ON THE COMBINATORICS OF RIGID OBJECTS IN 2-CALABI-YAU CATEGORIES

RAIKA DEHY AND BERNHARD KELLER

Abstract. Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T , the index of an object X of C is a certain element of the Grothendieck group of the additive category T . In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T′ form a basis of the Grothendieck group of T and that, if T and T′ are related by a mutation, then the indices with respect to T and T′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T . Conjecturally, these indices coincide with Fomin-Zelevinsky’s g-vectors.

1. Introduction

This note is motivated by the representation-theoretic approach to Fomin-Zelevinsky’s cluster algebras [6] [7] [4] [8] developed by Marsh-Reineke-Zelevinsky [18], Buan-Marsh-Reineke-Reiten-Todorov [3], Geiss-Leclerc-Schröer [11] [12] and many others, cf. [2] for a survey. In this approach, a central rôle is played by certain triangulated 2-Calabi-Yau categories and by combinatorial invariants associated with their rigid objects (we refer to [14] [5] for different approaches). Here, our object of study is the index, which is a certain ‘dimension vector’ associated with each object of the given Calabi-Yau category.

More precisely, we fix a Hom-finite 2-Calabi-Yau triangulated category C with split idempotents which admits a cluster-tilting subcategory T . It is known from [16] that for each object X of C, there is a triangle

T_{1} → T_{0} → X → ΣT_{1}

of C, where T_{1} and T_{0} belong to T . Following [19], we define the index of X to be the difference [T_{0}] − [T_{1}] in the split Grothendieck group K_{0}(T ) of the additive category T . We show that

- if X is rigid (i.e. C(X, ΣX) = 0), then it is determined by its index up to isomorphism;

- the indices of the direct factors of a rigid object all lie in the same hyperquadrant of K_{0}(T ) with respect to the basis given by a system of representatives of the isomorphism classes of the indecomposables of T ;

- the indices of the direct factors of a rigid object are linearly independent;

- the indices of a system of representatives of the indecomposable objects of any cluster-tilting subcategory T′ form a basis of K_{0}(T ). In particular, all cluster-tilting subcategories have the same (finite or infinite) number of pairwise non isomorphic indecomposable objects.

Date: May 10, 2007, last modified on May 14, 2008.

1991 Mathematics Subject Classification. 18E30, 16D90, 18G40, 18G10, 55U35.

Key words and phrases. Calabi-Yau category, Cluster algebra, Tilting.

Note that the last point was shown in Theorem I.1.8 of [1] under the additional assumption that C is a stable category. We then study how the index of an object transforms when we mutate the given cluster-tilting subcategory. We find that this transformation is given by the right hand side of Conjecture 7.12 of [8], cf. section 4. This motivates the definition of g^{†}-vectors as the combinatorial counterpart to indices. If, as we expect, Conjecture 7.12 of [loc. cit.] holds, then our g^{†}-vectors are identical with the g-vectors of [loc. cit.], whose definition we briefly recall below. We finally show that if C has a cluster-structure in the sense of [1], then we have a bijection between g^{†}-vectors and indecomposable rigid objects reachable from T and between g^{†}-clusters and cluster-tilting subcategories reachable from T.

Our results are inspired by and closely related to the conjectures of [8] and the results of section 15 in [10]. As a help to the reader not familiar with [8], we give a short summary of the notions introduced there which are most relevant for us: Let n ≥ 1 be an integer and B a skew-symmetric integer matrix. Let F be the field of rational functions ℚ(x_{1}, …, x_{n}, y_{1}, …, y_{n}) in 2n indeterminates. Let A ⊂ F be the cluster algebra with principal coefficients associated with the initial seed (x, y, B), where x = (x_{1}, …, x_{n}) and y = (y_{1}, …, y_{n}), cf. sections 1 and 2 of [8]. As shown in Proposition 3.6 of [8], each cluster variable of A lies in the ring ℤ[x_{1}^{±1}, …, x_{n}^{±1}, y_{1}, …, y_{n}]. Moreover, by Proposition 6.1 of [8], each cluster variable of A is homogeneous with respect to the ℤ^{n}-grading on ℤ[x_{1}^{±1}, …, x_{n}^{±1}, y_{1}, …, y_{n}] given by

deg(x_{i}) = e_{i} , deg(y_{j}) = − ∑_{i=1}^{n} b_{ij}e_{i} ,

where the e_{i} form the standard basis of ℤ^{n}. The g-vector associated with a cluster variable X is by definition the vector deg(X) of ℤ^{n}. More generally, the g-vector of a cluster monomial M is deg(M). Now we can state the conjectures of [8] which motivated the above statements on the combinatorics of rigid objects:

- different cluster monomials have different g-vectors (part (1) of Conjecture 7.10 of [8]);

- the g-vectors of the variables in a fixed cluster all lie in the same hyperquadrant of ℤ^{n} (Conjecture 6.13 of [8]);

- the g-vectors of the variables in a fixed cluster form a basis of ℤ^{n} (part (2) of Conjecture 7.10 of [8]);

- under a mutation of the initial cluster, the g-vector of a given cluster variable transforms according to a certain piecewise linear transformation, cf. section 4 (Conjecture 7.12 of [8]).

In [9], the results of this paper have been used to prove these conjectures for certain classes of cluster algebras.

2. A rigid object is determined by its index

Let k be an algebraically closed field and C a Hom-finite k-linear triangulated category with split idempotents. In particular, the decomposition theorem holds for C: Each object decomposes into finite sum of indecomposable objects, unique up to isomorphism, and indecomposable objects have local endomorphism rings. We write Σ for the suspension functor of C. We suppose that C is 2-Calabi-Yau, i.e. that the square of the suspension functor (with its canonical structure of triangle functor) is a Serre functor for C. This implies that we have bifunctorial isomorphisms

DC(X, Y) →^{∼} C (Y, Σ^{2}X) ,

where X and Y vary in C and D denotes the duality functor Hom_{k}(?, k) over the ground field. Moreover, we suppose that C admits a cluster-tilting subcategory T (called a maximal 1-orthogonal subcategory in [13]). Recall from [16] that this means that T is a full additive subcategory such that

- T is functorially finite in C, i.e. for all objects X of C, the restrictions of the functors C(X,?) and C(?, X) to T are finitely generated, and

- an object X of C belongs to T iff we have C(T, ΣX) = 0 for all objects T of T . We call an object X of C rigid if the space C(X, ΣX) vanishes.

2.1. Rigid objects yield open orbits. Let X be a rigid object of C. From [17], we know that there is a triangle

\xymatrix{T_{1} \ar[r]^{f} & T_{0} \ar[r]^{h} & X \ar[r] & ΣX} ,

where T_{0} and T_{1} belong to T . The algebraic group G = Aut(T_{0})×Aut(T_{1}) acts on C(T_{1}, T_{0}) via

(g_{0}, g_{1})f′ = g_{0}f′g_{1}^{−1}.

Lemma. The orbit of f under the action of G is open in C(T_{1}, T_{0}).

Proof. It suffices to prove that the differential of the map g ↦ gf is a surjection from Lie(G) to C(T_{1}, T_{0}). This differential is given by

(γ_{0}, γ_{1})f = γ_{0}f − fγ_{1}.

Let f′ be an element of C(T_{1}, T_{0}). Consider the following diagram

\xymatrix{Σ^{−1}X \ar[r]^{e} & T_{1} \ar@{-->}[dl]_{γ_{1}} \ar[d]^{f′} \ar[r]^{f} & T_{0} \ar@{-->}[dl]^{γ_{0}} \ar@{-->}[d]^{β_{0}} \ar[r]^{h} & X \\ T_{1} \ar[r]_{f} & T_{0} \ar[r]_{h} & X \ar[r] & ΣT_{1}.}

Since X is rigid, the composition hf′e vanishes. So there is a β_{0} such that β_{0}f = hf′. Now h is a right T -approximation. So there is a γ_{0} such that hγ_{0} = β_{0}. It follows that we have

h(γ_{0}f − f′) = 0.

So there is a γ_{1} such that

γ_{0}f − f′ = fγ_{1}.

This shows that the differential of the map g ↦ gf is indeed surjective. □

2.2. Rigid objects have disjoint terms in their minimal presentations. Let

F : C → mod T

be the functor taking an object Y of C to the restriction of C(?, Y) to T . Let X be a rigid object of C. Let

\xymatrix{T_{1} \ar[r] & T_{0} \ar[r]^{h} & X \ar[r]^{ε} & ΣT_{1}}

be a triangle such that T_{0} and T_{1} belong to T and h is a minimal right T -approximation.

Proposition. T_{0} and T_{1} do not have an indecomposable direct factor in common.

We give two proofs of the proposition. Here is the first one:

Proof. We know that

F T_{1} → F T_{0} → F X → 0

is a minimal projective presentation of F X. Since F induces an equivalence from T onto the category of projectives of mod T , it is enough to show that F T_{1} and F T_{0} do not have an indecomposable factor in common. For this, it suffices to show that no simple module S occuring in the head of F T_{0} also occurs in the head of F T_{1}. Equivalently, we have to show that if a simple S satisfies Hom(F X, S) ≠ 0, then we have Ext^{1}(F X, S) = 0. So let S be a simple admitting a surjective morphism

p : F X → S.

Let f : F T_{1} → S be a map representing an element in Ext^{1}(F X, S). Since F T_{1} is projective, there is a morphism f_{1} : F T_{1} → F X such that p ∘ f_{1} = f. Now using the fact that F is essentially surjective and full, we choose a preimage up to isomorphism S̃ of S and preimages f̃, p̃ and f̃_{1} of f, p and f_{1} in C as in the following diagram

\xymatrix{Σ^{−1}X \ar[r]^{Σ^{−1}ε} & T_{1} \ar[dl]_{f̃_{1}} \ar[d]^{f̃} \ar[r] & T_{0} \ar[r] & X \\ X \ar[r]_{p̃} & S̃}

Denote by mod T the category of finitely presented k-linear functors from T^{op} to the category of k-vector spaces. Since F induces a bijection

C(T, Y) → (mod T )(F T, F Y)

for all Y in C, we still have p̃∘ f̃_{1} = f̃. The composition f̃_{1} ∘ (Σ^{−1}ε) vanishes since we have C(Σ^{−1}X, X) = 0. Therefore, the composition

f̃∘ (Σ^{−1}ε) = p̃∘ f̃_{1} ∘ (Σ^{−1}ε)

vanishes. This implies that f̃ factors through the morphism T_{1} → T_{0}. But then f factors through the morphism F T_{1} → F T_{0} and f represents 0 in Ext^{1}(F X, S). □

Let us now give a second, more geometric, proof of the proposition:

Proof. Suppose that T_{0} and T_{1} have an indecomposable direct factor T_{2} so that we have decompositions

T_{0} = T′_{0} ⊕ T_{2} and T_{1} = T′_{1} ⊕ T_{2}.

For a morphism f : T_{1} → T_{0}, let

[ \begin{matrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{matrix} ]

be the matrix corresponding to f with respect to the given decompositions. Of course, up to isomorphism, the cone on f only depends on the orbit of f under the group Aut(T_{0}) × Aut(T_{1}). Suppose that the cone on f is isomorphic to X, which is rigid. Then we know that the orbit of f in C(T_{1}, T_{0}) is open. Hence there is some f′ in the orbit such that the component f′_{22} is invertible. But then, using elementary operations on the rows and columns of the matrix of f′, we see that the orbit of f contains a morphism f′′ whose matrix is diagonal with invertible component f′′_{22}. Clearly, the triangle on f′′ is not minimal. This shows that T_{1} and T_{0} do not have a common indecomposable factor if they are the terms of a minimal triangle whose third term is the rigid object X. □

2.3. A rigid object is determined by its index. The (split) Grothendieck group K_{0}(T ) of the additive category T is the quotient of the free group on the isomorphism classes [T] of objects T of T by the subgroup generated by the elements of the form

[T_{1} ⊕ T_{2}] − [T_{1}] − [T_{2}].

It is canonically isomorphic to the free abelian group on the isomorphism classes of the indecomposable objects of T . It contains a canonical positive cone formed by the classes of objects of T . Each element c of K_{0}(T ) can be uniquely written as

c = [T_{0}] − [T_{1}]

where T_{0} and T_{1} are objects of T without common indecomposable factors. Let X be an object of C. Recall that its index [19] is the element

ind(X) = [T_{0}] − [T_{1}]

of K_{0}(T ) where T_{0} and T_{1} are objects of T which occur in an arbitrary triangle

T_{1} → T_{0} → X → ΣT_{1}.

Now suppose that X is rigid. We know that if we choose the above triangle minimal, then T_{0} and T_{1} do not have common indecomposable factors. Thus they are determined by ind(X). Moreover, since the C(T_{1}, T_{0}) is an irreducible variety (like any finite-dimensional vector space), each morphism f : T_{1} → T_{0} whose orbit under the group Aut(T_{0})×Aut(T_{1}) is open yields a cone isomorphic to X. Thus up to isomorphism, X is determined by ind(X). In fact, X is isomorphic to the cone on a general morphism f : T_{1} → T_{0} between the objects T_{0} and T_{1} without a common indecomposable factor such that ind(X) = [T_{0}]−[T_{1}]. We have proved the

Theorem. The map X ↦ ind(X) induces an injection from the set of isomorphism classes of rigid objects of C into the set K_{0}(T ).

This theorem was inspired by part (1) of conjecture 7.10 in [8].

2.4. Direct factors of rigid objects have sign-coherent indices. Let A be a free abelian group endowed with a basis e_{i}, i ∈ I. A subset X ⊂ A is sign-coherent if, for all elements x, y ∈ X and for all i ∈ I, the sign of the component x_{i} in the decomposition

x = ∑ x_{i}e_{i}

agrees with the sign of y_{i}, cf. Definition 6.12 of [8]. This means that the set X is entirely contained in a hyperquadrant of A with respect to the given basis e_{i}, i ∈ I. Now consider the free abelian group K_{0}(T ) endowed with the basis formed by the classes of indecomposable objects of T . Suppose that X is a rigid object of C. We claim that the set of indices of the direct factors of X is sign-coherent. Indeed, let U and V be direct factors of X. Choose minimal triangles

T_{1}^{U} → T_{0}^{U} → U → ΣT_{1}^{U} and T_{1}^{V} → T_{0}^{V} → V → ΣT_{1}^{V} ,

where the T_{i}^{U} and T_{i}^{V} belong to T . Then the triangle

T_{1}^{U} ⊕ T_{1}^{V} → T_{0}^{U} ⊕ T_{0}^{V} → U ⊕ V → Σ(T_{1}^{U} ⊕ T_{1}^{V})

is minimal. Since U ⊕ V is rigid, the two terms T_{1}^{U} ⊕ T_{1}^{V} and T_{0}^{U} ⊕ T_{0}^{V} do not have indecomposable direct factors in common. In particular, whenever an indecomposable object occurs in T_{0}^{U} (resp. T_{1}^{U}), it does not occur in T_{1}^{V} (resp. T_{0}^{V}). This shows that ind(U) and ind(V) are sign-coherent. This property is to be compared with conjecture 6.13 of [8].

2.5. Indices of factors of rigid objects are linearly independent. Let X be a rigid object of C and let X_{i}, i ∈ I, be a finite family of indecomposable direct factors of X which are pairwise non isomorphic. We claim that the elements ind(X_{i}), i ∈ I, are linearly independent in K_{0}(T ). Indeed, suppose that we have a relation

∑_{i∈I_{1}} c_{i} ind(X_{i}) = ∑_{j∈I_{2}} c_{j} ind(X_{j})

for two disjoint subsets I_{1} and I_{2} of I and positive integers c_{i} and c_{j}. Then the rigid objects

⨁_{i∈I_{1}} X_{i}^{c_{i}} and ⨁_{j∈I_{2}} X_{j}^{c_{j}}

have equal indices. So they are isomorphic. Since I_{1} and I_{2} are disjoint, all the c_{i} and c_{j} have to vanish.

2.6. The indices of the indecomposables of a cluster tilting subcategory form a basis. The following theorem was inspired by part (2) of conjecture 7.10 of [8].

Theorem. Let T′ be another tilting subcategory of C. Then the elements ind(T′), where T′ runs through a system of representatives of the isomorphism classes of indecomposables of T′, form a basis of the free abelian group K_{0}(T ).

Proof. Indeed, we already know that the ind(T′) are linearly independent. So it is enough to show that the subgroup they generate contains ind(T) for each indecomposable T of T . Indeed, let T be an indecomposable of T and let

T → T′_{1} → T′_{0} → ΣT

be a triangle with T′_{i} in T′ (this triangle allows to compute the index of ΣT with respect to T′). Then the map F T′_{1} → F T′_{0} is surjective and therefore, we have

ind(T) − ind(T′_{1}) + ind(T′_{0}) = 0

by Proposition 6 of [19]. Thus, ind(T) is in the subgroup of K_{0}(T ) generated by the ind(T′), where T′ runs through the indecomposables of T′. □

3. How the index transforms under change of cluster-tilting subcategory

Let T′ be another cluster-tilting subcategory. Suppose that T and T′ are related by a mutation, i.e. there is an indecomposable S of T and an indecomposable S^{∗} of T′ such that, if indec denotes the set of isomorphism classes of indecomposables, we have

indec(T′) = indec(T ) \ {S} ∪ {S^{∗}},

and that there exist triangles

S^{∗} → B → S → ΣS^{∗} and S → B′ → S^{∗} → ΣS

with B and B′ belonging to T ∩ T′, cf. e.g. [3] [12] [15]. We define two linear maps

ϕ_{+} : K_{0}(T ) → K_{0}(T′) and ϕ_{−} : K_{0}(T ) → K_{0}(T′)

which both send each indecomposable T′′ belonging to both T and T′ to itself and such that

ϕ_{+}(S) = [B] − [S^{∗}] and ϕ_{−}(S) = [B′] − [S^{∗}].

For an object X of C, we denote by ind_{T} (X) the index of X with respect to T and by [ind_{T} (X) : S] the coefficient of S in the decomposition of ind_{T} (X) with respect to the basis given by the indecomposables of T . The following theorem is inspired by Conjecture 7.12 of [8].

Theorem. Let X be a rigid object of C. We have

ind_{T′}(X) = \{ \begin{matrix}ϕ_{+}(ind_{T} (X)) & if [ind_{T} (X) : S] ≥ 0 ; \\ ϕ_{−}(ind_{T} (X)) & if [ind_{T} (X) : S] ≤ 0. \end{matrix}

Proof. Let

T_{1} → T_{0} → X → ΣT_{1}

be a triangle with T_{0} and T_{1} in T . Suppose first that S occurs neither as a direct factor of T_{1} nor of T_{0}. Then clearly the triangle yields both the index of X with respect to T and with respect to T′ and we have

ϕ_{+}(ind_{T} (X)) = ϕ_{−}(ind_{T} (X)) = ind_{T′}(X).

Now suppose that the multiplicity [ind_{T} (X) : S] equals a positive integer i ≥ 1. This means that S occurs with multiplicity i in T_{0} but does not occur as a direct factor of T_{1}. Choose a decomposition T_{0} = T′′_{0} ⊕ S^{i}. From the octahedron constructed over the composition

T′′_{0} ⊕ B^{i} → T′′_{0} ⊕ S^{i} → X ,

we extract the following commutative diagram, whose rows and columns are triangles

\xymatrix{ΣS^{∗i} \ar[r]^{1} & ΣS^{∗i} \\ T_{1} \ar[u] \ar[r] & T′′_{0} ⊕ S^{i} \ar[u] \ar[r] & X \ar[r] & ΣT_{1} \\ T′_{1} \ar[u] \ar[r] & T′′_{0} ⊕ B^{i} \ar[r] \ar[u] & X \ar[u]_{1} \ar[r] & ΣT′_{1} \ar[u] \\ S^{∗i} \ar[u] \ar[r]^{1} & S^{∗i}. \ar[u]}

Since there are no non zero morphisms from T_{1} to ΣS^{∗i} (T_{1} and S^{∗} belong to T′), the leftmost column is a split triangle and T′_{1} is isomorphic to S^{∗i} ⊕ T_{1}. Thus, the third line yields the index of X with respect to T′, which equals

ind_{T′}(X) = [T′′_{0} ⊕ B^{i}] − [T′_{1}] = [T′′_{0}] − [T_{1}] + i([B] − [S^{∗}]) = ϕ_{+}(ind_{T} (X)).

Finally, suppose that the multiplicity [ind_{T} (X) : S] is equals a negative integer −i ≤ −1. This means that S occurs with multiplicity i in T_{1} but does not occur in T_{0}. Choose a decomposition T_{1} = T′′_{1} ⊕ S^{i}. From the octahedron over the composition

Σ^{−1}X → T′′_{1} ⊕ S^{i} → T′′_{1} ⊕ B^{′i} ,

we extract the following diagram, whose rows and columns are triangles

\xymatrix{ & Σ^{−1}S^{∗i} \ar[r]^{1} \ar[d] & Σ^{−1}S^{∗i} \ar[d] \\ Σ^{−1}X \ar[d]_{1} \ar[r] & T′′_{1} ⊕ S^{i} \ar[r] \ar[d] & T_{0} \ar[r] \ar[d] & X \ar[d]^{1} \\ Σ^{−1}X \ar[r] & T′′_{1} ⊕ B^{′i} \ar[r] \ar[d] & T′_{0} \ar[d] \ar[r] & X \\  & S^{∗i} \ar[r]_{1} & S^{∗i}}

Since there are no non zero morphisms from Σ^{−1}S^{∗i} to T_{0} (S^{∗} and T_{0} belong to T′), the object T′_{0} is isomorphic to T_{0} ⊕ S^{i} and we can read ind_{T′}(X) off the third line of the diagram:

ind_{T′}(X) = [T′_{0}]−[T′′_{1}⊕B^{′i}] = [T_{0}⊕S^{∗i}]−[T′′_{1}]−i[B′] = [T_{0}]−[T′′_{1}]−i([B′]−[S^{∗}]) = ϕ_{−}(ind_{T} (X)).

□

4. g^{†}-vectors and g^{†}-clusters

In this section, we recall fundamental constructions from [8] in a language adapted to our applications. We will define g^{†}-vectors using the right hand side of Conjecture 7.12 of [loc. cit.]. If, as we expect, this conjecture holds, then our g^{†}-vectors are identical with the g-vectors of [loc. cit.].

Let Q be a quiver. Thus Q is given by a set of vertices I = Q_{0}, a set of arrows Q_{1} and two maps s and t from Q_{1} to I = Q_{0} taking an arrow to its source, respectively its target. We assume that Q is locally finite, i.e. for each given vertex i of Q there are only finitely many arrows α such that s(α) = i or t(α) = i. Moreover, we assume that Q has no loops (i.e. arrows α such that s(α) = t(α)) and no 2-cycles (i.e. pairs of distinct arrows α ≠ β such that s(α) = t(β) and t(β) = s(α)). The quiver Q is thus determined by the set I and the skew-symmetric integer matrix B = (b_{ij})_{I×I} such that, whenever the coefficient b_{ij} is positive, it equals the number of arrows from i to j in Q. Notice that if, for an integer x, we write [x]_{+} = max(x, 0), then the number of arrows from i to j in Q is [b_{ij}]_{+}. The mutation μ_{k}(Q) of Q at a vertex k is by definition the quiver with vertex set I whose numbers of arrows are given by the mutated matrix B′ = μ_{k}(B) as defined, for example, in definition 2.4 of [8]:

b′_{ij} = \{ \begin{matrix}−b_{ij} & if i = k or j = k; \\ b_{ij} + sgn(b_{ik})[b_{ij}b_{kj}]_{+} & otherwise. \end{matrix}

As in definition 2.8 of [8], we let 𝕋 = 𝕋_{I} be the regular tree whose edges are labeled by the elements of I such that for each vertex t and each element k of I, there is precisely one edge incident with t and labeled by k. We fix a vertex t_{0} of 𝕋 and define Q_{t_{0}} = Q. Clearly, there is a unique map assigning a quiver Q_{t} to each vertex t such that if t and t′ are linked by an edge labeled by k, we have Q_{t′} = μ_{k}(Q_{t}). In analogy with the terminology of [8], we call the map t ↦ Q_{t} the quiver pattern associated with t_{0} and Q.

Now for each vertex t of 𝕋, we define K_{t} to be the free abelian group on the symbols e_{i}^{t}, i ∈ I. For two vertices t and t′ linked by an edge labeled k, we let

ϕ_{t′,t}^{+} : K_{t} → K_{t′} respectively ϕ_{t′,t}^{−} : K_{t} → K_{t′}

be the linear map sending e_{j}^{t} to e_{j}^{t′} for each j ≠ k and sending e_{k} to

−e_{k}^{t′} + Σ_{j}[b_{jk}^{t}]_{+} e_{j}^{t′} respectively − e_{k}^{t′} + Σ_{j}[b_{kj}^{t}]_{+} e_{j}^{t′} ,

where (b_{ij}^{t}) is the skew-symmetric matrix associated with the quiver Q_{t}. We define the piecewise linear transformation

ϕ_{t′,t} : K_{t} → K_{t′}

to be the map whose restriction to the halfspace of elements with positive e_{k}^{t}-coordinate is ϕ_{t′,t}^{+} and whose restriction to the opposite halfspace is ϕ_{t′,t}^{−}. Thus, the image of an element g with coordinates g_{j}, j ∈ I, is the element g′ with coordinates

g′_{j} = \{ \begin{matrix}−g_{j} & if j = k; \\ g_{j} + [b_{jk}^{t}]_{+} g_{k} & if j ≠ k and g_{k} ≥ 0; \\ g_{j} + [b_{kj}^{t}]_{+} g_{k} & if j ≠ k and g_{k} ≤ 0. \end{matrix}

It is easy to check that this rule agrees with formula (7.18) in Conjecture 7.12 of [8]. If t and t′ are two arbitrary vertices of 𝕋, there is a unique path

t = t_{1}  t_{2}  …  t_{N} = t′

of edges leading from t to t′ and we define ϕ_{t′,t} to be the composition

ϕ_{t_{N} ,t_{N−1}} ∘ ⋯ ∘ ϕ_{t_{2},t_{1}}.

For a vertex t of 𝕋 and a vertex l of Q, the g^{†}-vector g_{l,t}^{†} is the element of the abelian group K_{t_{0}} defined by

g_{l,t}^{†} = ϕ_{t_{0},t}(e_{l}^{t}).

The g^{†}-cluster associated with a vertex t of 𝕋 is the set of g^{†}-vectors g_{l,t}^{†}, l ∈ I. If Conjecture 7.12 of [8] holds for the cluster algebra with principal coefficients associated with the matrix B, then it is clear that in the notations of formula (6.4) of [8], we have

g_{l,t}^{†} = g_{l,t}

for all vertices t of 𝕋 and all l ∈ I, , i.e. the g^{†}-vectors equal the g-vectors for the cluster algebra with principal coefficients associated with the skew-symmetric matrix B.

5. Rigid objects in 2-Calabi-Yau categories with cluster structure

Let C be a Hom-finite 2-Calabi-Yau category with a cluster-tilting subcategory T . Let Q = Q(T ) be the quiver of T . Recall that this means that the vertices of Q are the isomorphism classes of indecomposable objects of T and that the number of arrows from the isoclass of T_{1} to that of T_{2} equals the dimension of the space of irreducible morphisms

irr(T_{1}, T_{2}) = rad(T_{1}, T_{2})/rad^{2}(T_{1}, T_{2}) ,

where rad denotes the radical of T , i.e. the ideal such that rad(T_{1}, T_{2}) is formed by all non isomorphisms from T_{1} to T_{2}.

We make the following assumption on C: For each cluster-tilting subcategory T′ of C, the quiver Q(T′) does not have loops or 2-cycles. We refer to section 1, page 11 of [1] for a list of classes of examples where this assumption holds. By theorem 1.6 of [1], the assumption implies that the cluster-tilting subcategories of C determine a cluster structure for C. Let us recall what this means:

1) For each cluster-tilting subcategory T′ of C and each indecomposable S of T′, there is a unique (up to isomorphism) indecomposable S^{∗} not isomorphic to M and such that the additive subcategory T′′ = μ_{S}(T′) of C with

indec(T′′) = indec(T′) \ {S} ∪ {S^{∗}}

is a cluster-tilting subcategory;

2) the space of morphisms from S to ΣS^{∗} is one-dimensional and in the non-split

triangles

S^{∗} → B → S → ΣS^{∗} and S → B′ → S^{∗} → ΣS

the objects B and B′ belong to T′ ∩ T′′;

3) the multiplicity of an indecomposable L of T′ ∩ T′′ in B equals the number of arrows from L to S in Q(T′) and that from S^{∗} to L in Q(T′′); the multiplicity of L in B′ equals the number of arrows from S to L in Q(T′) and that from L to S^{∗} in Q(T′′);

4) finally, we have Q(T′′) = μ_{S}(Q(T′)). Let Q = Q(T ) be the quiver of T . Notice that its set of vertices is the set Q_{0} = I of

isomorphism classes of indecomposables of T . Let 𝕋 be the regular tree associated with Q as in section 4. We fix a vertex t_{0} of 𝕋 and put T_{t_{0}} = T . For two cluster tilting subcategories T′ and T′′ as above, let ψ_{T′′,T′} : indec(T′) → indec(T′′) be the bijection taking S to S^{∗} and fixing all other indecomposables.

Thanks to point 1), with each vertex t of 𝕋, we can associate a) a unique cluster-tilting subcatgory T_{t} and

b) a unique bijection

ψ_{t,t_{0}} : indec(T_{t_{0}}) → indec(T_{t})

such that T_{t_{0}} = T and that, whenever two vertices t and t′ are linked by an edge labeled by an indecomposable S of T = T_{t_{0}}, we have

a) T_{t′} = μ_{S′}(T_{t}), where S′ = ψ_{t,t_{0}}(S), and

b) ψ_{t′,t_{0}} = ψ_{t′,t} ∘ ψ_{t,t_{0}}.

Moreover, thanks to point 4), the map t ↦ Q(T_{t}) is the quiver-pattern associated with Q and t_{0} in section 4. Notice that the group K_{0}(T ) with the basis formed by the isomorphism classes of indecomposables canonically identifies with the free abelian group K_{t_{0}} of section 4. We define a cluster-tilting subcategory T′ to be reachable from T if we have T′ = T_{t} for some vertex t of the tree 𝕋. We define a rigid indecomposable M to be reachable from T if it belongs to a cluster-tilting subcategory which is reachable from T . Theorem. a) The index ind(M) of a rigid indecomposable reachable from T is a g^{†}-

vector and the map M ↦ ind(M) induces a bijection from the set of isomorphism classes of rigid indecomposables reachable from T onto the set of g^{†}-vectors.

b) Under the bijection M ↦ ind(M) of a), the cluster-tilting subcategories reachable from T are mapped bijectively to the g^{†}-clusters.

Proof. a) By assumption, there is a vertex t of 𝕋 such that M belongs to T_{t}. Now we use theorem 3 and induction on the length of the path joining t_{0} to t in the tree 𝕋 to conclude that

ind(M) = g_{M′,t}^{†} , where M = ψ_{t,t_{0}}(M′).

This formula shows that the map M ↦ ind(M) is a well-defined surjection onto the set of g^{†}-vectors. By theorem 2.3, the map M ↦ ind(M) is also injective. b) By assumption, a reachable cluster-tilting subcategory T′ is of the form T_{t} for some vertex t of the tree 𝕋. Thus its image is the g^{†}-cluster associated with t. This shows that the map is well-defined and surjective. It follows from a) that it is also injective. □

Acknowledgments

The second-named author would like to thank Jan Schröer for stimulating discussions. Both authors are grateful to Andrei Zelevinsky for helpful comments on a previous version of this article.

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Université Cergy-Pontoise/Saint-Martin, Départment de Mathématiques, UMR 8088 du CNRS, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France

E-mail address: Raika.Dehy@math.u-cergy.fr

UFR de Mathématiques, Université Denis Diderot – Paris 7, Institut de Mathématiques, UMR 7586 du CNRS, 2, place Jussieu, 75251 Paris Cedex 05, France

E-mail address: keller@math.jussieu.fr