Quantized Chebyshev polynomials and cluster characters with coefficients

DOI 10.1007/s10801-009-0198-8

Quantized Chebyshev polynomials and cluster characters with coefficients

G. Dupont

Received: 21 November 2008 / Accepted: 30 July 2009 / Published online: 18 August 2009

© Springer Science+Business Media, LLC 2009

Abstract We introduce quantized Chebyshev polynomials as deformations of gen- eralized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomi- als arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of typeA. We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.

Keywords Cluster algebras·Quantized Chebyshev polynomials·Principal coefficients·Regular components·Orthogonal polynomials

1 Introduction

Normalized Chebyshev polynomials are elementary well known objects which can be defined as follows. For everyn≥1, then-th normalized Chebyshev polynomial of the first kindFnis characterized byFn(t+t⁻¹)=tⁿ+t⁻ⁿand then-th normal- ized Chebyshev polynomial of the second kindS_nis characterized byS_n(t+t⁻¹)= _n

k=0t^n−2k. These polynomials made their first appearance in the context of cluster algebras respectively in [23] and in [8].

Cluster algebras were introduced in the early 2000’s by Fomin and Zelevinsky in [14]. Since then, they found applications in many areas of mathematics including combinatorics, Lie theory, Poisson geometry and representation theory. In their most simple incarnation, cluster algebras are commutative algebras overZPwhereP is some tropical semi-field. The generators, called cluster variables, are gathered into sets of fixed finite cardinality called clusters. Monomials in variables belonging to

G. Dupont (

Institut Camille Jordan, Université Lyon 1, 43, Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France

e-mail:dupont@math.univ-lyon1.fr

a same cluster are called cluster monomials. The elements of a cluster algebra can always be expressed as Laurent polynomials in cluster variables belonging to any fixed cluster, this is referred to as the Laurent phenomenon [14]. An element in a cluster algebraAis called positive if it can be expressed as a Laurent polynomial with coefficients inZ_≥0Pin every cluster ofA. AZP-basisBinAis called canonically positive, if positive elements inAcoincide precisely withZ≥0P-linear combinations of elements ofB. It is not known if there is necessarily a canonically positive basis in a given cluster algebraA. Nevertheless, if such a basis exists, its elements are uniquely determined up to normalization by elements ofP. Canonically positive bases were investigated in particular cases in [5,23].

The research of bases, and especially canonically positive bases, in cluster algebras was one of the main motivation for their study. In the symmetric coefficient-free case, that is, when P= {1}, Caldero and Keller proved that if the cluster algebra A is simply-laced of finite type (ie if it has only finitely many cluster variables), then cluster monomials form aZ-basis inA[7].

For rank 2 cluster algebras of affine and finite type, Sherman and Zelevinsky managed to compute canonically positive bases with arbitrary coefficients [23]. In particular, the authors proved that ifA is a coefficient-free rank 2 cluster algebra of affine type, the canonically positive basis ofAisB(A)= {cluster monomials} {F_n(z)|n≥1} wherez is some well chosen particular positive element in A (see Section7for details). Using coefficient-free cluster characters, Caldero and Zelevin- sky managed to compute a slightly different basis for the coefficient-free clus- ter algebra associated to the Kronecker quiver [8]. Namely, this basis is given by {cluster monomials} {S_n(z)|n≥1}. The presence of normalized second kind Chebyshev polynomials in this case comes from the study of coefficient-free clus- ter characters associated to regular modules over the path algebra of the Kronecker quiver.

LetQ=(Q0, Q1)be an acyclic quiver, that is, a quiver without oriented cycles whereQ0is a finite set of vertices andQ1 a finite set of arrows. Letk=Cbe the field of complex numbers. We denote bykQthe path algebra ofQ, bykQ-mod the category of finite dimensional left-kQ-modules and byCQthe cluster category ofQ.

Let y be aQ₀-tuple of elements ofP. We denote byA(Q,y,x)the cluster algebra with principal coefficients at the initial seed (Q,y,x)where x=(xi|i∈Q0)and y=(y_i|i∈Q₀)is a minimal set of generators of the semifieldP.

Inspired by works on cluster characters for the coefficient-free case [3,6,7,19], Fu and Keller introduced in [13] cluster characters with coefficients in order to realize elements in the cluster algebraA(Q,y,x)from objects in the cluster categoryCQ. In particular cluster variables inA(Q,y,x)are characters associated to indecomposable rigid objects in the cluster categoryCQ. In this paper, we consider a more elementary description of cluster characters with coefficients than the one proposed in [13]. We will see in Section2.2that these two definitions coincide. The cluster character with coefficients onCQis a map

X^Q,y_? :Ob(CQ)−→Z[y][x^±1]

whose detailed definition will be given in section2. We denote by X_?^Q:Ob(CQ)−→Z[x^±1]

the usual Caldero-Chapoton map introduced in [3,7], which will also be referred to as the cluster character without coefficients onCQ.

In [10] (see also [20] for a similar description in Dynkin typeA), we introduced a generalization of Chebyshev polynomials of the second kind arising in cluster al- gebras associated to acyclic representation-infinite quivers. More precisely, ifQis a quiver of infinite representation type, the coefficient-free cluster characterX^Q_M of an indecomposable regular moduleM can be expressed as a polynomial with inte- gral coefficients evaluated at the characters of quasi-composition factors ofM. The polynomials appearing were called generalized Chebyshev polynomials.

In [5], Cerulli Irelli studied cluster algebras with coefficients associated to an affine quiver of typeA˜2,1. It turned out that if the coefficients are not specialized at 1, gen- eralized Chebyshev polynomials do not appear anymore. The aim of this paper is to introduce a certain deformation of generalized Chebyshev polynomials that allows to recover the polynomiality property for cluster characters with coefficients evaluated at indecomposable regular modules over the path algebra of a representation-infinite quiver.

Whereas the final goal of this paper is to give an efficient tool for calculations in cluster algebras, most of the results can be read independently of the theory of cluster algebras.

Our main results are the following: Consider a family q= {q_i|i∈Z}of indeter- minates overZand a family

x_i,1|i∈Z

of indeterminates overZ[q]. We define by induction a family

x_i,n|i∈Z, n≥1

Q(q)(xi,1|i∈Z) by

x_i,nx_i+1,n=x_i,n+1x_i+1,n−1+ n k=1

q_i+k (1.1)

with the convention thatxi,0=1 for alli∈Z.

The first result of this paper is a polynomial closed expression for thex_i,n: Theorem 1 For anyn≥1 and anyi∈Z, we have

xi,n=det

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

xi+n−1,1 1 (0) qi+n−1 . .. . ..

. .. . .. . .. . .. . .. 1 (0) q_i+1 xi,1

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦ .

In particular,x_i,nis a polynomial inZ[q_i+1, . . . , q_i+n−1, x_i,1, . . . , x_i+n−1,1].

Note that the well-known Dodgson’s determinant evaluation rule turns out to be a consequence of theorem 1 and equation (1.1) when all theq_is are specialized at 1.

Identifying naturally the ringZ[q_i+1, . . . , q_i+n−1, x_i,1, . . . , x_i+n−1,1]with a sub- ring ofq_i, . . . , q_i+n−1, x_i,1, . . . , x_i+n−1,1, we denote byP_nthe polynomial in 2nvari- ables such that

x_i,n=P_n(q_i, . . . , q_i+n−1, x_i,1, . . . , x_i+n−1,1)

andP_nis called then-th quantized Chebyshev polynomial of infinite rank. Note that the definition ofP_ndoes not depend oni.

For any p≥1, the abelian group pZ acts Z-linearly on Z[q_i, x_i,1|i∈Z] by kp.q_i=q_i+kpandkp.x_i=x_i+kpfor anyk∈Z. We denote by

π_p:Z[q_i, x_i,1|i∈Z]−→Z[q_i, x_i,1|i∈Z]/pZ

the canonical map. We setPn,pto be the unique polynomial such that for everyi∈Z andn≥1, we have

πp(xi,n)=Pn,p(πp(qi), . . . , πp(qi+p−1), πp(xi), . . . , πp(xi+p−1)).

The polynomialPn,pis called then-th quantized Chebyshev polynomial of rankp. If we denote byk[p]the remainder of the euclidean division of an integerkbyp,Pn,p

is the polynomial such thatPn,p(qi[p], . . . , qi+p−1[p], xi[p],1, . . . , xi+p−1[p],1)is the determinant

det

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

x_i+n−1[p],1 1 (0) q_i+n−1[p] . .. . ..

. .. . .. . .. . .. . .. 1 (0) q_i+1[p] x_i[p],1

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

In the sequel, we will use the following notation : ifJ is a set, a= {ai|i∈J}is a family of indeterminates overZandν= {νi|i∈J} ⊂Zhas finite support, we write a^ν=

i∈Ja_i^νi.

IfQis a representation-infinite quiver, any regular componentRin the Auslander- Reiten quiver (kQ) of kQ-mod is of the form ZA_∞/(p) for some p≥0 [1, Sect. VIII.4, Theorem 4.15]. We denote byR_i, i∈Z/pZ, the quasi-simple mod- ules inR, ordered such thatτ R_i R_i−1for alli∈Z/pZ. Fori∈Z/pZandn≥1, denote byR_i⁽ⁿ⁾the unique indecomposable module such that there exists a sequence of irreducible monomorphisms

Ri R_i⁽¹⁾−→R⁽²⁾_i −→ · · · −→R_i⁽ⁿ⁾.

We say thatR_i⁽ⁿ⁾has quasi-socleR_i and quasi-lengthn. By conventionR⁽⁰⁾_i denotes the zero module. The quotientsR_i^(k)/R_i^(k−1) for k=1, . . . , nare called the quasi- composition factors of the moduleM. Every indecomposable module inRcan be writtenR_i⁽ⁿ⁾for somei∈Z/pZandn≥1.

Our main result is that quantized Chebyshev polynomials appear naturally for cluster characters with coefficients associated to regular modules.

Theorem 2 LetQbe a quiver of infinite representation type,Rbe a regular compo- nent in(kQ)and letp≥0 be such thatRis of the formZA∞/(p). We denote by {R_i|i∈Z/pZ}the set of quasi-simple modules inR, ordered such thatτ R_i R_i−1 for alli∈Z/pZ. Then for everyn≥1 andi∈Z/pZ, we have

X^Q,y

R⁽ⁿ⁾_i =Pn(y^dimRi, . . . ,y^dimRi+n−1, X^Q,y_R

i , . . . , X_R^Q,y

i+n−1) or equivalently

X^Q,y

R⁽ⁿ⁾_i =det

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣ X^Q,y_R

i+n−1 1 (0)

y^dimRi+n−1 . .. . .. . .. . .. . ..

. .. . .. 1 (0) y^dimRi+1 X_R^Q,y

i

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ .

Moreover, ifp >0, we have X^Q,y

R_i⁽ⁿ⁾=P_n,p(y^dimRi, . . . ,y^dimRi+p−1, X_R^Q,y

i , . . . , X^Q,y_R

i+p−1).

We also prove that quantized Chebyshev polynomials arise in cluster algebras of Dynkin typeA. For any integerr≥1, letAbe the quiver of type −→Ar, that is, of Dynkin typeAr equipped with the following linear orientation:

0 1 2 · · · r−1

LetA(A,x,y)be the cluster algebra with principal coefficients at the initial seed (A,x,y)andX^A,ybe the cluster character with coefficients onCA.

For anyi∈ [0, r−1], we denote by Si the simplekA-module associated to the vertexiand for anyn∈ [1, r−i], we denote byS_i⁽ⁿ⁾the indecomposablekA-module with socleS_iand lengthn. We prove:

Theorem 3 Letr≥1 be an integer andAbe the above quiver of type−→

Ar. Then, for anyi∈ [0, r−1]andn∈ [1, r−i], we have

X^A,y

S_i⁽ⁿ⁾=P_n(y_i, . . . , y_i+n−1, X_S^A,y

i , . . . , X^A,y_S

i+n−1) or equivalently

X^A,y

S_i⁽ⁿ⁾=det

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣ X_S^A,y

i+n−1 1 (0)

y_i+n−1 . .. . .. . .. . .. . ..

. .. . .. 1 (0) y_i+1 X_S^A,y

i

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ .

Note that this result was obtained independently by Yang and Zelevinsky by consid- ering generalized minors [24].

The paper is organized as follows. In section2, we give definitions and properties of cluster characters with and without coefficients. In section3, we study in detail cluster characters with coefficients for equioriented Dynkin quivers of typeA. The study of Dynkin typeAallows to define quantized Chebyshev polynomials in section 4where Theorem 1 and Theorem 3 are proved. In section5, we prove Theorem 2 and give some explicit computations in cluster algebras of typeA˜2,1. In section6, we study algebraic properties of some particular quantized Chebyshev polynomi- als, namely the quantized versions of normalized Chebyshev polynomials of the first and second kinds. Finally, in section7, we give examples and conjectures for these polynomials to appear in bases, and especially canonically positive bases, in cluster algebras of affine types.

2 Cluster characters

2.1 Definitions and basic properties

LetQbe an acyclic quiver. We denote byA(Q,x,y)the cluster algebra with princi- pal coefficients at the initial seed(Q,x,y)where y= {y_i|i∈Q₀}is the initial coeffi- cientQ0-tuple and wherex= {x_i|i∈Q0}is the initial cluster. We simply denote by A(Q,x)the coefficient-free cluster algebra with initial seed(Q,x).

Letk=Cbe the field of complex numbers andkQ-mod be the category of finite dimensional left-modules over the path algebra ofQ. All along this paper, this cate- gory will be identified with the category rep(Q)of finite dimensional representations ofQover k. We denote by τ_kQ (or simply τ) the Auslander-Reiten translation on kQ-mod. LetD=D^b(kQ)be the bounded derived category ofQwith shift functor denoted by[1]kQ(or simply[1]). We denote byCQthe cluster category of the quiver Q, that is, the orbit categoryD/F of the auto-functorF =τ⁻¹[1]inD. This is an additive triangulated category [17], 2-Calabi-Yau whose indecomposable objects are given by indecomposablekQ-modules and shifts of indecomposable projectivekQ- modules [2]. This category was independently introduced by Caldero, Chapoton and Schiffler for the typeAcase [4].

For everyi∈Q0, we denote byS_i the simplekQ-module associated to the ver- tex i,Pi its projective cover andIi its injective hull. We denote by αi =dimSi

the dimension vector ofS_i. Since dim induces an isomorphism of abelian groups K₀(kQ)−→Z^Q0,α_i is identified with thei-th vector of the canonical basis ofZ^Q0.

As Q is acyclic, kQ is a finite dimensional hereditary algebra, we denote by

−,−the Euler form onkQ-mod. It is given by

M, N =dim HomkQ(M, N )−dim Ext¹_kQ(M, N )

for anykQ-modulesMandN. Note that−,−is well-defined on the Grothendieck group.

For anykQ-moduleMand any dimension vector e, we denote by Gre(M)= {N⊂M|dimN=e}

the grassmannian of submodules of dimension e ofM. This is a projective variety and we denote byχ (Gre(M)) its Euler characteristic with respect to the simplicial cohomology.

Roughly speaking, a cluster character evaluated at akQ-moduleMis some nor- malized generating series for Euler characteristics of grassmannians of submodules of the moduleM. More precisely :

Definition 2.1 The cluster character with coefficients on kQ-mod is the map Ob(CQ)−→Z[y][x^±1]defined as follows :

a. IfMis an indecomposablekQ-module, we set X_M^Q,y=

e∈N^Q0

χ (Gre(M))

i∈Q₀

x_i^{−e,αi−αi,dimM−e}y_i^ei; (2.1) b. ifM Pi[1]is the shift of an indecomposable projective module, we set

X_M^Q,y=xi; c. for any two objectsM, NinCQ, we set

X_M^Q,yX^Q,y_N =X_M^Q,y_⊕N.

It follows from the definition that equation (2.1) holds for anykQ-module. Note that cluster characters are invariant on isoclasses.

For any objectM inCQ, we denote byX_M^Q the value of the Caldero-Chapoton map atM. Equivalently,X^Q_M is the specialization ofX^Q,y_M aty_i=1 for alli∈Q0.

We now prove a multiplication formula on almost split sequences forX^Q,y_? . This is an analogue to [3, Proposition 3.10] for the Caldero-Chapoton map.

Proposition 2.2 LetQbe an acyclic quiver,N be an indecomposable non-projective module. Then

X_M^Q,yX_N^Q,y=X_B^Q,y+y^dimN

whereBis the uniquekQ-module such that there exists an almost split sequence 0−→M−→ⁱ B−→^p N−→0.

Proof The proof is almost the same as in [3] for the coefficient-free case. We give it for completeness. We write m=dimMand n=dimN. We thus have

X^Q,y_M X_N^Q,y=X^Q,y_M⊕N=

e∈N^Q0

χ (Gre(M⊕N ))

i

x⁻_i ^{e,αi−αi,m+n−e}y_i^ei.

Since the varieties Gre(M⊕N )and

f+g=eGrf(M)×Grg(N )are isomorphic, we get

X^Q,y_M⊕N=

f,g

χ (Grf(M))χ (Grg(N ))

i

x_i^{−f+g,αi−αi,m+n−f−g}y_i^fi+gi.

We now consider the case where f=0 and g=dimN. Since Gr0(M)×GrdimN(N )= {(0, N )} the corresponding Laurent monomial inX_M^Q,y_⊕Nis

i

x_i^{−n,αi−αi,m}yⁿ_iⁱ

but m=c(n) wherec is the Coxeter transformation induced on K0(kQ) by the Auslander-Reiten translation. Thus,n, αi = − αi,mand then

i

x_i^{−n,αi−αi,m}y_iⁿⁱ=

i

y_iⁿⁱ=y^dimN.

Now, since the sequence is almost split, for every e∈N^Q0, the map ζe:

Gre(B)−→

f+g=eGrf(M)×Grg(N ) L → (i⁻¹(L), p(L))

is an algebraic homomorphism such that the fiber of a point(A, C)is empty if and only if(A, C)=(0, N )and is an affine space otherwise. It thus follows that

X_M^Q,yX^Q,y_N =X_B^Q,y+y^dimN

and the proposition is proved.

2.2 Adding coefficients to cluster characters

We now prove that in order to compute cluster characters with coefficients associated to a quiver Q, it suffices to compute cluster characters without coefficients for a certainQobtained from the quiverQ. More precisely letQ=(Q₀, Q₁)be an acyclic quiver, we denote byQ=(Q0,Q1)the acyclic quiver with vertex set consisting of two copies ofQ0. The first copy is identified withQ0and the second copy is denoted by

Q₀= {σ (v)|v∈Q₀}

whereσis a fixed bijectionQ0−→Q₀. For anyv=w∈Q0, ifv, w∈Q0, the arrows fromvtowinQ1are given by the arrows fromv towinQ1, otherwise there are only arrowsv−→σ (v)inQ₁ wherev runs overQ₀. In particular, we can identify Q₁with a subset ofQ₁. The quiverQis called the framed quiver associatedQ. By construction, the framed quiver of an acyclic quiver is itself acyclic. Note that framed quivers are familiar objects in the context of quiver varieties (see e.g.[18]).

Given an acyclic quiverR=(R₀, R₁)we denote byB(R)the incidence matrix ofR. That is the skew-symmetric matrix(b_ij)∈M_R0(Z)whose entries are given by

bij= | {α:i−→j∈R1} | − | {α:j−→i∈R1} | for anyi, j∈R₀.

We thus have

B(Q) =

B(Q) I

−I 0

.

The categorykQ-mod can be canonically identified with a subcategory of kQ- mod. We denote byι:kQ-mod−→kQ-mod the corresponding embedding, realiz- ingkQ-mod as a full, exact, extension-closed subcategory of kQ-mod. Dimension vectors induce bijectionsK₀(kQ-mod) Z^Q0 and K₀(kQ-mod) Z^Q0. Identify- ingZ^Q0 withZ^Q0× {0} ⊂Z^Q0×Z^Q0 Z^Q0 we can identifyK₀(kQ-mod)with a subgroup ofK0(kQ-mod).

LetA(Q,u)be the coefficient-free cluster algebra with initial seed(Q, u)where u=

u_i|i∈Q₀

. According to the Laurent phenomenon, it is a subring of the ring Z[u^±1] of Laurent polynomials in u. We denote by X_?^Q:Ob(C_Q)−→Z[u^±1] the Caldero-Chapoton map on CQ. For any kQ-module M, the value of the Caldero- Chapoton map atMis thus given by :

X_M^Q=

e∈N^Q0

χ (Gre(M))

i∈Q0

u⁻_i ^{e,αi−αi,dimM−e}

where−,−denotes the Euler form onkQ-mod.

We consider the homomorphism ofZ-algebras

π:

⎧⎨

⎩

Z[u^±1] −→ Z[y^±1,x^±1]

u_i → x_i ifi∈Q₀, u_j → y_i ifj=σ (i)∈Q₀. Lemma 2.3 For anykQ-moduleM, we have

X^Q,y_M =π

X_ι(M)^Q

.

Proof LetMbe akQ-module which we consider as a representation ofQ. For any i∈Q0, we denote byM(i)the correspondingk-vector space at vertexiand for any α:i−→j ∈Q1, we denote by M(α):M(i)−→M(j )the corresponding k-linear map. Thus,ι(M)can be identified with the representation ofQgiven byι(M)(i)= M(i)ifi∈Q₀,ι(M)(i)=0 ifi∈Q₀andι(M)(α)=M(α)ifα∈Q₁,ι(M)(α)=0 ifα∈Q₁. In particular, dimι(M)=dimM. Moreover, Gre(ι(M))= ∅if e∈N^Q0 andιinduces an isomorphism Gre(M) Gre(ι(M))otherwise.

Note also that for anyi∈Q₀,j∈Q0, we have αi, αj

=0, αj, αi

=

−1 ifj=σ⁻¹(i), 0 otherwise, and for any i, j ∈Q₀, the form

α_i, α_j

is the same computed in kQ-mod and kQ-mod.

We thus have : X^Q_M=

e∈Q0

χ (Gre(ι(M)))

i∈Q0

u⁻_i ^{e,αi−αi,dimι(M)−e}

=

e∈N^Q0

χ (Gre(M))

i∈Q0

u⁻_i ^{e,αi−αi,dimM−e}

i∈Q₀

u⁻_i ^{e,αi−αi,dimι(M)−e}

=

e∈N^Q0

χ (Gre(M))

i∈Q0

u⁻_i ^{e,αi−αi,dimM−e}

i∈Q₀

u⁻_i ^e,αi

=

e∈N^Q0

χ (Gre(M))

i∈Q0

u⁻_i ^{e,αi−αi,dimM−e}

i∈Q₀

u^e_i^σ−1(i)

Applyingπ, we thus get π(X^Q_M)=

e∈N^Q0

χ (Gre(M))

i∈Q0

x_i^{−e,αi−αi,dimM−e}

i∈Q₀

y_i^ei

=X^Q,y_M

and the lemma is proved.

Remark 2.4 In [13], the authors gave a slightly different definition of the cluster characters with coefficients than the one we use here. We now prove that the definition we give in this paper is compatible with their definition.

LetQbe an acyclic quiver,Qthe corresponding framed quiver andQ=Q^op. Let mod-kQbe the category of finite dimensional right modules overkQconsidered in [13]. This category is equivalent to the categorykQ-mod of finite dimensional left- modules over the path algebra ofQ. It thus follows from [13] that the cluster category CQis equivalent to the category^⊥((kQ/ kQ))/(k Q/ kQ) wheredenotes the shift functor inCQ and where^⊥((kQ/ kQ)) denotes the full subcategory consisting of objectsMinCQsuch that Ext¹_C

Q(M, Pi)=0 for anyi∈Q₀. Thus objects inCQcan be identified with objectsM inCQsuch that Ext¹_C

Q(M, P_i)=0 for anyi∈Q₀and such thatM Pi for anyi∈Q₀.

Given an objectMinCQ, the cluster characterX_M∈Z[y,x^±1]associated toMby Fu and Keller is defined as follows. Using the above equivalence of categories,Mis viewed as an object in^⊥((kQ/ kQ))/(k Q/ kQ) and the characterX_M isπ(X_M^kQ) whereX_?^kQ:Ob(CQ)−→Z[u^±1]is the cluster character onCQassociated by Palu to the cluster-tilting objectkQinCQ(see [19] for details).

Fix thus an indecomposable objectMin^⊥((kQ/ kQ))/(k Q/ kQ). If Mis not a projectivekQ-module, then Mis akQ-module and

0=Hom_C

Q(Pi, M)=Hom_kQ(Pi, M)=dimM(i)

for anyi∈Q₀ so thatM can be viewed as a representation ofQ. In particular, there is somekQ-moduleM₀such that(M)=ι(M₀). Thus, we get equalities

X_M=π(X_M^kQ)=π(X^Q_M )=X^Q,y_ι(M

0)

where the second equality follows from [19, Section 5] and the last equality follows from Lemma2.3. IfMis a projective modulePjfor somej∈Q0, then

X_M =π(X^k_P^Q

j)=π(u_j)=x_j=X^Q,y_P

j[1].

Conversely, for any objectMinkQ-mod,ι(M)is an object inkQ-mod such that 0=Hom_kQ(P_i, ι(M))=Hom_C

Q(P_i, ι(M))for anyi∈Q₀ so that⁻¹ι(M)belongs to^⊥((kQ/ kQ))/(k Q/ kQ). Thus,

X^Q,y_M =π(X^Q_ι(M) )=π(X^k_ι(M)^Q )=X ₋1ι(M)

where the first equality follows from Lemma 2.3 and the second follows from [19, Section 5]. Thus, cluster characters with coefficients we defined coincide with those previously introduced by Fu and Keller. In particular, the cluster variables in A(Q,x,y)are precisely the charactersX^Q,y_M whenMruns over the indecomposable rigid objects inCQ[13].

3 Characters with coefficients in Dynkin typeA

Letr≥1 be an integer andAdenote the quiver of type−→Ar, that is, of Dynkin type Arequipped with the following orientation :

0 1 2 · · · r−1.

For anyi∈ [0, r−1],n∈ [1, r−i], we denote byS_i⁽ⁿ⁾the unique (up to isomorphism) indecomposablekA-module with socleSi and lengthn. By convention, for anyi∈ [0, r−1],S_i⁽⁰⁾denotes the zero module. For simplicity, we denote byi+rthe vertex σ (i)∈A₀for anyi∈ [0, r−1].

The following lemma is analogous to [9, Lemma 4.2.1] :

Lemma 3.1 For anyi∈ [0, r−2]andn∈ [1, r−1−i], the following holds:

X^A,y

S⁽ⁿ⁾_i X^A,y

S_i⁽ⁿ⁾₊₁=X^A,y

S⁽ⁿ_i⁺¹⁾X^A,y

S_i⁽ⁿ₊⁻₁¹⁾+y^dimSi+(n)1.

Proof For anyi∈ [0, r−2]andn∈ [1, r−1−i], there is an almost split sequence 0−→S_i⁽ⁿ⁾−→S_i⁽ⁿ⁺¹⁾⊕S⁽ⁿ_i+⁻₁¹⁾−→S_i+⁽ⁿ⁾₁−→0.

The lemma is thus a direct consequence of Proposition2.2.

We now prove a relation analogous to three terms recurrence relations in the con- text of orthogonal polynomials. This relation will be essential in order to extract quan- tized Chebyshev polynomials.

Lemma 3.2 For anyi∈ [0, r−2]andn∈ [1, r−1−i], we have X^A,y

S_i⁽ⁿ⁾X^A,y_S

i+n=X^A,y

S_i⁽ⁿ⁺¹⁾+y_i+nX^A,y

S⁽ⁿ_i ⁻¹⁾.

Proof Leti∈ [0, r−2]andn∈ [1, r−1−i]. We consider the indecomposablekA- modulesι(S_i⁽ⁿ⁾)andι(Si+n)in the cluster categoryCA. We thus have isomorphisms of vector spaces (see [2]) :

Ext¹_C

A(ι(Si+n), ι(S_i⁽ⁿ⁾)) Ext¹

kA(ι(Si+n), ι(S_i⁽ⁿ⁾))⊕Ext¹

kA(ι(S_i⁽ⁿ⁾), ι(Si+n)) Ext¹_kA(S_i+n, S_i⁽ⁿ⁾)⊕Ext¹_kA(S_i⁽ⁿ⁾, S_i+n)

Ext¹_kA(S_i+n, S_i⁽ⁿ⁾) HomkA(S⁽ⁿ⁾_i , S_i+n−1) k

So we can apply Caldero-Keller’s one-dimensional multiplication formula for cluster characters without coefficients [6] toι(Si+n)andι(S_i⁽ⁿ⁾)inC_A. We get :

X_ι(S^A

i+n)X^A

ι(S⁽ⁿ⁾_i )=X^A

ι(S_i⁽ⁿ⁺¹⁾)+X_B^A

whereB=kerfˆ⊕cokerfˆ[−1]_Afor any 0= ˆf ∈Hom_kA(ι(S_i⁽ⁿ⁾), τ_A(ι(S_i+n−1))) k.

We now have to compute Hom_kA(ι(S_i⁽ⁿ⁾), τ_A(ι(Si+n−1))). For this, we first com- pute τ_A(ι(S_i+n−1)) taking care of the fact that ι does not commute with the Auslander-Reiten translation.

In order to fix notations, we drawAas follows :

r r+1 · · · 2r−2 2r−1

0 1 · · · r−2 r−1

We compute that a projective resolution ofS_i+n−1is given by P_i+n−1⊕P_i+n+r−→^f P_i+n−→S_i+n−→0.

Applying the Nakayama functorνwe get

I_i+n−1⊕I_i+n+r−−→^{ν(f )} I_i+n

where ν(f ) is surjective since I_i+n−1−→I_i+n is onto. It follows from [16] that τ_kA(ι(S_i+n)) kerν(f )and thus kerν(f )is the representation given by

0 · · · 0 0 k 0 · · · 0

0 · · · 0 k k k · · · k

i+n−1

where the arrows are obviously zero or identity maps.

Sinceι(S_i⁽ⁿ⁾)is the representation given by

0 · · · 0 0 · · · 0 0 · · · 0

0 · · · 0 k · · · k 0 · · · 0

i i+n−1

we get that for any non-zero morphismfˆ, the kernel kerfˆis given by

0 · · · 0 0 · · · 0 0 · · · 0

0 · · · 0 k · · · k 0 · · · 0

i i+n−2

which is isomorphic toι(S_i⁽ⁿ⁻¹⁾)and cokerfˆis

0 · · · 0 k 0 · · · 0 · · · 0

0 · · · 0 k k · · · k · · · k

i+n

which is isomorphic to the injectivekA-module I_i+n+r. It thus follows that B ι(S_i⁽ⁿ⁻¹⁾)⊕P_i+n+r[1]

and

X^A_B=X^A

ι(S_i⁽ⁿ⁻¹⁾)u_i+n+r. Thus,

X^A_ι(S

i+n)X^A

ι(S_i⁽ⁿ⁾)=X^A

ι(S⁽ⁿ_i ⁺¹⁾)+u_i+n+rX^A

ι(S_i⁽ⁿ⁻¹⁾). Applying the homomorphismπ of Lemma2.3to this identity, we get

X^A,y_S

i+nX^A,y

S_i⁽ⁿ⁾=X^A,y

S_i⁽ⁿ⁺¹⁾+yi+nX^A,y

S⁽ⁿ_i ⁻¹⁾

and the lemma is proved.

Lemma 3.3 LetAbe a quiver of type−→

Ar withreven. Then the set

X^A,y_S

i |i∈ [0, r−1]

is algebraically independent overZ[y].

Proof Denote byBthe incidence matrix ofA. Asris even,Bis of full rank and thus there exists aZ-linear formonZ^Q0 such that(Bα_i) <0 for everyi∈ [0, r−1]. It thus follows from [7] that

F_n=

⎛

⎝

(ν)≤n

Z

i∈Q₀

x_i^νi

⎞

⎠∩Z[X_S^A

i|i∈ [0, r−1]]

defines a filtration onZ[X^A_S

i|i∈ [0, r−1]and in the associated graded algebra, we have

gr(X_M^A)=gr

r−1 i=0

x_i^−Si,M

for everykA-moduleM. We now consider the grading on Z[X^A,y_S

i |i∈ [0, r−1]]

given the grading onZ[X^A_S

i|i∈ [0, r−1]]and deg(yi)=0 for everyi∈Q0. We thus have that

gr(X^A,y_M )=gr

r−1

i=0

x_i^−Si,M

and thus, since(S_i, M)_i=0...r−1=(S_i, N)_i=0...r−1if dimM=dimN, it follows that any finite set{X_M^A,y

i |i∈J}with dimM_i=dimM_j is linearly independent over Z[y].