = T_{0}^{min} (γ) ⊕ T and T_{1} = T_{1}^{min} (γ) ⊕ T. Let ϕ ∈ Hom_{C}(T_{1}, T_{0}) be a generic element. We now prove that there exists (g_{0}, g_{1}) ∈ Aut(T_{0} ⊕ T) × Aut(T_{1} ⊕ T)^{op} such that ϕ = (g_{0}, g_{1}).St_{T} (ϕ^{min} ) where ϕ^{min} ∈ Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ)).

The element ϕ ∈ Hom_{C}(T_{1}, T_{0}) can be viewed as a 2 × 2 matrix

ϕ = [ \begin{matrix}f & h \\ g & r \end{matrix} ]

with f ∈ Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ)), g ∈ Hom_{C}(T_{1}^{min} (γ), T), h ∈ Hom_{C}(T, T_{0}^{min} (γ)) and r ∈ End_{C}(T). Since ϕ is generic, r is generic in End_{C}(T) so that it is invertible. Thus, we get

ϕ = [ \begin{matrix}f & h \\ g & r \end{matrix} ] = [ \begin{matrix}1_{T_{0}} & hr^{−1} \\ 0 & 1_{T} \end{matrix} ] [ \begin{matrix}f − hr^{−1}g & 0 \\ 0 & 1_{T} \end{matrix} ] [ \begin{matrix}1_{T_{1}} & 0 \\ g & r \end{matrix} ]

so that

ϕ = (g_{0}, g_{1}) [ \begin{matrix}f − hr^{−1}g & 0 \\ 0 & 1_{T} \end{matrix} ] = (g_{0}, g_{1})St_{T} (ϕ^{min} )

where

g_{0} = [ \begin{matrix}1_{T_{0}} & hr^{−1} \\ 0 & 1_{T} \end{matrix} ] ∈ Aut_{C}(T_{0} ⊕ T),

g_{1} = [ \begin{matrix}1_{T_{1}} & 0 \\ g & r \end{matrix} ] ∈ Aut_{C}(T_{1} ⊕ T)^{op}

and ϕ^{min} = f − hr^{−1}g ∈ Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ)). This proves the theorem. □

Corollary 3.5. For any T_{0}, T_{1} ∈ T , X(T_{1}, T_{0}) only depends on [T_{0}] − [T_{1}].

Proof. Let T_{0}, T_{1} be objects in T and set γ = [T_{0}] − [T_{1}]. It follows from the stability theorem that X(T_{1}, T_{0}) = X(St_{T} (ϕ^{min} )) where ϕ^{min} is a generic element in Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ)). In particular, X(T_{1}, T_{0}) = X(T_{1}^{min} (γ), T_{0}^{min} (γ)) only depends on γ. □

Definition 3.6. For any γ ∈ K_{0}(T ), a generic morphism of index γ is a morphism in the dense open subset U_{(T_{1}^{min} (γ),T_{0}^{min} (γ))} ⊂ Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ)).

If T_{0}, T_{1} are objects in T such that γ = [T_{0}] − [T_{1}] ∈ K_{0}(T ), we will sometimes abuse terminology and view the generic morphism of index γ as an element of Hom_{C}(T_{1}, T_{0}) by considering its image under the stabilization map.

We now define the generic characters as the images under X of the generic morphisms.

Definition 3.7. For any γ ∈ K_{0}(T ), the generic character of index γ is

X(γ) = X(T_{0}^{min} (γ), T_{1}^{min} (γ)).

The set

G^{T} (C) = {X(γ)|γ ∈ K_{0}(T )}

is called the set of generic characters associated to T in C.

3.3. Generic morphism and general presentations in B_{T} -mod . The classical generic variables introduced in [Dup08] for an acyclic quiver Q were closely related to the generic representation theory of the finite dimensional hereditary algebras kQ developed in [Kac80, Kac82, Sch92]. In general, a 2-Calabi-Yau tilted algebra B_{T} is a basic finite dimensional k-algebra but is not hereditary [KR07]. In [DF09], the authors develop a generic representation theory for any finite dimensional basic k-algebra. This is done by replacing the usual notion of generic representation by the notion of generic presentation. We recall briefly this notion.

Given a finite dimensional basic k-algebra B, we denote by ϵ_{1}, …, ϵ_{n} the idempotents of B. The indecomposable projective B-modules are _{B}P_{i} = Bϵ_{i} for i ∈ {1, …, n}. We denote by K_{0}(B-proj ) the Grothendieck group of the additive category B-proj = add (_{B}P_{1} ⊕ ⋯ ⊕_{B}P_{n}) and for any B-module M, we denote by [M] its class in K_{0}(B-proj ).

For any projective B-modules M_{1}, M_{0}, and any morphism f ∈ Hom_{B}(M_{1}, M_{0}), the δ-vector of f is [M_{0}] − [M_{1}]. Note that, identifying K_{0}(B-proj ) with ℤ^{n} by sending each [_{B}P_{i}] to the i-th vector α_{i} of the canonical basis of ℤ^{n}, this coincides with the definition provided in [DF09].

The group Aut_{B}(M_{0}) × Aut_{B}(M_{1})^{op} acts on Hom_{B}(M_{1}, M_{0}) by (g_{0}, g_{1}).f = g_{0}fg_{1}. A morphism f ∈ Hom_{B}(M_{1}, M_{0}) is called a generic if its Aut_{B}(M_{0}) × Aut_{B}(M_{1})^{op} -orbit is a Zariski dense open subset in Hom_{B}(M_{1}, M_{0}). If M is a B_{T} -module, a general presentation of M is a generic morphism in Hom_{B_{T}} (M_{1}, M_{0}) where M_{0}, M_{1} are projective B_{T} -modules such that there is a projective resolution M_{1}⟶ M_{0}⟶ M⟶ 0 of B_{T} -modules.

We now prove that, under the functor F_{T} : C⟶ B_{T} -mod , generic T -morphisms in the category C correspond to generic morphisms in B_{T} -mod .

Lemma 3.8. For any T_{0}, T_{1} in T and any f ∈ Hom_{C}(T_{1}, T_{0}), the following are equivalent :

(1) f is a generic T -morphism in Hom_{C}(T_{1}, T_{0}) ;

(2) F_{T} f is a general presentation in Hom_{B_{T}} (F_{T} T_{1}, F_{T} T_{0}).

Proof. The functor F_{T} = Hom_{C}(T, −) induces a k-linear equivalence of categories

F_{T} : C/T [1] ⟶^{∼} B_{T} -mod .

Let T_{0}, T_{1} be objects in T . Since T is a cluster-tilting object, for any X in T [1], Hom_{C}(T_{1}, X) = 0 so that there are no morphisms from T_{1} to T_{0} in C factorizing through T [1]. In particular, F_{T} induces an isomorphism of k-vector spaces

Hom_{C}(T_{1}, T_{0}) ≃ Hom_{B_{T}} (F_{T} T_{1}, F_{T} T_{0}).

Thus, f is generic in Hom_{C}(T_{1}, T_{0}) if and only if F_{T} f is generic Hom_{B_{T}} (F_{T} T_{1}, F_{T} T_{0}) □

Corollary 3.9. A presentation of B_{T} -modules is generic if and only if it is the image under F_{T} of a generic T -morphism.

Proof. The projective B_{T} -modules are the F_{T} (M) where M is an object in T . Fix thus two projective B_{T} -modules F_{T} T_{0}, F_{T} T_{1} and fix g ∈ Hom_{B_{T}} (F_{T} T_{1}, F_{T} T_{0}). The functor F_{T} induces an isomorphism of k-vector spaces

Hom_{C}(T_{1}, T_{0}) ≃ Hom_{B_{T}} (F_{T} T_{1}, F_{T} T_{0})

so that a g = F_{T} f for some f ∈ Hom_{C}(T_{1}, T_{0}). The corollary thus follows from Lemma 3.8. □

Remark 3.10. Note that for any f ∈ Hom_{C}(T_{1}, T_{0}), the δ-vector of F_{T} f is [F_{T} T_{0}]− [F_{T} T_{1}]. Thus, using the above identifications with ℤ^{n},

δ(F_{T} f) = [F_{T} T_{0}] − [F_{T} T_{1}] = [T_{0}] − [T_{1}] = ind_{T} (Cone (f)).

Thus F_{T} induces a 1-1 correspondence between generic T -morphism and general presentations in B_{T} -mod. Under this correspondence, the generic T -morphism of index γ corresponds to the generic presentation of δ-vector γ.

4. Cluster monomials and generic characters

We now prove that cluster monomials in A(B_{T} , x) are generic cluster characters. We will see in Corollary 7.5 that the converse is not true in general.

Theorem 4.1. Let C be a triangulated 2-Calabi-Yau category with constructible cones and T be a cluster tilting object in C. Then, the following hold :

(1) If M is a rigid object in C, then

X_{M}^{T} = X(ind_{T} (M)) ;

(2) If M is rigid object in C, reachable from T, then

X(ind_{T} (M)) ∈ M(Q_{T} , x) ;

(3) If M is an indecomposable rigid in C, reachable from T, then

X(ind_{T} (M)) ∈ Cl(Q_{T} , x) ;

(4)

M(Q_{T} , x) ⊂ G^{T} (C).

Proof. Let M be an object in C and

T_{1}^{M} ⟶^{f_{M}} T_{0}^{M}⟶ M⟶ T_{1}^{M}[1]

be a triangle in C_{Q} with T_{0}^{M}, T_{1}^{M} ∈ add T. Set γ = ind_{T} (M) = [T_{0}^{M}] − [T_{1}^{M}]. If M is rigid in C, then T_{0}^{M} and T_{1}^{M} have no common direct summands [DK08, Proposition 2.2] so that T_{0}^{M} = T_{0}^{min} (γ) and T_{1}^{M} = T_{1}^{min} (γ). Moreover, the Aut_{C}(T_{0}) × Aut_{C}(T_{1})^{op} -orbit of f_{M} is open and dense in Hom_{C}(T_{1}, T_{0}) [DK08, §2.1]. Since X is invariant under the action of Aut_{C}(T_{1}) × Aut_{C}(T_{0})^{op} and Hom_{C}(T_{1}, T_{0}) is an irreducible variety, it follows that X(f_{M}) is the generic value of X on Hom_{C}(T_{1}, T_{0}). Thus,

X(f_{M}) = X([T_{0}^{M}] − [T_{1}^{M}]) = X(ind_{T} (M)).

Since Cone (f_{M}) ≃ M, it follows from the definition of X that X(f_{M}) = X_{M}^{T}. This proves the first point.

By [Pal08], X_{?}^{T} induces a 1-1 correspondence from the set of reachable indecomposable rigid (resp. rigid) objects to cluster variables (resp. cluster monomials) in A(Q_{T} , x). Together with the first point, this proves the second, third and fourth points. □

5. Indices and dimension vectors in cluster categories

From now on, we assume that C = C_{Q} is the cluster category of an acyclic quiver Q and that T = kQ is the canonical cluster-tilting object in C_{Q}. We recall that the cluster category is the orbit category in D^{b}(kQ-mod ) of the functor τ^{−1}[1] where [1] is the suspension functor in the bounded derived category D^{b}(kQ-mod ) and τ is the Auslander-Reiten translation. It is a canonically triangulated 2-Calabi-Yau category [Kel05] with constructible cones [Pal09].

For any i ∈ Q_{0}, we denote by S_{i} the simple kQ-module at vertex i and by P_{i} its projective cover. Indecomposable objects in C_{Q} can be identified with indecomposable kQ-modules and shifts of indecomposable projective modules. The canonical cluster-tilting object T can thus be written as T = kQ = ⨁_{i∈Q_{0}} P_{i}.

5.1. Dimension vectors and index. We shall now compare the notion of index in the cluster category C_{Q} to the notion of dimension vector in the underlying module category kQ-mod.

For any kQ-module M, the dimension vector of M is the element dim M = (dim_{k}Hom_{kQ}(P_{i}, M))_{i∈Q_{0}} ∈ ℕ^{Q_{0}}. Let K_{0}(kQ-mod ) denote the Grothendieck group of kQ-mod. It is known that dim induces an isomorphism of abelian groups K_{0}(kQ) ⟶^{∼} ℤ^{Q_{0}} sending the isoclass of S_{i} to the i-th vector α_{i} of the canonical basis of ℤ^{Q_{0}} for any i ∈ Q_{0}.

As usual, we identify kQ-mod with the category rep(Q) of finite dimensional representations of Q over k. We recall that a representation M of Q is a pair M = ((M(i))_{i∈Q_{0}}, (M(α))_{α∈Q_{1}}) such that each M(i) is a finite dimensional kvector space and each M(α) is a k-linear map M(i)⟶ M(j) where α : i⟶ j ∈ Q_{1}. Note that dim M = (dim_{k}M(i))_{i∈Q_{0}} for any representation M of Q.

For any d ∈ ℕ^{Q_{0}}, we denote by rep(Q, d) the set of representations M of Q such that dim M = d which can is identified with the irreducible affine variety

rep(Q, d) = ∏_{α:i⟶ j} Hom_{k}(k^{d_{i}}, k^{d_{j}}),

called representation space of dimension d.

We define the dimension vector dim_{C} M of an object M in the cluster category by

dim_{C} M = \{ \begin{matrix}dim M & if M is an indecomposable kQ-module ; \\ −(E^{t})^{−1}dim S_{i} & if M ≃ P_{i}[1] ; \\ dim_{C} M_{1} + dim_{C} M_{2} & if M = M_{1} ⊕ M_{2}. \end{matrix}

Since dim_{C} M = dim M for any kQ-module M, we will simply write dim M for the dimension vector of an arbitrary object M in C.

Remark 5.1. Note that our convention for dimension vectors of objects in C_{Q} agrees with the one considered in [IOTW09] but differs from the one taken in [CK06, Dup08]. Indeed, in [CK06, Dup08], the dimension vector of P_{i}[1] was set to −dim S_{i}. This was more accurate from the point of view of denominator vectors of the corresponding cluster characters (see for instance [CK06, Theorem 3] and [Dup08, Lemma 3.9]). Nevertheless, as we shall see, it appears that the convention of [IOTW09] we use here is more natural from the point of view of indices, g-vectors, virtual generic decompositions and generic characters.

Since Q is acyclic, kQ is hereditary and thus, the bilinear form ⟨−, −⟩ is well defined on ℤ^{Q_{0}} ≃ K_{0}(kQ-mod ). The Euler matrix of Q is thus the matrix E ∈ M_{Q_{0}}(ℤ) of the (non-symmetric) bilinear form ⟨−, −⟩ on K_{0}(kQ-mod ). We refer the reader to [ASS05, §III.3] for classical properties of E.

Lemma 5.2. Let C = C_{Q} be the cluster category of an acyclic quiver Q. Let T = kQ be the canonical cluster tilting object in C and T = add T. Then, for any object M be an object in C, we have

ind_{T} (M) = E^{t}dim M,
coind_{T} (M) = Edim M.

Proof. Without loss of generality, we can assume that X is indecomposable. If M is a kQ-module, then it follows from [Pal08, Lemma 2.3] that

coind_{T} (M) = (⟨S_{i}, M⟩)_{i∈Q_{0}}
= (α_{i}Edim M)_{i∈Q_{0}}
= Edim M

and

ind_{T} (M) = (⟨M, S_{i}⟩)_{i∈Q_{0}}
= (dim M^{t}Eα_{i})_{i∈Q_{0}}
= E^{t}dim M.
□

5.2. Presentation spaces and index. As we mentioned, classical approaches in geometric representation theory considered objects rather than morphisms. In particular, the geometric representation theory of acyclic quivers focuses on representation spaces. Nevertheless, generalizations can be carried out to arbitrary k-algebras B if one focuses on morphisms in B-mod instead of objects in B-mod. In this settings, Kac’s concept of generic representations of quivers [Kac80, Kac82] is replaced by Derksen-Fei’s concept of general presentations of kQ-modules [DF09]. Following the approach of [IOTW09], we will thus consider presentation spaces in kQ-mod instead of representations spaces in rep(Q).

Given an element α ∈ ℤ^{Q_{0}}, a projective decomposition of α is a pair (γ_{0}, γ_{1}) ∈ ℕ^{Q_{0}} × ℕ^{Q_{0}} such that E^{t}α = γ_{0} − γ_{1}. It is called minimal if γ_{0} and γ_{1} have disjoint support. A minimal decomposition of a given element γ ∈ ℕ^{Q_{0}} is unique.

For any γ ∈ ℕ^{Q_{0}}, we set P (γ) = ⨁_{i∈Q_{0}} P_{i}^{⊕γ_{i}}. For any α ∈ ℤ^{Q_{0}}, the presentation space associated to a projective decomposition (γ_{0}, γ_{1}) of α is

R(γ_{0}, γ_{1}) = Hom_{kQ}(P (γ_{1}), P (γ_{0})).

The minimal presentation space R^{min} (α) is the representation space associated to the minimal projective decomposition of α.

Let α be an element ℤ^{Q_{0}}, which shall be thought as a dimension vector of an object in C_{Q}. Then, by Lemma 5.2, γ = E^{t}α ∈ ℤ^{Q_{0}} may be thought as an index vector of an object in C_{Q}. Let (γ_{0}, γ_{1}) be a projective decomposition of α then γ = γ_{0} − γ_{1} = [P (γ_{0})] − [P (γ_{1})]. The objects T_{0} = P (γ_{0}) and T_{1} = P (γ_{1}) belong to T = add (kQ) = kQ-proj and the representation space R(γ_{0}, γ_{1}) is thus isomorphic to Hom_{kQ}(T_{1}, T_{0}) which is isomorphic to Hom_{C}(T_{1}, T_{0}) since T_{1} is a projective kQmodule [BMR^{+}06].

If (γ_{0}^{min} , γ_{1}^{min} ) is the minimal projective decomposition of α, we have T_{0}^{min} (γ) = P (γ_{0}^{min} ) and T_{1}^{min} (γ) = P (γ_{1}^{min} ). Thus, the minimal presentation space R^{min} (α) is isomorphic to the space of T -morphisms Hom_{C}(T_{1}^{min} (γ), T_{0}^{min} (γ)). In particular, the generic morphism of index γ can be viewed as a generic element of the minimal presentation space R^{min} ((E^{t})^{−1}γ) .

5.3. g-vectors. We now prove that generic cluster characters are naturally parameterized by their g-vectors, up to a Coxeter transformation. For details concerning g-vectors, we refer the reader to [FZ07] in general and [FK10] for g-vectors in the context of cluster characters.

We denote by C = −E^{t}E^{−1} the Coxeter matrix of kQ. If a Laurent polynomial L ∈ ℤ[x^{±1}] has a g-vector, we denote it by g_{L}. For any object M in C, X_{M}^{T} has a g-vector ([FK10]) and we denote this g-vector by g_{M}.

Proposition 5.3. Let C be the cluster category of an acyclic quiver Q and T = kQ be the canonical cluster-tilting object. Then, for any γ ∈ ℤ^{n},

g_{X(γ)} = C^{−1}γ.

Proof. It follows from [FK10], that g_{M} = −coind_{T} (M) but Lemma 5.2 implies that −coind_{T} (M) = −Edim M. Fix γ ∈ ℤ^{n}. Then X(γ) = X_{Cone (f)}^{T} for some T -morphism f such that γ = ind_{T} (Cone (f)). Let M = Cone (f). By Lemma 5.2,

γ = ind_{T} (M) = E^{t}dim M.

It follows that

γ = E^{t}dim M
= E^{t}(−E^{−1})g_{M}
= Cg_{M}

but the Coxeter transformation is invertible so that

g_{X(γ)} = g_{M} = C^{−1}γ.
□

Corollary 5.4. Let M be a kQ-module, and P_{0}^{M} ⟶^{f_{M}} P_{1}^{M}⟶ M⟶ 0 be a projective presentation of kQ-modules. Then,

δ(f_{M}) = Cg_{M}.

Proof. It follows from Remark 3.10 that δ(f_{M}) = ind_{T} (Cone (f_{M})) = ind_{T} (M) = E^{t}dim M = Cg_{M}. □

6. Generic cluster characters and virtual generic decomposition

In this section, we still assume that C = C_{Q} is the cluster category of an acyclic quiver Q and that T = kQ is the canonical cluster-tilting object in C.

Following [Sch92], given two elements β, γ ∈ ℕ^{Q_{0}}, we say that Ext_{kQ}^{1}(β, γ) vanishes generally if there exists M_{β} ∈ rep(Q, β), M_{γ} ∈ rep(Q, γ) such that Ext_{kQ}^{1}(M_{β}, M_{γ}) = 0. This is equivalent to the fact that there exists dense open subsets O_{β} ⊂ rep(Q, β), O_{γ} ⊂ rep(Q, γ) such that Ext_{kQ}^{1}(M_{β}, M_{γ}) = 0 for any M_{β} ∈ O_{β} and M_{γ} ∈ O_{γ}.

Given an element α ∈ ℕ^{Q_{0}}, Kac proved that there exists a unique decomposition

α = β_{1} + ⋯ + β_{k}

such that :

(1) Ext_{kQ}^{1}(β_{i}, β_{j}) vanishes generally if i ≠ j ;

(2) each β_{i} is a Schur root of Q. This decomposition is called the generic (or canonical) decomposition of α.

This was generalized in [IOTW09] to arbitrary elements ℤ^{Q_{0}}. Namely, for any α ∈ ℤ^{Q_{0}} there exists a unique decomposition

α = β_{1} + ⋯ + β_{k} − (E^{t})^{−1}γ

such that :

(1) β_{1}, …, β_{k}, γ ∈ ℕ^{Q_{0}} ;

(2) β_{i}, γ have disjoint support for all i ;

(3) Ext_{kQ}^{1}(β_{i}, β_{j}) vanishes generally if i ≠ j ;

(4) each β_{i} is a Schur root of Q. This decomposition is called the virtual generic decomposition of α. Note that if α ∈ ℕ^{n}, generic and virtual generic decomposition coincide [IOTW09].

Theorem 6.1. Let C be the cluster category of an acyclic quiver Q and T = kQ be the canonical cluster-tilting object in C. Let α ∈ ℤ^{n} with virtual generic decomposition

α = β_{1} + ⋯ + β_{k} − (E^{t})^{−1}γ.

Then,

X(E^{t}α) = X(E^{t}β_{1}) ⋯ X(E^{t}β_{k})X(−γ).

Proof. Let α ∈ ℤ^{n} with virtual canonical decomposition α = β_{1}+⋯+β_{k}−(E^{t})^{−1}γ. For any i = 1, …, k, let M_{i} be a generic representation in rep(Q, β_{i}). Let p_{M_{i}} be the canonical projective presentation of M_{i} and 0_{γ} be the zero map 0_{γ} : P (γ)⟶ 0. Then, it follows from [IOTW09, Theorem 6.3.1] that X(E^{t}α) = X(f) where f = ⨁_{i=1}^{k} p_{M_{i}} ⊕ 0_{γ}. Thus,

X(E^{t}α) = X(f)

= X(⨁_{i=1}^{k} p_{M_{i}} ⊕ 0_{γ})

= ∏_{i=1}^{k} X(p_{M_{i}})X(0_{γ})

but it follows from [IOTW09] that p_{M_{i}} is generic for every i = 1, …, k so that X(p_{M_{i}}) = X(ind_{T} (M_{i})) = X(E^{t}dim M_{i}) = X(E^{t}β_{i}). Also, 0_{γ} is generic so that X(0_{γ}) = X(ind_{T} (P_{γ}[1])) = X(−γ). Thus,

X(E^{t}α) = X(E^{t}β_{1}) ⋯ X(E^{t}β_{k})X(−γ).
□

7. Generic cluster characters and classical generic variables

In this section, C still denotes the cluster category of an acyclic quiver Q and T = kQ is the canonical cluster-tilting object in C. In this case, the cluster character X^{T} coincides with the Caldero-Chapoton map CC : Ob(C)⟶ ℤ[x^{±1}] introduced in [CC06]. In [Dup08], we introduced a family of Laurent polynomials in ℤ[x^{±1}], called generic variables, by considering generic values of the Caldero-Chapoton map. We shall now see that generic characters coincide with these generic variables when C = C_{Q} and T = kQ. First, we briefly review the construction of [Dup08].

For any α ∈ ℕ^{Q_{0}}, there exists a dense open subset U_{α} ⊂ rep(Q, α) such that CC is constant over U_{α}. The value of CC on U_{α} does not depend on the chosen open set U_{α} and is denoted by CC(α) [Dup08, Lemma 3.1].

More generally, if α ∈ ℤ^{n}, set α = α_{+} − α_{−} with α_{±} ∈ ℕ^{Q_{0}} having disjoint supports and set

CC(α) = CC(α_{+}) ∏_{i∈Q_{0}} x_{i}^{(α_{−})_{i}} = CC(α_{+})CC(P (α_{−})[1]).

The set

G(Q) = {CC(α)|α ∈ ℤ^{Q_{0}}}

is called the set of generic variables in A(Q) and for any α ∈ ℤ^{Q_{0}}, CC(α) is called the generic variable of dimension α.

Generic variables satisfy multiplicative properties with respect to Kac’s generic decomposition [Dup08, Proposition 3.7]. Namely, for α ∈ ℕ^{n}, if

α = β_{1} + ⋯ + β_{k}

is the generic decomposition of α, then

CC(α) = CC(β_{1}) ⋯ CC(β_{k}).

Lemma 7.1. For any α ∈ ℕ^{n},

X(E^{t}α) = CC(α).

Proof. Let α ∈ ℕ^{n}. For any projective decomposition E^{t}α = γ_{0} − γ_{1}, X(E^{t}α) = X(f) for a generic element f ∈ Hom_{kQ}(P (γ_{1}), P (γ_{0})). We can thus assume that E^{t}α = γ_{0} − γ_{1} is the projective canonical decomposition of [IOTW09]. Consider the triangle

P (γ_{1}) ⟶^{f} P (γ_{0})⟶ Cone (f)⟶ P (γ_{1})[1].

Applying F_{T}, we get

P (γ_{1}) ⟶^{F_{T} f} P (γ_{0})⟶ F_{T} (Cone (f))⟶ 0

so that X(f) = X_{Cone (F_{T} (f))}^{T} = X_{CokerF_{T} (f)}^{T}. According to Corollary 3.9, F_{T} f is generic in Hom_{kQ}(P (γ_{1}), P (γ_{0})). Thus, we get

X(f) = X^{T} (Cone f) = CC(CokerF_{T} f).

Consider the map κ : g ↦ Cokerg on Hom_{kQ}(P (γ_{1}), P (γ_{0})). For any morphism g ∈ Hom_{kQ}(P (γ_{1}), P (γ_{0})), we have dim Coker(g) = dim P (γ_{0}) − dim P (γ_{1}) = (E^{t})^{−1}(γ_{0} − γ_{1}) = α so that κ is a map

κ : Hom_{kQ}(P (γ_{1}), P (γ_{0}))⟶ rep(Q, α).

Now, it follows from [IOTW09, Proposition 4.1.7] that there exists a Zariski dense open subset U ⊂ Hom_{kQ}(P (γ_{1}), P (γ_{0})) such that κ_{|U} : U⟶ rep(Q, α) is algebraic and injective. Thus, dim κ(U) ≥ dim U > 0 and thus κ(U) ∩ U_{α} ≠ ∅ where U_{α} denotes, as before, the Zariski dense open subset for the Caldero-Chapoton map. It follows that κ^{−1}(U_{α}) is a Zariski dense open subset in Hom_{kQ}(P (γ_{1}), P (γ_{0})) and we can thus assume that the generic element f belongs to κ^{−1}(U_{α}). It follows that

X(E^{t}α) = X(f) = CC(CokerF_{T} f) = CC(α)

which proves the lemma. □

Lemma 7.2. For any α ∈ (−ℕ)^{n},

X(α) = CC(α).

Proof. Let γ ∈ ℕ^{n} such that α = −γ. Consider the morphism 0_{γ} : Hom_{kQ}(P (γ), 0). We have the triangle

P (γ) ⟶^{0_{γ}} 0⟶ P (γ)[1] ⟶^{∼} P (γ)[1]

so that

X(0_{γ}) = X_{P (γ)[1]}^{T} = CC(P (γ)[1]) = CC(−γ) = CC(α).

Now, by definition, X(f) is the generic character of index α, that is,

X(α) = X(0_{γ}) = CC(α).
□

Let α, β ∈ ℤ^{n} such that α = α_{+} − α_{−} (resp. β = β_{+} − β_{−}) where α_{−}, α_{+} (resp. β_{−}, β_{+}) have disjoint support. Following [Dup08], we say that Ext_{C_{Q}}^{1}(α, β) vanishes generally if there exists M_{α} ∈ rep(Q, α_{+}), M_{β} ∈ rep(Q, β_{+}) such that

Ext_{C_{Q}}^{1}(M_{α} ⊕ P (α_{−})[1], M_{β} ⊕ P (β_{−})[1]) = 0.

It follows from [Dup08, Lemma 3.9] that if Ext_{C_{Q}}^{1}(α, β) vanishes generally, then

CC(α + β) = CC(α)CC(β).

Theorem 7.3. Let C = C_{Q} be the cluster category of an acyclic quiver Q and T = kQ be the canonical cluster-tilting object in C. Let α ∈ ℤ^{n} with virtual generic decomposition

α = β_{1} + ⋯ + β_{k} − (E^{t})^{−1}γ.

Then,

X(E^{t}α) = CC(β_{1} + ⋯ + β_{k} − γ).

Proof. According to Theorem 6.1, we have

X(E^{t}α) = X(E^{t}β_{1}) ⋯ X(E^{t}β_{k})X(−γ).

For any i = 1, …, k, it follows from 7.2 that X(E^{t}β_{i}) = CC(β_{i}). Also, it follows from 7.2 that X(−γ) = CC(−γ). Now, β_{1} + ⋯ + β_{k} is the (classical) generic decomposition of ∑_{i=1}^{k} β_{i} so that

CC(β_{1}) ⋯ CC(β_{k}) = CC(β_{1} + ⋯ + β_{k}).

Moreover, β_{i} and γ have disjoint support for any i ∈ {1, …, k} so that γ and ∑_{i=1}^{k} β_{i} have disjoint support and Ext_{C}^{1}(∑_{i=1}^{k} β_{i}, −γ) vanishes generally. Thus,

CC(∑_{i=1}^{k} β_{i})CC(−γ) = CC(∑_{i=1}^{k} β_{i} − γ)

which proves the theorem. □

Theorem 7.4. Let C = C_{Q} be the cluster category of an acyclic quiver Q and T = kQ be the canonical cluster-tilting object in C. Then,

G(Q) = G^{T} (C).

Proof. E^{t} is an invertible matrix so E^{t}α runs over ℤ^{n} when α runs over ℤ^{n}. It thus follows from Theorem 7.3 that G(Q) ⊃ G^{T} (C).

Conversely, if α ∈ ℤ^{n}, we can write α = α_{+} − α_{−} where α_{±} ∈ ℕ^{n} have disjoint supports. Considering the classical canonical decomposition

α_{+} = β_{1} + ⋯ + β_{k}

we get α = β_{1}+⋯+β_{k}−α_{−}. For any i = 1, …, k, consider a generic representation M_{i} ∈ rep(Q, β_{i}). It follows from [Dup08] that

CC(α) = ∏_{i=1}^{k} CC(M_{i})CC(P (α_{−})[1]).

Considering the minimal projective resolutions, we obtain triangles in C

P_{1}^{M_{i}} ⟶^{p_{M_{i}}} P_{0}^{M_{i}}⟶ M_{i}⟶ P_{1}^{M_{i}}[1]

for any i = 1, …, k. Moreover we have the triangle in C

P (α_{−}) ⟶^{0} 0⟶ P (α_{−})[1] ≃ P (α_{−})[1].

So, we get a triangle

⨁_{i=1}^{k} P_{1}^{M_{i}} ⊕ P (α_{−}) ⟶^{f} ⨁_{i=1}^{k} P_{0}^{M_{i}}⟶ ⨁_{i=1}^{k} M_{i} ⊕ P (α_{−})[1]⟶ ⨁_{i=1}^{k} P_{1}^{M_{i}}[1] ⊕ P (α_{−})[1]

where f = ⨁_{i=1}^{k} p_{M_{i}} ⊕ 0. By definition of X, we have

X(f) = X_{Cone (f)}^{T} = ∏_{i=1}^{k} CC(M_{i})CC(P (α_{−})[1]) = CC(α).

But, it follows from [IOTW09, Theorem 6.3.1] that f is generic, thus

X(f) = X(ind_{T} (⨁_{i=1}^{k} M_{i}) + ind_{T} (P (α_{−})[1]))

= X(∑_{i=1}^{k}(E^{t})dim M_{i} + (E^{t})dim P (α_{−})[1])

= X(∑_{i=1}^{k}(E^{t})β_{i} − α_{−})

and thus G^{T} (C) ⊂ G(Q). □

Combining with known results on classical generic variables, we get :

Corollary 7.5. Let C = C_{Q} be the cluster category of an acyclic quiver and T = kQ be the canonical cluster-tilting object in C. Then the following hold :

(1) G^{T} (C) is a ℂ-linear basis in ℂ ⊗_{ℤ} A(Q, x) ;

(2) if moreover Q_{T} is a Dynkin or an affine quiver, then G^{T} (C) is a ℤ-linear basis in A(Q, x) ;

(3) G^{T} (C) = M(Q, x_{T}) if and only if Q_{T} is a Dynkin quiver.

Proof. Let Q be an acyclic quiver. In [GLS10] the authors construct a ℂ-basis S^{∗}(Q) of ℂ ⊗_{ℤ} A(Q, x). According to [Sch09], S^{∗}(Q) coincides with the set G(Q) of generic variables in A(Q, x) which coincides with G^{T} (C) according to Theorem 7.4. This proves the first point. The second point is a consequence of [Dup08, Theorem 4.20, Theorem 4.21] and [DXX09, Theorem 8.2].The third point is a consequence of [Dup08, Lemma 3.10]. □

More generally, we conjecture :

Conjecture 7.6. Let C be a Hom-finite triangulated 2-Calabi-Yau category with constructible cones and let T be a cluster-tilting object in C. We conjecture :

(1) G^{T} (C) is a ℤ-basis in A(Q_{T} , x) ;

(2) if T′ is another cluster-tilting object reachable from T, then G^{T} (C) coincides with G^{T′}(C) under the canonical isomorphism of cluster algebras A(Q_{T} , x) ≃ A(Q_{T′}, x′).

Acknowledgments

The author would like to thank Bernhard Keller for interesting discussions on this topic. He is also grateful to Ibrahim Assem, Thomas Brüstle, Virginie Charette and the CRM-ISM for their financial support.

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Université de Sherbrooke, 2500, Boul. de l’Université, J1K 2R1, Sherbrooke, QC, Canada.

   E-mail address: gregoire.dupont@usherbrooke.ca
   URL: http://pages.usherbrooke.ca/gdupont2