TRANSVERSE QUIVER GRASSMANNIANS AND BASES IN AFFINE CLUSTER ALGEBRAS

G. DUPONT

Abstract. Sherman-Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. These constructions involve cluster monomials and Chebyshev polynomials of the first kind evaluated at a certain “imaginary” element in the cluster algebra.

For any affine quiver Q, we provide a geometric realization of these elements in terms of the representation theory of Q. This is done by introducing an analogue of the Caldero-Chapoton cluster character where the usual quiver grassmannian is replaced by a constructible subset called transverse quiver grassmannian.

Contents

1. Introduction 1 2. Background, notations and terminology 3 3. Difference properties of higher orders 9 4. Integrable bundle on rep(Q) and their characters 15 5. A geometrization of B(Q) 18 6. Examples 20 Acknowledgements 25 References 25

1. Introduction

Cluster algebras were introduced by Fomin and Zelevinsky in order to define a combinatorial framework for studying positivity in algebraic groups and canonical bases in quantum groups [FZ02, FZ03, BFZ05, FZ07]. Since then, cluster algebras found applications in various areas of mathematics like Lie theory, combinatorics, Teichmüller theory, Poisson geometry or quiver representations.

A (coefficient-free) cluster algebra A is a commutative ℤ-algebra equipped with a distinguished set of generators, called cluster variables, gathered into possibly overlapping sets of fixed cardinality, called clusters. Monomials in variables belonging to a same cluster are called cluster monomials. According to the so-called Laurent phenomenon [FZ02], it is known that A ⊂ ⋂_{c} ℤ[c^{±1}] where c runs over the clusters in A. An element y ∈ A is called positive if y ∈ ⋂_{c} ℤ_{≥0}[c^{±1}]\{0} where c runs over the clusters in A. A ℤ-basis B ⊂ A is called canonically positive if the semi-ring of positive elements in A coincides with the set of ℤ_{≥0}-linear combinations of elements of B. Note that if such a basis exists, it is unique.

2000 Mathematics Subject Classification. 13F99, 16G99, 16G20 .

Key words and phrases. Cluster algebras, Canonically positive basis, Chebyshev Polynomials, Cluster categories, Cluster characters.

The problems of both the existence and the description of a canonically positive basis in an arbitrary cluster algebra are still widely open. This was first solved in the particular case of cluster algebras of finite type 𝔸_{2} and affine type 𝔸̃_{1,1} by Sherman and Zelevinsky [SZ04]. It was later extended by Cerulli for cluster algebras of affine type 𝔸̃_{2,1} [Cer09]. At the best of our knowledge, these are the only known constructions of canonically positive bases in cluster algebras.

Using categorifications of acyclic cluster algebras with cluster categories and cluster characters, it is possible to rephrase Sherman-Zelevinsky and Cerulli constructions in order to fit into the more general context of acyclic cluster algebras associated to arbitrary affine quivers.

If Q is an acyclic quiver and u is a Q_{0}-tuple of indeterminates over ℤ, we denote by A(Q) the acyclic cluster algebra with initial seed (Q, u). We denote by C_{Q} the associated cluster category and by X_{?} : Ob(C_{Q})⟶ ℤ[u^{±1}] the Caldero-Chapoton map on C_{Q}, also called cluster character (see Section 2 for details). When Q is an affine quiver with positive minimal imaginary root δ, we set

B(Q) = M(Q) ⊔ {F_{n}(X_{δ})X_{R}|n ≥ 1, R is regular and rigid in kQ-mod}

where M(Q) denotes the set of cluster monomials in A(Q), F_{n} denotes the n-th normalized Chebyshev polynomials of the first kind and X_{δ} is the evaluation of X_{?} at any quasi-simple module in an homogeneous tube of the Auslander-Reiten quiver Γ(kQ) of kQ-mod.

If Q is of type 𝔸̃_{1,1} (respectively 𝔸̃_{2,1}), the set B(Q) coincides with the canonically positive basis constructed in [SZ04] (respectively [Cer09]). It was conjectured in [Dup09b, Conjecture 7.10] that, for any affine quiver Q, the set B(Q) is the canonically positive basis of A(Q). Using the so-called generic basis, it is possible to prove that, for any affine quiver Q, the set B(Q) is a ℤ-basis in A(Q) [Dup08, DXX09]. Nevertheless, it is not known if this basis is the canonically positive basis in general.

An essential problem for investigating this question is due to the fact that the elements of the form F_{n}(X_{δ})X_{R} are defined combinatorially and have no representationtheoretic or geometric interpretation. The aim of this article is to provide such an interpretation.

Extending the idea of Caldero and Chapoton [CC06], for any integrable bundle F on rep(Q) (see Section 4 for detailed definitions), we define a character θ_{F} from the set of objects in C_{Q} to the set ℤ[u^{±1}]. With this terminology, the Caldero-Chapoton map X_{?} corresponds to the character θ_{Gr} where Gr : M ↦ Gr(M) denotes the integrable bundle of quiver grassmannians.

We introduce a constructible subset Tr(M) ⊂ Gr(M), called transverse quiver grassmannian. We prove that the bundle M ↦ Tr(M) is integrable on rep(Q) and that the elements in B(Q) can be described using the associated character θ_{Tr}. More precisely, we prove that

F_{l}(X_{δ}) = θ_{Tr}(M)

for any indecomposable object M with dimension vector lδ. As opposed to θ_{Gr}, it turns out that θ_{Tr} is independent on the tube containing M. In particular, it takes the same values if M belongs to an homogeneous or to an exceptional tube. This is surprising since the usual quiver grassmannians of two indecomposable modules of dimension lδ belonging to tubes of different rank are in general completely different.

Moreover, if R is an indecomposable regular rigid kQ-module, then

F_{l}(X_{δ})X_{R} = θ_{Tr}(M)

where M is the unique indecomposable kQ-module of dimension lδ + dim R. As a consequence, we obtain the following description of the set B(Q) :

B(Q) = {θ_{Tr}(M ⊕ R)|M is an indecomposable (or zero) regular kQ-module,
R is any rigid object in C_{Q} such that Ext_{C_{Q}}^{1}(M, R) = 0} .

The paper is organized as follows. In Section 2, we start by recalling several results concerning Chebyshev and generalized Chebyshev polynomials. Then, we recall necessary background on cluster categories and cluster characters associated to acyclic and especially affine quiver. Finally, we recall the known results concerning construction of bases in affine cluster algebras.

In Section 3, we use the combinatorics of generalized Chebyshev polynomials in order to prove relations for cluster characters associated to regular kQ-modules when Q is an affine quiver with minimal imaginary root δ. These relations are generalizations of the so-called difference property, introduced previously in [Dup08] in order to compute the difference between cluster characters evaluated at indecomposable modules of dimension vector δ in different tubes.

Section 4 introduces the notions of integrable bundles on rep(Q) and associated characters for any acyclic quiver. In this terminology, the Caldero-Chapoton map is the character associated to the quiver grassmannian bundle. For affine quivers, we introduce the integrable bundle Tr of grassmannian of transverse submodules and prove that it coincides with the Caldero-Chapoton map on rigid objects in the cluster category.

In Section 5, we prove that the elements in B(Q) can be expressed as values of the character θ_{Tr} associated to the integrable bundle Tr of rep(Q). This provides a geometrization of the set B(Q).

In Section 6, we illustrate some of our results for quivers of affine types 𝔸̃_{1,1} and 𝔸̃_{2,1}, putting into context the results of [SZ04] and [Cer09].

2. Background, notations and terminology

Throughout the paper, k will denote the field ℂ of complex numbers. Given a quiver Q, we denote by Q_{0} its set of arrows, Q_{1} its set of vertices. We always assume that Q_{0}, Q_{1} are finite sets and that the underlying unoriented graph of Q is connected. A quiver is called acyclic if it does not contain any oriented cycles.

We now fix an acyclic quiver Q and a Q_{0}-tuple u = (u_{i}|i ∈ Q_{0}) of indeterminates over ℤ. We denote by A(Q) the coefficient-free cluster algebra with initial seed (Q, u).

2.1. Chebyshev polynomials and their generalizations. Chebyshev and generalized Chebyshev polynomials are orthogonal polynomials in one or several variables playing an important role in the context of cluster algebras associated to representation-infinite quivers [SZ04, CZ06, Dup09a, Dup09b]. Before introducing cluster categories and cluster characters, we recall some basic results concerning these polynomials.

For any l ≥ 0, the l-th normalized Chebyshev polynomial of the first kind is the polynomial F_{l} in ℤ[x] defined inductively by

F_{0}(x) = 2, F_{1}(x) = x,
F_{l}(x) = xF_{l−1}(x) − F_{l−2}(x) for any l ≥ 2.

F_{l} is characterized by the following identity in ℤ[t, t^{−1}] :

F_{l}(t + t^{−1}) = t^{l} + t^{−l}.

These polynomials first appeared in the context of cluster algebras in [SZ04].

For any l ≥ 0, the l-th normalized Chebyshev polynomial of the second kind is the polynomial S_{l} in ℤ[x] defined inductively by

S_{0}(x) = 1, S_{1}(x) = x,
S_{l}(x) = xS_{l−1}(x) − S_{l−2}(x) for any l ≥ 2.

S_{l} is characterized by the following identity in ℤ[t, t^{−1}] :

S_{l}(t + t^{−1}) = ∑_{k=0}^{n} t^{n−2k}.

Second kind Chebyshev polynomials first appeared in the context of cluster algebras in [CZ06]. For any l ≥ 1, S_{l}(x) is the polynomial given by

S_{l}(x) = det [ \begin{matrix}x & 1 & & & (0) \\ 1 & x & ... \\ & ... & ... & ... \\ & & ... & ... & 1 \\ (0) & & & 1 & x \end{matrix} ]

where the matrix is tridiagonal in M_{l}(ℤ[x]). First kind and second kind Chebyshev polynomials are related by :

F_{l}(x) = S_{l}(x) − S_{l−2}(x)

for any l ≥ 1 with the convention that S_{−1}(x) = 0.

Fix {x_{i}|i ≥ 1} a family of indeterminates over ℤ. For any l ≥ 0, the l-th generalized Chebyshev polynomial is the polynomial in ℤ[x_{1}, …, x_{l}] defined inductively by

P_{0} = 1, P_{1}(x_{1}) = x_{1},
P_{l}(x_{1}, …, x_{l}) = x_{l}P_{l−1}(x_{1}, …, x_{l−1}) − P_{l−2}(x_{1}, …, x_{l−2}) for any l ≥ 2.

Equivalently,

P_{l}(x_{1}, …, x_{l}) = det [ \begin{matrix}x_{l} & 1 & & & (0) \\ 1 & x_{l−1} & ... \\ & ... & ... & ... \\ & & ... & ... & 1 \\ (0) & & & 1 & x_{1} \end{matrix} ]

where the matrix is tridiagonal in M_{l}(ℤ[x_{1}, …, x_{l}]). These polynomials first appeared in the context of cluster algebras in [Dup09a] under the name of generalized Chebyshev polynomials of infinite rank and similar polynomials also arose in the context of cluster algebras in [YZ08, Dup09b].

2.2. Cluster categories and cluster characters. Let kQ-mod be the category of finitely generated left-modules over the path algebra kQ of Q. As usual, this category will be identified with the category rep(Q) of finite dimensional representations of Q over k.

For any vertex 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 ⟨−, −⟩ the homological Euler form defined on kQ-mod by

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

for any two kQ-modules M, N. Since Q is acyclic, kQ is a finite dimensional hereditary algebra so that ⟨−, −⟩ is well defined on the Grothendieck group K_{0}(kQ).

For any kQ-module M, the dimension vector of M is

dim M = (dim Hom_{kQ}(P_{i}, M))_{i∈Q_{0}} ∈ ℕ^{Q_{0}}.

Viewed as a representation of Q, dim M = (dim M(i))_{i∈Q_{0}} where M(i) is the kvector space at vertex i in the representation M of Q. The dimension vector map dim induces an isomorphism of abelian group

dim : K_{0}(kQ) ⟶^{∼} ℤ^{Q_{0}}

sending the simple S_{i} to the i-th vector of the canonical basis of ℤ^{Q_{0}}.

The cluster category was introduced in [BMR^{+}06] (see also [CCS06]) in order to define a categorical framework for studying the cluster algebra A(Q). Let D^{b}(kQ) be the bounded derived category of kQ-mod with shift functor [1] and Auslander-Reiten translation τ. The cluster category is the orbit category C_{Q} of the autofunctor τ^{−1}[1] in D^{b}(kQ). It is a 2-Calabi-Yau triangulated category whose indecomposable objects are either indecomposable kQ-modules or shifts of indecomposable projective modules [Kel05, BMR^{+}06]. In particular, every object M in C_{Q} can be decomposed into

M = M_{0} ⊕ P_{M}[1]

where M_{0} is a kQ-module and P_{M} is a projective kQ-module.

Given a representation M of Q, the quiver grassmannian of M is the set Gr(M) of subrepresentations of M. For any element e ∈ ℤ^{Q_{0}}, the set

Gr_{e}(M) = {N ⊂ M|dim N = e}

is a projective variety. We denote by χ(Gr_{e}(M)) its Euler characteristic with respect to the simplicial cohomology.

Definition 2.1 ([CC06]). The Caldero-Chapoton map is the map

X_{?} : rep(Q)⟶ ℤ[u^{±1}]

defined by :

(a) If M is an indecomposable kQ-module, then

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

where u^{d} = ∏_{i∈Q_{0}} u_{i}^{d_{i}} for any d = (d_{i})_{i∈Q_{0}} ∈ ℤ^{Q_{0}}.

(b) For any i ∈ Q_{0},

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

(c) For any two objects M, N in C_{Q},

X_{M⊕N} = X_{M}X_{N}.

Note that equation (1) also holds for decomposable modules.

Caldero and Keller proved that X_{?} induces a 1-1 correspondence between indecomposable rigid objects in C_{Q} and cluster variables in A(Q). Moreover, X_{?} induces a 1-1 correspondence between cluster-tilting objects in C_{Q} and clusters in A(Q) [CK06, Theorem 4]. In particular, it is possible to have the following description of cluster monomials in A(Q) :

M(Q) = {X_{M}|M is rigid in C_{Q}} .

For any Laurent polynomial L ∈ ℤ[u^{±1}], the denominator vector of L it the Q_{0}- tuple den (L) ∈ ℤ^{Q_{0}} such that there exists a polynomial P (u_{i}|i ∈ Q_{0}) not divisible by any u_{i} such that

L = \frac{P (u_{i}|i ∈ Q_{0})}{u^{den (L)}}.

We define the dimension vector map dim_{C_{Q}} on C_{Q} by setting dim_{C_{Q}}M = dim M if M is a kQ-module and dim_{C_{Q}}P_{i}[1] = −dim S_{i} and extending by additivity. Note that, for any kQ-module M, we have dim M = dim_{C_{Q}}(M) so that, we will abuse notations and write dim M for any object in C_{Q}. Caldero-Keller’s denominators theorem [CK06, Theorem 3] relates the denominator vector of the character with the dimension vector of the corresponding object in the cluster category, namely

den (X_{M}) = dim M

for any object M in C_{Q}.

Moreover, it was proved in [CK08, CK06] that X_{?} induces on A(Q) a structure of Hall algebra on the cluster category C_{Q}. We will use a particular case of this Hall structure in the following (Theorem 2.2).

2.3. Representation theory of affine quivers. We shall briefly recall some wellknown facts concerning the representation theory of affine quivers. We refer the reader to [SS07, Rin84] for details.

We now fix an affine quiver, that is, an acyclic quiver of type 𝔸̃_{n}, 𝔻̃_{n}, 𝔼̃_{6}, 𝔼̃_{7} of 𝔼̃_{8} where n ≥ 1. We will sometimes say that a quiver is of affine type 𝔸̃_{r,s} if it is an orientation of an affine diagram of affine type 𝔸̃_{r+s−1} with r arrows going clockwise and s arrows going counterclockwise. Let 𝔤_{Q} denote the Kac-Moody algebra associated to Q.

We denote by Φ_{>0} the set of positive roots of 𝔤_{Q}, by Φ_{>0}^{re} the set of positive real roots and by Φ_{>0}^{im} the set of positive imaginary roots. There exists a unique δ ∈ Φ_{>0} such that Φ_{>0}^{im} = ℤ_{>0}δ. We always identify the root lattice of 𝔤_{Q} with ℤ^{Q_{0}} by sending the i-th simple root of 𝔤_{Q} to the i-th vector of the canonical basis of ℤ^{Q_{0}}.

According to Kac’s theorem, for any d ∈ ℕ^{Q_{0}}, there exists an indecomposable representation M such that dim M = d if and only if d ∈ Φ_{>0}. Moreover, this representation is unique up to isomorphism if and only if d ∈ Φ_{>0}^{re}. A positive root d is called a Schur root if there exists a (necessarily indecomposable) representation M of Q such that dim M = d and End_{kQ}(M) ≃ k.

We define a partial order ≤ on the root lattice by setting

e ≤ f ⇔ e_{i} ≤ d_{i} for any i ∈ Q_{0}.

and we set

e ≨ f if e ≤ f and e ≠ f.

The Auslander-Reiten quiver Γ(kQ) of kQ-mod contains infinitely many connected components. There exists a connected component containing all the projective (resp. injective) modules, called preprojective (resp. preinjective) component of Γ(kQ) and denoted by P (resp. I). The other components are called regular. A kQ-module M is called preprojective (resp. preinjective, regular) if each indecomposable direct summand of M belong to a preprojective (resp. preinjective, regular) component.

It is convenient to introduce the so-called defect form on ℤ^{Q_{0}}. It is given by

∂_{?} : \{ \begin{matrix}ℤ^{Q_{0}} & ⟶ & ℤ \\ e & ↦ & ∂_{e} = ⟨δ, e⟩ \end{matrix}

By definition, the defect ∂_{M} of a kQ-module M is the defect ∂_{dim M} of its dimension vector. It is well-known that an indecomposable kQ-module M is preprojective (resp. preinjective, regular) if and only if ∂_{M} < 0 (resp. > 0, = 0).

The regular components in Γ(kQ) form a ℙ^{1}(k)-family of tubes. Thus, for every tube T , there exists an integer p ≥ 1, called rank of T such that T ≃ ℤ𝔸_{∞}/(τ^{p}). The tubes of rank one are called homogeneous, the tubes of rank p > 1 are called exceptional. At most three tubes are exceptional in Γ(kQ). It is well-known that the full subcategory of kQ-mod generated by the objects in any tube T is standard, that is, isomorphic to the mesh category of T . It is also known that there are neither morphisms nor extensions between pairwise distinct tubes.

An indecomposable regular module M is called quasi-simple if it is at the mouth of the tube, or equivalently, if it does not contain any proper regular submodule. For any quasi-simple module R in a tube T and any integer l ≥ 1, we denote by R^{(l)} the unique indecomposable kQ-module with quasi-socle R and quasi-length l. For any indecomposable regular kQ-module R^{(l)}, we denote by

q.socR^{(l)} = R

the quasi-socle of M and by

q.radR^{(l)} = R^{(l−1)}

the quasi-radical of M with the convention that R^{(0)} = 0.

For any indecomposable regular kQ-module M, we have the following :

M is rigid ⇔ dim M ≨ δ ;

End_{kQ}(M) ≃ k ⇔ dim M ≤ δ.

Cluster characters in tubes are known to be governed by the combinatorics of generalized Chebyshev polynomials. More precisely, it is proved in [Dup09a, Theorem 5.1] that for any quasi-simple module M in a tube T , we have

X_{M^{(l)}} = P_{l}(X_{M}, X_{τ^{−1}M}, …, X_{τ^{−l+1}M}).

In particular, if T is homogeneous, we get

X_{M^{(l)}} = S_{l}(X_{M})

recovering a result of [CZ06].

In [Dup09a, Theorem 7.2], generalized Chebyshev polynomials provide multiplication formulas for cluster characters associated to indecomposable regular kQmodules. The following theorem will be essential in the proofs :

Theorem 2.2 ([Dup09a]). Let Q be an affine quiver and T be a tube of rank p in Γ(kQ). Let R_{i}, i ∈ ℤ denote the quasi-simple modules in T ordered such that τR_{i} ≃ R_{i−1} and R_{i+p} ≃ R_{i} for any i ∈ ℤ. Let m, n > 0 be integers and j ∈ [0, p−1]. Then, for every k ∈ ℤ such that 0 < j + kp < n and m > n − j − kp, we have the following identity :

X_{R_{j}^{(m)}}X_{R_{0}^{(n)}} = X_{R_{0}^{(m+j+kp)}}X_{R_{j}^{(n−j−kp)}} + X_{R_{0}^{(j+kp−1)}}X_{R_{n+1}^{(m+j+kp−n−1)}}.

2.4. Bases in affine cluster algebras. We shall now review some results concerning the construction of ℤ-bases in cluster algebras associated to affine quivers. In this section, Q still denotes a quiver of affine type with positive minimal imaginary root δ.

It is known that if M, N are quasi-simple modules in distinct homogeneous tubes, then X_{M} = X_{N} (see e.g. [Dup08]). We denote by X_{δ} this common value. Following the terminology of [Dup08], X_{δ} is called the generic variable of dimension δ.

The following holds :

Theorem 2.3 ([Dup08, DXX09]). Let Q be an affine quiver, then

G(Q) = M(Q) ⊔ {X_{δ}^{l}X_{R}|l ≥ 1, R is regular and rigid in kQ-mod} .

Moreover, den induces a 1-1 correspondence from G(Q) to ℤ^{Q_{0}}.

The set G(Q) is called the generic basis of A(Q).

Since F_{l} and S_{l} are monic polynomials of degree l, it follows that, for any affine quiver Q, the sets

B(Q) = M(Q) ⊔ {F_{l}(X_{δ})X_{R}|l ≥ 1, R is regular and rigid in kQ-mod}

and

C(Q) = M(Q) ⊔ {S_{l}(X_{δ})X_{R}|l ≥ 1, R is regular and rigid in kQ-mod}

are ℤ-bases of the cluster algebra A(Q).

When Q is the Kronecker quiver B(Q) coincides with the canonically positive basis constructed by Sherman and Zelevinsky [SZ04] and C(Q) coincides with the basis constructed by Caldero and Zelevinsky [CZ06]. When Q is a quiver of affine type 𝔸̃_{2,1}, the basis B(Q) is the canonically positive basis of A(Q) constructed by Cerulli [Cer09].

Since X_{δ} = X_{M} for any quasi-simple module in an homogeneous tube, it follows that S_{l}(X_{δ}) = X_{M^{(l)}} so that the set C(Q) has an interpretation in terms of the cluster character X_{?}. No such interpretation is known for the set B(Q). The aim of this paper is to provide one.

The map ϕ : G(Q)⟶ B(Q) preserving cluster monomials and sending X_{δ}^{l}X_{R} to F_{l}(X_{δ})X_{R} for any l ≥ 1 and any rigid regular module R is a 1-1 correspondence. We denote by

𝔟_{?} : \{ \begin{matrix}ℤ^{Q_{0}} & ⟶^{1:1} & B(Q) \\ d & ↦ & 𝔟_{d} \end{matrix}

the 1-1 correspondence obtained by composing the above bijection with the one provided in Theorem 2.3.

3. Difference properties of higher orders

In this section, Q still denotes an affine quiver with positive minimal imaginary root δ.

3.1. The difference property. In [Dup08] was introduced the difference property which relates the possibly different values of cluster characters evaluated at different indecomposable representations of dimension δ. This difference property is crucial in [Dup08]. It is also an essential ingredient in the present article since the transverse grassmannians will precisely arise from difference properties of higher orders.

This difference property was established in [Dup08] for affine type 𝔸̃ and in [DXX09] for the other types. It can be expressed as follows :

Theorem 3.1 ([Dup08, DXX09]). Let Q be an affine quiver and M be any indecomposable module of dimension δ, then

𝔟_{δ} = X_{δ} = X_{M} − X_{q.radM/q.socM}

with the convention that X_{q.radM/q.socM} = 0 if M is quasi-simple.

3.2. Higher difference properties. The aim of this section is to provide a analogous of Theorem 3.1 for 𝔟_{d} when d is any positive root with zero defect. We will first consider the imaginary roots and then the real roots of defect zero.

We fix a tube T in Γ(kQ) of rank p ≥ 1. The quasi-simples of T are denoted by R_{0}, …, R_{p−1} and are ordered such that τR_{i} ≃ R_{i−1} for any i ∈ ℤ/pℤ. Note that for any l ≥ 1, 0 ≤ k ≤ p − 1 and any i ∈ ℤ/pℤ, dim R_{i}^{(lp)} = lδ ∈ Φ_{>0}^{im} and dim R_{i}^{(lp+k)} ∈ Φ_{>0}^{re} if k ≠ 0.

The following technical lemma will be useful in the proof of Proposition 3.3 :

Lemma 3.2. With the above notations, for any l ≥ 1

X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}} = X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}}

Proof. We first notice that generalized Chebyshev polynomials are symmetric in the sense that for every i ∈ ℤ and n ≥ 1,

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

If l = 1, the result is obvious. We thus fix some l ≥ 2. For technical convenience, we denote by R_{i}, i ∈ ℤ the quasi-simple modules in T and we assume that R_{i} ≃ R_{i+p} for every i ∈ ℤ. Consider the morphism of ℤ-algebras

ϕ : \{ \begin{matrix}ℤ[X_{R_{0}}, …, X_{R_{lp−1}}] & ⟶ & ℤ[X_{R_{0}}, …, X_{R_{lp−1}}] \\ X_{R_{i}} & ↦ & X_{R_{lp−1−i}} & for all i = 0, …, lp − 1. \end{matrix}

It is well defined since X_{R_{0}}, …, X_{R_{p−1}} are known to be algebraically independent over ℤ (see e.g. [Dup09a]).

According to Theorem 2.2, we have

X_{R_{1}^{(p−1)}}X_{R_{0}^{(lp−1)}} = X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}} − X_{R_{p+1}^{((l−1)p−2)}}.

According to [Dup09a, Theorem 5.1], each of the X_{R_{j}}^{(k)} appearing above lies in ℤ[X_{R_{0}}, …, X_{R_{lp−1}}]. We can thus apply ϕ and we get

(2)  ϕ(X_{R_{1}^{(p−1)}})ϕ(X_{R_{0}^{(lp−1)}}) = ϕ(X_{R_{0}^{(p)}})ϕ(X_{R_{1}^{(lp−2)}}) − ϕ(X_{R_{p+1}^{((l−1)p−2)}}).

10 G. DUPONT

We now compute these images under ϕ.

ϕ(X_{R_{1}^{(p−1)}}) = ϕ(P_{p−1}(X_{R_{1}}, …, X_{R_{p−1}}))

= P_{p−1}(ϕ(X_{R_{1}}), …, ϕ(X_{R_{p−1}})) = P_{p−1}(X_{R_{lp−2}}, …, X_{R_{(l−1)p}}) = P_{p−1}(X_{R_{(l−1)p}}, …, X_{R_{lp−2}}) = X_{R_{(l−1)p}^{(p−1)}} = X_{R_{0}^{(p−1)}}

ϕ(X_{R_{0}^{(lp−1)}}) = ϕ(P_{lp−1}(X_{R_{0}}, …, X_{R_{lp−2}}))

= P_{lp−1}(ϕ(X_{R_{0}}), …, ϕ(X_{R_{lp−2}})) = P_{lp−1}(X_{R_{lp−1}}, …, X_{R_{1}}) = P_{lp−1}(X_{R_{1}}, …, X_{R_{lp−1}}) = X_{R_{1}^{(lp−1)}}

ϕ(X_{R_{0}^{(p)}}) = ϕ(P_{p}(X_{R_{0}}, …, X_{R_{p−1}}))

= P_{p}(ϕ(X_{R_{0}}), …, ϕ(X_{R_{p−1}})) = P_{p}(X_{R_{lp−1}}, …, X_{R_{(l−1)p}}) = P_{p}(X_{R_{(l−1)p}}, …, X_{R_{lp−1}}) = X_{R_{(l−1)p}^{(p)}} = X_{R_{0}^{(p)}}

ϕ(X_{R_{1}^{(lp−2)}}) = ϕ(P_{lp−2}(X_{R_{1}}, …, X_{R_{lp−2}}))

= P_{lp−2}(ϕ(X_{R_{1}}), …, ϕ(X_{R_{lp−2}})) = P_{lp−2}(X_{R_{lp−2}}, …, X_{R_{1}}) = P_{lp−2}(X_{R_{1}}, …, X_{R_{lp−2}}) = X_{R_{1}^{(lp−2)}}

ϕ(X_{R_{p+1}^{((l−1)p−2)}}) = ϕ(P_{(l−1)p−2}(X_{R_{p+1}}, …, X_{R_{lp−2}}))

= P_{(l−1)p−2}(ϕ(X_{R_{p+1}}), …, ϕ(X_{R_{lp−2}})) = P_{(l−1)p−2}(X_{R_{(l−1)p−2}}, …, X_{R_{1}}) = P_{(l−1)p−2}(X_{R_{1}}, …, X_{R_{(l−1)p−2}}) = X_{R_{1}^{(l−1)p−2}} Replacing in Equation (2), we get

X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} = X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}} − X_{R_{1}^{((l−1)p−2)}}
= X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}} − X_{R_{p+1}^{((l−1)p−2)}}
= X_{R_{1}^{(p−1)}}X_{R_{0}^{(lp−1)}}

which proves the lemma. □

We can now prove some higher difference properties for imaginary roots.

Proposition 3.3. Fix l ≥ 1. Then, for any indecomposable representation M in rep(Q, lδ), we have

𝔟_{lδ} = F_{l}(X_{δ}) = X_{M} − X_{q.radM/q.socM}

with the convention that X_{q.radM/q.socM} = 0 if M is quasi-simple.

Proof. We first treat the case where M is an indecomposable representation of dimension lδ in an homogeneous tube. This is not necessary but in this particular case, the proof is straightforward. We write M = R^{(l)} for some quasi-simple module R in an homogeneous tube. If l = 1, the proposition follows from Theorem 3.1. If l ≥ 2, q.radM/q.socM ≃ R^{(l−2)} so that

X_{M} − X_{q.radM/q.socM} = X_{R^{(l)}} − X_{R^{(l−2)}}
= S_{l}(X_{R}) − S_{l−2}(X_{R})
= F_{l}(X_{R})
= F_{l}(X_{δ}).

We now assume that M is an indecomposable representation of dimension lδ in an exceptional tube T of rank p ≥ 2. We denote R_{0}, …, R_{p−1} the quasi-simples in T ordered such that τR_{i} ≃ R_{i−1} for any i ∈ ℤ/pℤ. We can thus write M ≃ R_{i}^{(lp)} for some i ∈ ℤ/pℤ. Without loss of generality, we assume that i = 0. In order to simplify notations, for any l ≥ 1, we denote by

Δ_{l} = X_{R_{0}^{(lp)}} − X_{R_{1}^{(lp−2)}}.

We thus have to prove that for any

Δ_{l} = F_{l}(X_{δ}).

The central tool in this proof is Theorem 2.2. According to theorem 3.1, we have

X_{δ} = X_{R_{0}^{(p)}} − X_{R_{1}^{(p−2)}}

so that the proposition holds for l = 1.

We now prove it for l = 2. We have

F_{2}(X_{δ}) = X_{δ}^{2} − 2
= (X_{R_{0}^{(p)}} − X_{R_{1}^{(p−2)}})^{2} − 2
= X_{R_{0}^{(p)}}^{2} + X_{R_{1}^{(p−2)}}^{2} − 2X_{R_{0}^{(p)}}X_{R_{1}^{(p−2)}} − 2

but according to the almost split multiplication formula ([CC06, Proposition 3.10]), we have

X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}} = X_{R_{0}^{(p)}}X_{R_{1}^{(p−2)}}

so that

F_{2}(X_{δ}) = X_{R_{0}^{(p)}}^{2} − 2X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}} + X_{R_{1}^{(p−2)}}^{2}.

But, according to Theorem 2.2, we have

X_{R_{0}^{(p)}}^{2} = X_{R_{0}^{(p)}}X_{R_{0}^{(p)}}
= X_{R_{0}^{(2p)}} + X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}}

so that finally

F_{2}(X_{δ}) = X_{R_{0}^{(2p)}} − X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}} + X_{R_{1}^{(p−2)}}^{2}.

Thus,

F_{2}(X_{δ}) = Δ_{2}
⇔ − X_{R_{1}^{(2p−2)}} = X_{R_{1}^{(p−2)}}^{2} − X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}}
⇔X_{R_{1}^{(2p−2)}} + X_{R_{1}^{(p−2)}}^{2} − X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}} = 0

But

X_{R_{1}^{(2p−2)}} = −X_{R_{1}^{(p−3)}}X_{R_{0}^{(p−1)}} + X_{R_{1}^{(p−2)}}X_{R_{p−1}^{(p)}}

So that,

F_{2}(X_{δ}) = Δ_{2}
⇔X_{R_{1}^{(p−2)}}^{2} + X_{R_{1}^{(p−2)}}X_{R_{p−1}^{(p)}} − X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−1)}} − X_{R_{0}^{(p−1)}}X_{R_{1}^{(p−3)}} = 0
⇔X_{R_{1}^{(p−2)}} [X_{R_{1}^{(p−2)}} + X_{R_{p−1}^{(p)}}] − X_{R_{0}^{(p−1)}} [X_{R_{1}^{(p−3)}} + X_{R_{1}^{(p−1)}}] = 0.

But Theorem 2.2 gives

X_{R_{p−1}}X_{R_{0}^{(p−1)}} = X_{R_{1}^{(p−2)}} + X_{R_{p−1}^{(p)}}
X_{R_{1}^{(p−2)}}X_{R_{p−1}} = X_{R_{1}^{(p−3)}} + X_{R_{1}^{(p−1)}}

so that we get

F_{2}(X_{δ}) = Δ_{2}

and the proposition is proved for l = 2.

For l > 2, we will use the three terms relations for first kind Chebyshev polynomials

F_{l}(x) = xF_{l−1}(x) − F_{l−2}(x).

Thus, it is enough to prove that for any l ≥ 2,

Δ_{l+1} = Δ_{1}Δ_{l} − Δ_{l−1}.

In order to simplify notations, we by (LHS) the left-hand side of the equality and by (RHS) the right-hand side of the above equality. We thus have

(RHS) = (X_{R_{0}^{(p)}} − X_{R_{1}^{(p−2)}})(X_{R_{0}^{(lp)}} − X_{R_{1}^{(lp−2)}}) − (X_{R_{0}^{((l−1)p)}} − X_{R_{1}^{((l−1)p−2)}})
= X_{R_{0}^{(p)}}X_{R_{0}^{(lp)}} − X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}} − X_{R_{1}^{(p−2)}}X_{R_{0}^{(lp)}}
+ X_{R_{1}^{(p−2)}}X_{R_{1}^{(lp−2)}} − X_{R_{0}^{((l−1)p)}} + X_{R_{1}^{((l−1)p−2)}}.

But, according to the multiplication theorem, we get

X_{R_{0}^{(p)}}X_{R_{0}^{(lp)}} = X_{R_{0}^{((l+1)p)}} + X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}}

TRANSVERSE GRASSMANNIANS AND BASES 13

So that

(LHS) = (RHS) ⇔ X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} − X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}} − X_{R_{1}^{(p−2)}}X_{R_{0}^{(lp)}}
+ X_{R_{1}^{(p−2)}}X_{R_{1}^{(lp−2)}} − X_{R_{0}^{((l−1)p)}} + X_{R_{1}^{((l−1)p−2)}} + X_{R_{1}^{((l+1)p−2)}} = 0.

Applying the multiplication theorem, we get

X_{R_{0}^{(lp−2)}}X_{R_{lp−2}^{(p)}} = X_{R_{0}^{((l+1)p−2)}} + X_{R_{0}^{(lp−3)}}X_{R_{lp−1}^{(p−1)}}

so that

X_{R_{1}^{((l+1)p−2)}} = X_{R_{1}^{(lp−2)}}X_{R_{p−1}^{(p)}} − X_{R_{1}^{(lp−3)}}X_{R_{0}^{(p−1)}}

and thus

(LHS) = (RHS) ⇔ X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} − X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}}
− X_{R_{1}^{(p−2)}}X_{R_{0}^{(lp)}} + X_{R_{1}^{(p−2)}}X_{R_{1}^{(lp−2)}}
− X_{R_{0}^{((l−1)p)}} + X_{R_{1}^{((l−1)p−2)}}
+ X_{R_{1}^{(lp−2)}}X_{R_{p−1}^{(p)}} − X_{R_{1}^{(lp−3)}}X_{R_{0}^{(p−1)}} = 0

but

X_{R_{0}^{(p)}}X_{R_{1}^{(lp−2)}} = X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}} + X_{R_{1}^{((l−1)p−2)}}.

Thus,

(LHS) = (RHS) ⇔ X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} − X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}}
− X_{R_{1}^{(p−2)}}X_{R_{0}^{(lp)}} + X_{R_{1}^{(p−2)}}X_{R_{1}^{(lp−2)}}
− X_{R_{0}^{((l−1)p)}} + X_{R_{1}^{(lp−2)}}X_{R_{p−1}^{(p)}} − X_{R_{1}^{(lp−3)}}X_{R_{0}^{(p−1)}} = 0
⇔ X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} − X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}}
− X_{R_{1}^{(p−2)}}X_{R_{0}^{(lp)}} − X_{R_{0}^{((l−1)p)}} − X_{R_{1}^{(lp−3)}}X_{R_{0}^{(p−1)}}
+ X_{R_{1}^{(lp−2)}} (X_{R_{1}^{(p−2)}} + X_{R_{p−1}^{(p)}}) = 0.

Theorem 2.2 gives

X_{R_{p−1}}X_{R_{0}^{(p−1)}} = X_{R_{1}^{(p−2)}} + X_{R_{p−1}^{(p)}}

so that

(LHS) = (RHS) ⇔ X_{R_{0}^{(p−1)}} (X_{R_{1}^{(lp−1)}} − X_{R_{1}^{(lp−3)}} + X_{R_{1}^{(lp−2)}}X_{R_{p−1}})
− X_{R_{0}^{((l−1)p)}} − X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}} − X_{R_{1}^{(p−2)}}X_{R_{0}^{(lp)}} = 0

The three terms relation for generalized Chebyshev polynomials gives

X_{R_{1}^{(lp−1)}} = X_{R_{p−1}}X_{R_{1}^{(lp−2)}} − X_{R_{1}^{(lp−3)}}.

Thus

(LHS) = (RHS) ⇔ 2X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}}
− (X_{R_{0}^{((l−1)p)}} + X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}} + X_{R_{0}^{(lp)}}X_{R_{1}^{(p−2)}}) = 0

Theorem 2.2 gives

X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} = X_{R_{0}^{(lp)}}X_{R_{1}^{(p−2)}} + X_{R_{0}^{((l−1)p)}}

so that we finally get

(LHS) = (RHS) ⇔ X_{R_{0}^{(p−1)}}X_{R_{1}^{(lp−1)}} − X_{R_{0}^{(lp−1)}}X_{R_{1}^{(p−1)}} = 0.

The second equality holds by lemma 3.2 so that we have proved that for any l ≥ 2,

Δ_{l+1} = Δ_{1}Δ_{l} − Δ_{l−1}.

Since we know that Δ_{1} = X_{δ} and Δ_{2} = F_{2}(X_{δ}), it follows that for any l ≥ 1, Δ_{l} = F_{l}(X_{δ}) which proves the proposition. □

We are now able to prove the general difference property :

Theorem 3.4. Let Q be an affine quiver, T be a tube of rank p ≥ 1 in Γ(kQ). Then for any l ≥ 1 and any 0 ≤ k ≤ p − 1, we have

𝔟_{lδ+dim R_{0}^{(k)}} = X_{R_{0}^{(k)}}F_{l}(X_{δ}) = X_{R_{0}^{(lp+k)}} − X_{R_{k+1}^{(lp−k−2)}}.

Proof. The first equality follows from the fact that

den (X_{R_{0}^{(k)}}X_{δ}^{l}) = dim R_{0}^{(k)} + lδ

so that

𝔟_{lδ+dim R_{0}^{(k)}} = X_{R_{0}^{(k)}}F_{l}(X_{δ}).

We now prove that

X_{R_{0}^{(k)}}F_{l}(X_{δ}) = X_{R_{0}^{(lp+k)}} − X_{R_{k+1}^{(lp−k−2)}}.

We denote by (LHS) the left-hand side and by (RHS) the right-hand side.

(LHS) = X_{R_{0}^{(k)}} (X_{R_{k}^{(lp)}} − X_{R_{k+1}^{(lp−2)}})
= X_{R_{0}^{(k)}}X_{R_{k}^{(lp)}} − X_{R_{0}^{(k)}}X_{R_{k+1}^{(lp−2)}}
= X_{R_{0}^{(lp+k)}} + X_{R_{0}^{(k−1)}}X_{R_{k+1}^{(lp−1)}} − X_{R_{0}^{(k)}}X_{R_{k+1}^{(lp−2)}}

so that (LHS) = (RHS) if and only if

(3)  X_{R_{k+1}^{(lp−k−2)}} = X_{R_{0}^{(k)}}X_{R_{k+1}^{(lp−2)}} − X_{R_{0}^{(k−1)}}X_{R_{k+1}^{(lp−1)}}

holds.

Using the three terms recurrence relations for generalized Chebyshev polynomials, we have

X_{R_{0}^{(k)}} = X_{R_{k−1}}X_{R_{0}^{(k−1)}} − X_{R_{0}^{(k−2)}}

and

X_{R_{k+1}^{(lp−1)}} = X_{R_{lp+k−1}}X_{R_{k+1}^{(lp−2)}} − X_{R_{k+1}^{(lp−3)}}

so that, replacing in the right-hand side of equality (3), we get :

X_{R_{0}^{(k)}}X_{R_{k+1}^{(lp−2)}} − X_{R_{0}^{(k−1)}}X_{R_{k+1}^{(lp−1)}} = X_{R_{0}^{(k−1)}}X_{R_{k+1}^{(lp−3)}} − X_{R_{0}^{(k−2)}}

Thus, by induction, we get

X_{R_{0}^{(k)}}X_{R_{k+1}^{(lp−2)}} − X_{R_{0}^{(k−1)}}X_{R_{k+1}^{(lp−1)}} = X_{R_{0}}X_{R_{k+1}^{(lp−k−1)}} − X_{R_{k+1}^{(lp−k)}}.

Now, the three terms recurrence relation gives

X_{R_{k+1}^{(lp−k)}} = X_{R_{k+1+lp−k−1}}X_{R_{k+1}^{(lp−k−1)}} − X_{R_{k+1}^{(lp−k−2)}}
= X_{R_{0}}X_{R_{k+1}^{(lp−k−1)}} − X_{R_{k+1}^{(lp−k−2)}}

and thus

X_{R_{0}}X_{R_{k+1}^{(lp−k−1)}} − X_{R_{k+1}^{(lp−k)}} = X_{R_{k+1}^{(lp−k−2)}}

so that equality (3) holds. □

As a corollary, for any positive root d with defect zero, we obtain a description of 𝔟_{d} as a certain difference of cluster characters :

Corollary 3.5. Let Q be an affine quiver, d be a positive root with defect zero. Let M be any indecomposable representation of dimension d. Then, there exists a quasi-simple module R_{0} in a tube of rank p ≥ 1, an integer 0 ≤ k ≤ p − 1 and an integer l ≥ 0 such that d = lδ + dim R_{0}^{(k)}. Moreover, for any such R_{0}, k, l, we have

𝔟_{d} = X_{R_{0}^{(k)}}F_{l}(X_{δ}) = X_{R_{0}^{(lp+k)}} − X_{R_{k+1}^{(lp−k−2)}}.

where R_{0}, …, R_{p−1} are the quasi-simple modules in T ordered such that τR_{i} ≃ R_{i−1} for every i ∈ ℤ/pℤ.

4. Integrable bundle on rep(Q) and their characters

In the previous section, we obtained a realization of the elements 𝔟_{d} associated to defect zero roots as differences of cluster characters. The aim of this section is to introduce a new character θ_{Tr} such that these elements correspond precisely to values of θ_{Tr}.

Unless it is otherwise specified, Q denotes an arbitrary acyclic quiver in this section.

4.1. Integrable bundles. For any d ∈ ℕ^{Q_{0}}, the representation variety rep(Q, d) of dimension d is the set of all representations M of Q with dimension vector d. Note that

rep(Q, d) ≃ ∏_{i⟶ j∈Q_{1}} Hom_{k}(k^{d_{i}}, k^{d_{j}})

so that rep(Q, d) is an affine irreducible variety. We identify the set of objects in the category rep(Q) with

rep(Q) = ⨁_{d∈ℕ^{Q_{0}}} rep(Q, d).

Definition 4.1. Let Q be any acyclic quiver. An integrable bundle on rep(Q) is a map

F : M ↦ F(M)

from rep(Q) to the set of subsets in rep(Q), such that, for any M ∈ rep(Q) :

(1) the set F_{e}(M) = F(M) ∩ rep(Q, e) is a constructible subset of rep(Q, e) for any e ∈ ℕ^{Q_{0}} ;

(2) the family (χ(F_{e}(M)))_{e∈ℕ^{Q_{0}}} has finite support.

Example 4.2. The map M ↦ Gr(M) is an integrable bundle called quiver grassmannian bundle.

For any kQ-module M and any submodule U ⊂ M, we set

Gr^{U}(M) = {N ∈ Gr(M)|U is a submodule of N}

the set of submodules of M containing U as a submodule. This is a constructible subset in the quiver grassmannian Gr(M).

If Q is an affine quiver, we will define another integrable bundle Tr as follows. Let M be an indecomposable kQ-module. If M is rigid, we set Tr(M) = Gr(M). If M is not rigid, it is regular and we can thus write M = R_{0}^{(lp+k)} for some quasisimple module R_{0} in a tube of rank p ≥ 1 and 0 ≤ k ≤ p − 1. In this case, we set

Tr(M) = Gr(M) \ Gr^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)}).

Note that if l = 0, M is rigid and Tr(M) = Gr(M). Finally, if M = ⨁_{i=1}^{n} M_{i} is decomposable, we define Tr(M) as the image of Tr(M_{1}) × ⋯ × Tr(M_{n}) under the algebraic isomorphism

\begin{matrix}Gr(M_{1}) × ⋯ × Gr(M_{n}) & ⟶ & Gr(M) \\ (U_{1}, ⋯, U_{n}) & ↦ & U_{1} ⊕ ⋯ ⊕ U_{n}. \end{matrix}

For every dimension vector e ∈ ℕ^{Q_{0}}, we set Tr_{e}(M) = Tr(M)∩rep(Q, e). It follows directly from the definition that Tr_{e}(M) is a constructible subset of Gr_{e}(M) called transverse quiver grassmannian of M (of dimension e). Since Tr_{e}(M) ⊂ Gr_{e}(M) for any e ∈ ℕ^{Q_{0}}, it follows that Tr_{e}(M) is empty for all but finitely many e ∈ ℕ^{Q_{0}} so that

Tr : M ↦ Tr(M)

is an integrable bundle on rep(Q).

4.2. Character associated to an integrable bundle. Extending an idea of Caldero and Chapoton, we associate to any integrable bundle on rep(Q) a map from the set of objects in C_{Q} to the ring ℤ[u^{±1}] of Laurent polynomials in the initial cluster of A(Q).

Definition 4.3. Let F be an integrable bundle on rep(Q). The character associated to F is the map

θ_{F}(?) : Ob(C_{Q})⟶ ℤ[u^{±1}]

given by

θ_{F}(M) = u^{−dim P_{M} [1]} ( ∑_{e∈ℕ^{Q_{0}}} χ(F_{e}(M_{0})) ∏_{i∈Q_{0}} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim M_{0}−e⟩})

for any object M = M_{0} ⊕ P_{M}[1] in C_{Q} where M_{0} is a kQ-module and P_{M} is a projective kQ-module.

Example 4.4. In this terminology, the Caldero-Chapoton map is the character associated to the quiver grassmannian bundle. More precisely,

θ_{Gr}(M) = X_{M}

for any object M in C_{Q}.

We thus set the following definition :

Definition 4.5. The transverse character is the map

θ_{Tr} : \{ \begin{matrix}Ob(C_{Q}) & ⟶ & ℤ[u^{±1}] \\ M & ↦ & θ_{Tr}(M) \end{matrix}

We first prove a multiplicative property of θ_{Tr}.

Lemma 4.6. Let M, N be two objects in C_{Q}, then

θ_{Tr}(M ⊕ N) = θ_{Tr}(M)θ_{Tr}(N).

Proof. For any g ∈ ℕ^{Q_{0}}, there is an isomorphism of varieties

⨆_{e+f=g} Gr_{e}(M) × Gr_{f}(N) ⟶ Gr_{g}(M ⊕ N)
(U, V) ↦ U ⊕ V.

By definition of Tr, this restricts to an isomorphism

⨆_{e+f=g} Tr_{e}(M) × Tr_{f}(N) ⟶ Tr_{g}(M ⊕ N)
(U, V) ↦ U ⊕ V.

It follows easily that

θ_{Tr}(M ⊕ N) = θ_{Tr}(M)θ_{Tr}(N).
□

We now prove that θ_{Tr} coincides with X_{?} on the set of rigid objects in C_{Q}. In particular, this will allow to realize cluster monomials in terms of θ_{Tr}.

Lemma 4.7. Let Q be an affine quiver. Then, for any rigid object M in C_{Q}, we have θ_{Tr}(M) = X_{M}. In particular,

M(Q) = {θ_{Tr}(M)|M is rigid in C_{Q}} .

Proof. We set M = M_{0} ⊕ P_{M}[1]. Since M is rigid in C_{Q}, M_{0} is a rigid kQmodule. Thus, it follows from the definition of the transverse grassmannian that Tr(M_{0}) = Gr(M_{0}). Then,

θ_{Tr}(M) = u^{−dim P_{M} [1]} ( ∑_{e∈ℕ^{Q_{0}}} χ(Tr_{e}(M_{0})) ∏_{i∈Q_{0}} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim M_{0}−e⟩})

= X_{P_{M} [1]} ( ∑_{e∈ℕ^{Q_{0}}} χ(Tr_{e}(M_{0})) ∏_{i∈Q_{0}} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim M_{0}−e⟩})

= X_{P_{M} [1]} ( ∑_{e∈ℕ^{Q_{0}}} χ(Gr_{e}(M_{0})) ∏_{i∈Q_{0}} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim M_{0}−e⟩})

= X_{P_{M} [1]}X_{M_{0}} = X_{P_{M} [1]⊕M_{0}} = X_{M}.

The second assertion follows directly from Caldero-Keller’s realization of cluster monomials :

M(Q) = {X_{M}|M is rigid in C_{Q}}
= {θ_{Tr}(M)|M is rigid in C_{Q}} .
□

5. A geometrization of B(Q)

We will now relate the transverse character with the difference properties obtained in Section 3. This will provide a realization of the elements in B(Q) in terms of θ_{Tr}.

5.1. From difference properties to θ_{Tr}. Using Theorem 3.4, we can now deduce a realization in terms of θ_{Tr} of the elements in B(Q) corresponding to positive roots with zero defect :

Theorem 5.1. Let d be any positive root. Then

𝔟_{d} = θ_{Tr}(M)

where M is any indecomposable representation of dimension d.

Proof. If d is a positive root with non-zero defect. Then, d is real and there exists a unique indecomposable representation M in rep(Q, d). Moreover, this representation has to be preprojective or preinjective. In both cases, it is rigid and thus 𝔟_{d} = X_{M} = θ_{Tr}(M). We can thus assume that d ∈ ℕ^{Q_{0}} is a root with zero defect.

Let M be an indecomposable representation in rep(Q, d). It is necessarily contained in a tube T of rank p ≥ 1. We denote by R_{0}, …, R_{p−1} the quasi-simple modules in T ordered such that τR_{i} ≃ R_{i−1} for any i ∈ ℤ/pℤ. We can write d = lδ+n where n is either a real Schur root or zero. If n ≠ 0, there exists a unique indecomposable representation N in rep(Q, n). More precisely, if M ≃ R_{0}^{(lp+k)} with l ≥ 0 and 0 ≤ k ≤ p−1, N is the rigid representation R_{0}^{(k)} (still with the convention that R_{0}^{(0)} = 0) and

𝔟_{d} = X_{R_{0}^{(k)}}F_{l}(X_{δ}).

Now according to Theorem 3.4, we have

X_{R_{0}^{(k)}}F_{l}(X_{δ}) = X_{R_{0}^{(lp+k)}} − X_{R_{k+1}^{(lp−k−2)}}

For any e ∈ ℕ^{Q_{0}}, the map

\{ \begin{matrix}Gr_{e}^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)}) & ⟶ & Gr_{e−dim R_{0}^{(k+1)}}(R_{k+1}^{(lp−k−2)}) \\ U & ↦ & U/R_{0}^{(k+1)} \end{matrix}

is an algebraic isomorphism and we denote by c_{e} ∈ ℤ the common value of the Euler characteristics of these constructible sets. Fix now some e ∈ ℕ^{Q_{0}}, the monomial corresponding to e in X_{R_{0}^{(lp+k)}} is

c_{e} ∏_{i} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim R_{0}^{(lp+k)}−e⟩}

and the monomial corresponding to e − dim R_{0}^{(k+1)} in X_{R_{k+1}^{(lp−k−2)}} is

c_{e} ∏_{i} u_{i}^{−⟨e−dim R_{0}^{(k+1)},S_{i}⟩−⟨S_{i},dim R_{k+1}^{(lp−k−2)}+dim R_{0}^{(k+1)}−e⟩}.

We now prove that these monomials are the same. For any i = 0, …, p − 1, we set by r_{i} = dim R_{i} and we denote by c the Coxeter transformation on ℤ^{Q_{0}} induced by the Auslander-Reiten translation. We recall that for any β, γ ∈ ℤ^{Q_{0}}, we have ⟨γ, c(β)⟩ = − ⟨β, γ⟩. With these notations, we have

dim R_{0}^{(k+1)} = r_{0} + ⋯ + r_{k}
dim R_{k+1}^{(lp−k−2)} = (l − 1)δ + r_{k+1} + ⋯ r_{p−2}

so that

dim R_{0}^{(k+1)} + dim R_{k+1}^{(lp−k−2)} = lδ − r_{p−1}.

We now compute the exponents :

− ⟨e, S_{i}⟩ − ⟨S_{i}, dim R_{0}^{(lp+k)} − e⟩
= − ⟨e, S_{i}⟩ − ⟨S_{i}, lδ + r_{0} + ⋯ + r_{k−1} − e⟩
= − ⟨e, S_{i}⟩ − ⟨S_{i}, lδ − e⟩ − ⟨S_{i}, r_{0} + ⋯ + r_{k−1}⟩
= − ⟨e, S_{i}⟩ − ⟨S_{i}, lδ − e⟩ − ⟨r_{1} + ⋯ + r_{k}, S_{i}⟩

and

− ⟨e − dim R_{0}^{(k+1)}, S_{i}⟩ − ⟨S_{i}, dim R_{k+1}^{(lp−k−2)} + dim R_{0}^{(k+1)} − e⟩
= − ⟨e, S_{i}⟩ + ⟨r_{0} + ⋯ + r_{k}, S_{i}⟩ + ⟨S_{i}, r_{p−1}⟩ − ⟨S_{i}, lδ − e⟩
= − ⟨e, S_{i}⟩ − ⟨S_{i}, lδ − e⟩ + ⟨r_{1} + ⋯ + r_{k}, S_{i}⟩

so that the two monomials are the same. Thus,

X_{R_{0}^{(lp+k)}} − X_{R_{k+1}^{(lp−k−2)}}

= ∑_{e}χ(Gr_{e}(R_{0}^{(lp+k)})) ∏_{i} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},lδ−e⟩−⟨r_{1}+···+r_{k},S_{i}⟩}

− ∑_{e}χ(Gr_{e−dim R_{0}^{(k+1)}}(R_{k+1}^{(lp−k−2)})) ∏_{i} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},lδ−e⟩−⟨r_{1}+···+r_{k},S_{i}⟩}

= ∑_{e}χ (Gr_{e}(R_{0}^{(lp+k)}) \ Gr_{e}^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)})) ∏_{i} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},lδ−e⟩−⟨r_{1}+···+r_{k},S_{i}⟩}

= ∑_{e}χ(Tr_{e}(R_{0}^{(lp+k)})) ∏_{i} u_{i}^{−⟨e,S_{i}⟩−⟨S_{i},dim R_{0}^{(lp+k)}−e⟩}.

This proves the corollary. □

5.2. Realization of B(Q) in terms of θ_{Tr}. Summing up the previous results, we deduce the following geometric description of B(Q) :

Theorem 5.2. Let Q be an affine quiver, then

B(Q) = {θ_{Tr}(M ⊕ R)|M is an indecomposable (or zero) regular kQ-module,
R is any rigid object in C_{Q} such that Ext_{C_{Q}}^{1}(M, R) = 0} .

Proof. We denote by S the right-hand-side of the claimed equality. By definition, we have

B(Q) = M(Q) ⊔ {F_{l}(X_{δ})X_{R}|l ≥ 1, R is a regular rigid kQ-module} .

We first prove that every S ⊂ B(Q). Let R be a rigid object in C_{Q}, then θ_{Tr}(R) is a cluster monomial by Lemma 4.7. Fix now M to be an indecomposable regular kQ-module in a tube T such that Ext_{C_{Q}}^{1}(M, R) = 0. If M is rigid, then M ⊕ R is rigid in C_{Q} and θ_{Tr}(M)θ_{Tr}(R) = θ_{Tr}(M ⊕ R) is a cluster monomial by Lemma 4.7. Now, if M is non-rigid, then d = dim M is a positive root of defect zero, thus, θ_{Tr}(M) = 𝔟_{d} by Theorem 5.1. Thus, there exists l ≥ 1 and E a indecomposable rigid (or zero) module in T such that d = lδ + dim E. According to Theorem 5.1, we have

θ_{Tr}(M ⊕ R) = θ_{Tr}(M)θ_{Tr}(R)
= 𝔟_{lδ+dim E}θ_{Tr}(R)
= F_{l}(X_{δ})X_{E}θ_{Tr}(R)
= F_{l}(X_{δ})θ_{Tr}(E)θ_{Tr}(R)
= F_{l}(X_{δ})θ_{Tr}(E ⊕ R).

Since Ext_{C_{Q}}^{1}(M, R) = 0, we have Ext_{kQ}^{1}(M, R) = 0 and Ext_{kQ}^{1}(R, M) = 0. Thus, it follows easily that Ext_{kQ}^{1}(E, R) = 0 and Ext_{kQ}^{1}(R, E) = 0 so that E ⊕ R is a rigid regular kQ-module. In particular, θ_{Tr}(E ⊕ R) = X_{E⊕R} and thus

θ_{Tr}(M ⊕ R) = F_{l}(X_{δ})X_{E⊕R} ∈ B(Q).

Conversely, fix an elements in B(Q). If x is a cluster monomial, then according to Lemma 4.7, there exists some rigid object M in C_{Q} such that x = θ_{Tr}(M). Thus, x ∈ S. Fix now some regular rigid kQ-module R and some integer l ≥ 1. Then the direct summands of R belong to exceptional tubes. We fix an indecomposable kQmodule M of dimension vector lδ in an homogeneous tube. Then, Ext_{C_{Q}}^{1}(M, R) = 0. According to Theorem 5.1, we have F_{l}(X_{δ})X_{R} = θ_{Tr}(M)X_{R} but R is rigid so that X_{R}θ_{Tr}(R). Thus,

F_{l}(X_{δ})X_{R} = θ_{Tr}(M)θ_{Tr}(R) = θ_{Tr}(M ⊕ R) ∈ S.

This proves the theorem. □

6. Examples

We shall now study two examples corresponding to cases where it is known that B(Q) is the canonically positive basis in A(Q).

6.1. The 𝔸̃_{1,1} case. Let Q be the Kronecker quiver, that is, the affine quiver of type 𝔸̃_{1,1} with the following orientation :

\xymatrix{Q : & 1 \ar[r] \ar[r] & 2}

The minimal imaginary root of Q is δ = (11).

For any λ ∈ k, we set

\xymatrix{M_{λ} : & k \ar[r]^{1} \ar[r]_{λ} & k}

and

\xymatrix{M_{∞} : & k \ar[r]^{0} \ar[r]_{1} & k.}

It is well-known that every tube in Γ(kQ) is homogeneous and that the family {M_{λ}|λ ∈ k ⊔ {∞}} is a complete set of representatives of pairwise non-isomorphic quasi-simple kQ-modules.

For any n ≥ 1, the indecomposable representations of quasi-length n are given by

\xymatrix{M_{λ}^{(n)} : & k^{n} \ar[r]^{1} \ar[r]_{J_{n}(λ)} & k^{n}}

for any λ ∈ k and

\xymatrix{M_{∞}^{(n)} : & k^{n} \ar[r]^{J_{n}(0)} \ar[r]_{1} & k^{n}}

where J_{n}(λ) ∈ M_{n}(k) denotes the Jordan block of size n associated to the eigenvalue λ. Quiver grassmannians and transverse quiver grassmannians of indecomposable representations with quasi-length 2 are described in Figure 1 below.

Note that kQ-mod contains no regular rigid modules. It follows that in this case

B(Q) = M(Q) ⊔ {θ_{Tr}(M)|M is an indecomposable (or zero) regular kQ-module} .

According to [SZ04], this set is the canonically positive basis of A(Q). In Figure 1, we see that, for any λ ∈ k ⊔ {∞},

θ_{Dr}(M_{λ}^{(2)}) = θ_{Gr}(M_{λ}^{(2)}) − 1 = X_{M_{λ}^{(2)}} − 1 = S_{2}(X_{M_{λ}}) − 1 = F_{2}(X_{M_{λ}}) = 𝔟_{2δ}.

This illustrates Theorem 5.1.

e  Gr_{e}(M_{0}^{(2)})  Tr_{e}(M_{0}^{(2)})  Gr_{e}(M_{λ}^{(2)})  Tr_{e}(M_{λ}^{(2)})  Gr_{e}(M_{∞}^{(2)})  Tr_{e}(M_{∞}^{(2)})  u^{⟨−e,S_{i}⟩−⟨S_{i},2δ−e⟩}
(00)  \{0\}  \{0\}  \{0\}  \{0\}  \{0\}  \{0\}  \frac{u_{1}^{2}}{u_{2}^{2}}

(01)  ℙ^{1} × \{S_{2}\}  ℙ^{1} × \{S_{2}\}  ℙ^{1} × \{S_{2}\}  ℙ^{1} × \{S_{2}\}  ℙ^{1} × \{S_{2}\}  ℙ^{1} × \{S_{2}\}  \frac{1}{u_{2}^{2}}

(02)  \{S_{2} ⊕ S_{2}\}  \{S_{2} ⊕ S_{2}\}  \{S_{2} ⊕ S_{2}\}  \{S_{2} ⊕ S_{2}\}  \{S_{2} ⊕ S_{2}\}  \{S_{2} ⊕ S_{2}\}  \frac{1}{u_{1}^{2}u_{2}^{2}}
(11)  {M_{0}}  ∅  {M_{λ}}  ∅  {M_{∞}}  ∅  1
(12) \{P_{1}, M_{0} ⊕ S_{2}\} \{P_{1}, M_{0} ⊕ S_{2}\} ℙ^{1} × \{M_{λ} ⊕ S_{2}\} ℙ^{1} × \{M_{λ} ⊕ S_{2}\} \{P_{1}, M_{∞} ⊕ S_{2}\} \{P_{1}, M_{∞} ⊕ S_{2}\}  \frac{1}{u_{1}^{2}}

(22)  \{M_{0}^{(2)}\}  \{M_{0}^{(2)}\}  \{M_{λ}^{(2)}\}  \{M_{λ}^{(2)}\}  \{M_{∞}^{(2)}\}  \{M_{∞}^{(2)}\}  \frac{u_{2}^{2}}{u_{1}^{2}}

Figure 1. Grassmannians and transverse grassmannians of indecomposable modules of quasi-length 2 6.2. The 𝔸̃_{2,1} case. We now consider the quiver of affine type 𝔸̃_{2,1} equipped with the following orientation :

\xymatrix{ &  & 2 \ar[dr] \\ Q & 1 \ar[rr] \ar[ur] &  & 3}

The minimal imaginary root of Q is δ = (111). For any λ ∈ k, we set

\xymatrix{ &  & k \ar[dr]^{λ} \\ M_{λ} & k \ar[rr]^{1} \ar[ur]^{1} &  & k}

and

\xymatrix{ &  & k \ar[dr]^{1} \\ M_{∞} & k \ar[rr]^{0} \ar[ur]^{1} &  & k.}

Γ(kQ) contains exactly one exceptional tube T of rank 2 whose quasi-simples are

\xymatrix{ &  & 0 \ar[dr] \\ R_{0} & k \ar[rr]^{1} \ar[ur]^{0} &  & k}

and

\xymatrix{ &  & k \ar[dr]^{0} \\ R_{1} ≃ S_{2} & 0 \ar[rr] \ar[ur] &  & 0.}

The set {M_{λ}|λ ∈ k ⊔ {∞}} is a complete set of representatives of pairwise nonisomorphic indecomposable representations in rep(Q, δ). For any λ ≠ 0, ∞, M_{λ} is a quasi-simple kQ-module in an homogeneous tube. Moreover, M_{0} = R_{1}^{(2)} and M_{∞} = R_{0}^{(2)}.

Quiver grassmannians and transverse quiver grassmannians of indecomposable representations of dimension δ are described in Figure 2 below. For simplicity, we only listed the dimension vectors corresponding giving non-empty quiver grassmannians.

e Gr_{e}(M_{λ}) Tr_{e}(M_{λ}) Gr_{e}(M_{0}) Tr_{e}(M_{0}) Gr_{e}(M_{∞}) Tr_{e}(M_{∞}) u^{⟨−e,S_{i}⟩−⟨S_{i},δ−e⟩}
(000)  \{0\}  \{0\}  \{0\}  \{0\}  \{0\}  \{0\}  \frac{u_{1}}{u_{3}}

(001)  \{S_{3}\}  \{S_{3}\}  \{S_{3}\}  \{S_{3}\}  \{S_{3}\}  \{S_{3}\}  \frac{1}{u_{2}u_{3}}
(010)  ∅  ∅  {S_{2}}  ∅  ∅  ∅  1
(011)  \{P_{2}\}  \{P_{2}\} \{S_{2} ⊕ S_{3}\} \{S_{2} ⊕ S_{3}\}  \{P_{2}\}  \{P_{2}\}  \frac{1}{u_{1}u_{2}}
(101)  ∅  ∅  ∅  ∅  {M_{∞}}  ∅  1
(111) \{M_{λ}\}  \{M_{λ}\}  \{M_{0}\}  \{M_{0}\}  \{M_{∞}\}  \{M_{∞}\}  \frac{u_{3}}{u_{1}}

Figure 2. Grassmannians and transverse grassmannians for quasi-length 2

In 2, we observe that X_{M_{λ}} = X_{M_{0}} − 1 = X_{M_{∞}} − 1, illustrating Theorem 3.1. Also, we see that θ_{Tr}(M_{λ}) = θ_{Tr}(M_{0}) = θ_{Tr}(M_{∞}) for any λ ∈ k \ {0} so that the transverse character does not depend on the chosen tube. Moreover,

θ_{Tr}(M_{λ}) = X_{M_{λ}} = F_{1}(X_{δ})

illustrating Theorem 5.1.

Remark 6.1. Figure 2 justifies in some sense the terminology “transverse submodule”. Indeed, we see that, given two indecomposable regular modules M and N having the same dimension vectors, the submodules U in Tr(M) are those having a corresponding submodule in Gr(N). In some sense, we can see U as a submodule “common” to M and N. This is why we call it transverse.

As suggested by Bernhard Keller, this notion of transversality should have a more precise meaning in the context of deformation theory. Some connections are known at this time, this should be discussed in a forthcoming article.

Acknowledgements

The author is grateful to the CRM-ISM, the University of Sherbrooke, Ibrahim Assem, Thomas Brüstle and Virginie Charette for their financial support. He would also like to thank Giovanni Cerulli Irelli for motivating the investigation of higher difference properties during his stay at the University of Padova. This was the starting point of this work. He would also like to thank the rest of the algebra group of Padova for their kind hospitality. Finally, he would like to thank Bernhard Keller, Philippe Caldero and Frédéric Chapoton for interesting discussions on the topic.

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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