Tensor products and <span class="mathjax-formula">$q$</span>-characters of HL-modules and monoidal categorifications

Tensor products and -characters of HL-modules and monoidal categorifications

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TENSOR PRODUCTS AND q -CHARACTERS OF HL-MODULES AND MONOIDAL CATEGORIFICATIONS

by Matheus Brito & Vyjayanthi Chari

Abstract. — We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite-dimensional representations of a quantum affine algebra of typeA.

We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors.

As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of typeAwith coefficients.

Résumé(Produits tensoriels etq-caractères de HL-modules et catégorifications monoïdales) Dans ce travail, nous étudions certaines sous-catégories monoïdales (introduites par David Hernandez et Bernard Leclerc) de représentations de dimension finie d’une algèbre affine de type A. Nous classifions l’ensemble des représentations premières de ces sous-catégories, et donnons des conditions nécessaires et suffisantes pour que le produit tensoriel des deux repré- sentations premières soit irréductible. Dans le cas où le produit tensoriel est réductible, nous décrivons une factorisation en modules premiers des facteurs simples. En conséquence, nous prouvons que ces sous-catégories monoïdales sont des catégorifications monoïdales d’algèbres amassées de typeAavec coefficients.

Contents

Introduction. . . 582

1. The main results. . . 584

2. Proof of Proposition 1.5 and aq-character formula.. . . 591

3. Irreducible tensor products.. . . 598

4. Identities inK0(Fξ). . . 604

5. Proof of Proposition 4.4. . . 611

References. . . 618

Mathematics Subject Classification (2010). — 17B37, 20G42, 13F60.

Keywords. — Cluster algebra, monoidal categorification, prime representations.

MB was partially supported by CNPq, grant 205281/2014-1. VC was partially supported by DMS 1719357.

Introduction

The study of the categoryF finite-dimensional representations of a quantum affine algebra goes back nearly thirty years and continues to be of significant interest. The irreducible objects in this category are indexed by elements of a free abelian monoid (denoted P⁺) with generatorsωi,a where i varies over the index set for the simple roots and avaries over non-zero elements of the field of rational functions in a vari- ableq. The category is not semisimple and there are many interesting indecomposable objects in it. In recent years, there has been a new insight in the study ofF coming from connections with cluster algebras through the work of [19], [18], [21] and also from KLR algebras through the work of [20].

The categoryF is a monoidal tensor category and an interesting feature is that a tensor product of generic simple objects is simple. An obviously related notion is that of a prime simple object; this is one which cannot be written in a non-trivial way as a tensor product of objects ofF. An open and very difficult question is the following:

classify prime simple objects inF and describe the factorization of an arbitrary simple object as a tensor product of primes. The answer to this question for sl2 was given in [8] where it was also proved that the factorization was unique. In higher rank the question along with that of uniqueness remains unanswered. However, in [16] and [17]

an important result was established which greatly simplifies the problem by reducing it to following: give a necessary and sufficient condition for the tensor product of a pair of prime simple objects to be simple.

In this paper we focus on this question for certain subcategories of F associated with quantum affinesln+1. These subcategories were introduced by David Hernandez and Bernard Leclerc ([19], [18]) and the definition has its roots in the theory of cluster algebras. The remarkable insight was that prime representations were analogous to cluster variables and the irreducibility of a tensor product of prime objects was analo- gous to the idea of two elements belonging to the same cluster. The role of the quiver in the theory of cluster algebras is played by the height function; a height function (of typeA_n) is a functionξ: [1, n]→Zsatisfying the condition |ξ(i)−ξ(i+ 1)|= 1 for 1 6i 6 n−1. Define P_ξ⁺ to be the submonoid of P⁺ generated by elements ω_i,qξ(i)±1 and letFξ be the full subcategory ofF consisting of objects whose Jordan- Holder constituents are indexed by elements ofPξ⁺. It was proved in [18] thatFξ is a monoidal tensor category and we letK0(Fξ)be the Grothendieck ring ofFξ. In the case whenξis the bipartite height function, i.e.,ξ(i−1) =ξ(i+ 1)for26i6n−2or the monotonic functionξ(i) =i they showed thatK0(Fξ)is isomorphic to a cluster algebra with coefficients of typeA.

In this paper we prove the result for all height functions of type Aby representa- tion theoretic methods. We define a subsetPrξ of P_ξ⁺ such that the corresponding irreducible representations (which we call HL-modules) are prime. Working entirely in Fξ we show that the HL-modules are precisely all the prime objects in this cate- gory. To do this, we establish necessary and sufficient conditions for a tensor product of HL-modules to be irreducible. In the case when the tensor product is reducible we

describe the Jordan-Holder constituents and their factorization as a tensor product of HL-modules.

The connection with cluster algebras is then made as follows. We define a quiverQ_ξ associated with ξ; since we are working in the general case the quiver we use is a mutation of the quivers in [19] and [18]. This mutation allows us to map a non-frozen variable in the initial seed of the cluster algebra to the class of the irreducible module corresponding to eitherω_i,ξ(i)+1orω_i,ξ(i)−1. The first mutation at any element of the initial seed is easily described; however is not necessarily of the form ω_i,ξ(i)±1. Our tensor product formulas now allow us to prove the existence of an algebra isomorphism between the cluster algebra with n frozen variables and K0(Fξ). The isomorphism maps a cluster variable to an HL-module and we identify this module explicitly. We also show that the isomorphism maps cluster monomials to simple tensor products of HL-modules. As a consequence of this result we give an alternative proof for the product of a pair of cluster variables to be a cluster monomial; equivalently we give an alternative proof of the criterion for a pair of roots to be compatible. In Proposition 2.5 we give a closed formula for a cluster variable in terms of the original seed. In terms of representation theory this can be interpreted as giving aq-character formula for the prime representations inFξ. It is useful to remark here that other explicit formulas for cluster variables can be found in the literature see for instance, [2], [4], [11], [14]. Not all these papers deal with frozen variables and even those that do impose conditions on the frozen variables which are not satisfied by the quivers considered in this paper. The role of the frozen variable in the connection with representation theory is important and motivates our formulas.

The paper is organized as follows. In Section 1 we recall the definition of the height function ξ and introduce the associated quiverQ_ξ. We then state and prove our main result modulo the key Propositions 1.5, 1.6 and 1.7. In Section 2 we prove Proposition 1.5 which gives a recursive formula for a cluster variable. This is done by a simple analysis of the quiver obtained by mutating at successive nodes. The answer we obtain is in a form which is well adapted to the representation theory of quantum affine algebras and can be viewed as an analog of Pieri’s rule in classical representation theory. We then solve the recursion to give a closed formula for the cluster variable in terms of the initial cluster which includes the frozen variables. In Sections 3, 4 and 5 we provide sufficient and necessary conditions, for the tensor product of two HL-modules to be irreducible. We also analyze the Jordan-Holder series of a reducible tensor product of HL-modules. The proof of Propositions 1.6 and 1.7 can be found in Section 4.

Acknowledgements. — MB is grateful to the Department of Mathematics, UCR, for their hospitality during a visit when part of this research was carried out. He also thanks David Hernandez for supporting his visit to Paris 7 and many helpful discus- sions. VC thanks David Hernandez, Bernard Leclerc and Salvatore Stella for helpful conversations. The authors are grateful to the referees for several helpful comments and references.

1. The main results

Throughout the paper we denote byC,Z,Z+ andNthe set of complex numbers, integers, non-negative and positive integers respectively. Fori, j ∈Z+ with i6j we let[i, j] ={i, i+ 1, . . . , j}. Given a commutative ringAwe denote byA[q](resp.A(q)) the ring (resp. quotient field) of polynomials in an indeterminate qwith coefficients in A.

1.1. The cluster algebra A(x, Q_ξ). — Let ξ : [1, n] → Z be a height function;

namely a function which satisfies the conditions

|ξ(i)−ξ(i−1)|= 1, 26i6n.

It will be convenient to extendξ to[0, n+ 1] by settingξ(0) =ξ(2) andξ(n−1) = ξ(n+ 1).

Remark. — Although trivial, it is useful to note that

{ξ(i+ 1), ξ(i−1)} ⊂ {ξ(i) + 1, ξ(i)−1}

and that the inclusion can be strict.

For i ∈ [1, n−2], let i ∈ [i, n], i > i be minimal such that ξ(i) = ξ(i+ 2) and set(n−1) = (n−1) andn =n. Let Qξ be a quiver with2nvertices labeled {1, . . . , n,1⁰, . . . , n⁰}and with the set of edges given as follows:

– there are no edges between the primed vertices; in other words the vertices {1⁰, . . . , n⁰}are frozen,

– if16j6n−1 andξ(j) =ξ(j+ 1) + 1, the edges atj are:

(j−1)oo j

1−δ_j,j '' δj,j

))

(j+ 1) 1−δj,j

gg

j⁰

OO

(j+ 1)⁰

and the reverse orientations ifξ(j) =ξ(j+ 1)−1, whereδj,j is the Kronecker delta function and we adopt the convention that a labeled edge exists iff the label is one,

– at the vertexnwe have edges(n−1)→n→n⁰ ifξ(n−1) =ξ(n) + 1and the reverse orientation otherwise.

Clearlyj is a sink or source ofQ_ξ (where we ignore the frozen vertices) iffj = 1 or j=j. For16j61 setj•= 0and forj >1 letj•be the maximal sink or source ofQξ satisfyingj•< j.

Fix a set x = {x1, . . . , xn, f1, . . . , fn} of algebraically independent variables and letA(x, Q_ξ)be the cluster algebra (with coefficients) with initial seed (x, Q_ξ). The definition of a cluster algebra is recalled briefly in Section 2.1; for the rest of this section we shall freely use the language of cluster algebras. Since the principal unfrozen part ofQξ is a quiver of typeAn, the set of non-frozen cluster variables inA(x, Qξ)

are indexed by the set Φ_>−1 of almost positive roots of a root system of type An. In other words if we let {αi : 1 6 i 6n}, be a set of simple roots for An and set α_i,j=α_i+· · ·+α_j,16i6j6n, then

Φ_>−1={−αi, αi,j: 16i6j6n}, and the cluster variables are denoted

{xi:=x[−αi], x[αi,j], fi: 16i6j6n}.

Moreover, the cluster variablex[α_i,j]is obtained by applying the sequencei, i+1, . . . , j of mutations at the original cluster.

1.2. The category Fξ. — Let Ub_q be the quantum loop algebra over C(q) associ- ated to sln+1 and let F be the monoidal tensor category whose objects are finite- dimensional representations ofUbq. Given a height functionξ: [1, n]→Zwe takePξ⁺

to be the free abelian monoid with generators {ω_i,ξ(i)±1 : i ∈ [1, n]}. It is known that P_ξ⁺ is the index set for a (sub)-family of isomorphism classes of irreducible ob- jects of F. We define Fξ to be the full subcategory of F consisting of objects all of whose Jordan-Holder constituents are indexed by elements ofPξ⁺. It was proved in [18] that Fξ is a monoidal category and we let K0(Fξ) be the corresponding Grothendieck ring. For ω ∈P_ξ⁺ let [ω] ∈K0(Fξ)be the isomorphism class of the corresponding object inFξ.

Remark. — It is important to keep in mind that the assignment ω →[ω] is not a morphism of monoidsP_ξ⁺→K0(Fξ), i.e.,[ω][ω⁰]is not always equal to[ωω⁰]. One of the goals of this paper is to determine a necessary and sufficient condition for equality to hold.

Fori∈[1, n] set

f_i=ωi,ξ(i)+1ω_i,ξ(i)−1.

If16i < j 6n, leti2<· · ·< ik−1 be an ordered enumeration of the subset {p:i < p < j, ξ(p−1) =ξ(p+ 1)},

i1=i,ik =j and define an elementω(i, j)∈P_ξ⁺ by:

ω(i, j) =ω_i1,a1· · ·ω_ik,ak,

wherea₁=ξ(i)±1ifξ(i+ 1) =ξ(i)∓1anda_m=ξ(i_m)±1ifξ(i_m) =ξ(i_m−1)±1, form>2. Set

Pr_ξ ={ω_i,ξ(i)±1, ω(i, j) : 16i6j6n, i6=j}.

Clearly the setPrξ has the same cardinality as the set of unfrozen cluster variables in A(x, Q_ξ).

Recall that an object ofF is said to be prime if it cannot be written in a non-trivial way as a tensor product of objects ofF. The following is a special case of the main result of [6].

Lemma. — The irreducible object ofFξ associated to an element

ω∈Pr_ξ∪ {f_i: 16i6n}

is prime.

1.3. Main Theorem. — Recall that by definitionn=nand(n−1) = (n−1)which in particular implies thatn•=n−1. Fork>1, set

(1.1) k= (k+ 1)(1−δk,k) + (k•+ 1)δk,k.

Theorem1. — Letξ: [1, n]→Zbe a height function. The assignment ι(xi) = [ω_i,ξ(i+1)], ι(fi) = [f_i],

extends to an isomorphism of rings ι : A(x, Q_ξ) →K0(Fξ) such that for 1 6i 6 k6n,

ι(x[αi,i]) = [ω_i,ξ(i+1)±2], ξ(i) =ξ(i+ 1)±1, (1.2)

ι(x[αi,k]) = [ω(i, k)], k6=i, (1.3)

ι(f_px[α]) = [f_pω] p∈[1, n], α∈Φ_>−1, [ω] =ι(x[α]).

(1.4)

In particular ι maps cluster variable to a prime object of Fξ. Moreover, ifβ1, β2 ∈ Φ_>−1 are such thatx[β₁]x[β₂] is a cluster monomial then

[ω₁ω₂] = [ω₁][ω₂], [ω_s] =ι(x[β_s]), s= 1,2.

Corollary. — The homomorphism ι sends a cluster monomial to the equivalence class of an irreducible object of Fξ. In particular any irreducible prime object is of the form [ω], ω∈ Prξ ∪ {f_i : 16i6n} and Fξ is a monoidal categorification of A(x, ξ).

Proof of Corollary. — Letx[β1]· · ·x[βr]be a cluster monomial for someβ1, . . . , βr∈

Φ_>−1and set[ωi] =ι(x[βi])for16i6r. Then the pairsx[βj]x[βp],16j6=p6r

are cluster monomials and hence using the Theorem 1 we haveι(x[βj]x[βp]) = [ωjωp] for16j6=p6r. It follows from the main result of [16] (see Section 3 of this paper for the statement) that ι(x[β1]· · ·x[βr]) = [ω1· · ·ωr] and the first assertion of the corollary is established. Suppose that[ω]is an irreducible object ofFξ. Thenι⁻¹[ω]

is a sum of cluster monomials. Hence [ω] can be written as a linear combination of elements[π]where eachπis a product of elements fromPr_ξ∪ {f_i: 16i6n}. Since irreducible modules are a basis K0(Fξ) it follows that[ω] is a product of elements fromPrξ∪ {f_i: 16i6n}and the corollary is established.

Remark. — Suppose thatξsatisfiesξ(i−1) =ξ(i+ 1)for all16i6nor thatξ(j) = ξ(i) + (j−i)for all16i6j 6n. In these two cases the existence ofιwas established in [19],[18] by very different methods. As was noted in [18] the categoriesFξ are not necessarily equivalent for different height functions.

1.4. — In Theorem 3 of this paper we give conditions for the equality[ππ⁰] = [π][π⁰] when π,π⁰ ∈ Prξ to hold in K0(Fξ). The translation to the language of cluster algebra gives the conditions for describing when two roots are compatible. Thus our theorem gives a proof of the following assertion (compare with the description in [18,

§10.2.3] where a similar description in the case of the bipartite height function).

Assume thati6j,k6`andi6k. Ifj6=i the rootsαi,j,αk,`are compatible iff:

– k=ior k > j+ 1, – j=j andj•+ 16k6j, – `6=k and

– eitherj=`,

– ori < k < j < `, and #{k6m < j−1 :m=m} ∈2Z++ 1, – ori < k < ` < j, and #{k6m < `−1 :m=m} ∈2Z+. The rootsαi,i andαk,` withi6kare compatible iff :

k•6=k−1, or (k−1)•>i or `6=k andi=k.

The roots−αi andαk,` are in the same cluster iff eitherk > i or` < i.

In Theorem 4 we write down the Jordan-Holder series for a reducible tensor product of objects. This amounts to writing down all the non-trivial exchange relations for cluster variables including the frozen variables and is not hard to do using the analysis above.

1.5. — The proof of the theorem involves three principal steps. For16j6n, set dj=δj,j=δξ(j),ξ(j+2).

The first step is the following proposition which gives a recursive formula for the cluster variables. We adopt the convention thatα_i,m=α_m, m6i.

Proposition. — For16i < j 6n we have x_ix[α_i] =f_ix^1−d_i+1ⁱ+f_i+1^1−dix_i−1x^d_i+1ⁱ ,

xjx[αi,j] =f_j^dj−1x[α_i,j−1]x^1−d_j+1^j +f_j+1^1−djx^d_j+1^j

(δi_•,j_•+δi,j_•)f_i^δi,j•x^1−δ_i−1^i,j•

+ (1−δi•,j•−δi,j•)f_j^d_•^{j• −1}x[αi,j•−1] .

The proof of this proposition is in Section 2 where we also give a closed formula forx[αi,j]as a Laurent polynomial in the variables {x1, . . . , xn, f1, . . . , fn}.

1.6. — The second step in the proof of the theorem is the following. We adopt the convention that we takeω_i,ξ(i+1)+2 ifξ(i) =ξ(i+ 1) + 1 and we take ω_{i,ξ(i+1)−2} if ξ(i) =ξ(i+ 1)−1.

Proposition. — The following equalities hold in K0(Fξ)for16i6j6n.

(i) We have

[ω_i,ξ(i+1)][ω(i, i+ 1)]^1−di[ω_i,ξ(i+1)±2]^di

= [f_i] [ω_i+1,ξ(i+2)]^1−di+ [f_i+1]^1−di[ω_i+1,ξ(i+2)]^di [ω_i−1,ξ(i)].

(ii) If j•6i < j then

[ω_j,ξ(j+1)][ω(i, j)]^1−δj,i[ω_i,ξ(i+1)±2]^δj,i

= [f_j]^dj−1[ω(i, j)]^1−dj−1[ω_i,ξ(i+1)±2]^dj−1[ω_j+1,ξ(j+2)]^1−dj

+ [f_j+1]^1−dj[f_i]^δi,j•[ω_j+1,ξ(j+2)]^dj[ω_i−1,ξ(i)]^1−δi,j•. (iii) If i < j• choose z ∈ {ξ(i) + 1, ξ(i)−1} so that ω⁻¹_i,zω(i, j•) ∈ Pξ⁺ and set k= (j•)•. Then,

[ω_j,ξ(j+1)][ω(i, j)]

= [f_j]^dj−1[ω_j+1,ξ(j+2)]^1−dj[ω(i, j−1)]^{1−δ(j−1)•,i•}[ωi,z]^{δ(j−1)•,i•}

+ [f_j+1]^1−dj[f_j•]^{dj• −1}[ω_j+1,ξ(j+2)]^dj[ω(i, j•−1)]^{1−δi•,k•dj• −1}[ωi,z]^{δi•,k•dj• −1}.

The proof of this proposition can be found in Section 4.

1.7. — Proposition 1.5 and Proposition 1.6 are enough to establish the existence ofι and to identify the image of a cluster variable. The third step needed to establish the theorem is to show thatι maps a cluster monomial to the isomorphism class of an irreducible representation. To do this we will need the following result.

Proposition. — Letω,ω⁰∈Prξ. Then either[ω][ω⁰] = [ωω⁰]or[ω][ω⁰] = [ω1]+[ω2] where[ω₁] and[ω₂] are the images underι of cluster monomials.

A much more precise statement can be found in Theorem 3 and Theorem 4 in Sections 3 and 4. In the rest of this section we assume Proposition 1.5, Proposition 1.6, Proposition 1.7 and prove Theorem 1.

1.8. Existence ofι. — Recall [1] that an element ofA(x, Qξ)is said to be a standard monomial if it is a monomial in the elements {xi, x[α_i] : i ∈ [1, n]} and does not involve any product of the formxix[αi],i∈[1, n]. It was proved in [1] that standard monomials are aZ[fi:i∈I]-basis ofA(x, ξ).

On the other hand consider the quotient of the polynomial ring (with integer coefficients) in variablesX_i, X[α_i], F_i,i∈[1, n]subject to the first relation in Propo- sition 1.5. It is not hard to show that this ring is theZ[Fi :i∈I]span of monomials in Xi, X[αi], i ∈ [1, n] which do not involve products of XiX[αi] for any i ∈[1, n].

It follows thatA(x, Qξ)is is isomorphic to this quotient (compare with [19, Lem. 4.4]).

Using Proposition 1.6(i) we have [ω_{i, ξ(i+1)}][ω(i, i+ 1)]^1−di[ω_i,ξ(i+1)±2]^di

= [f_i][ωi+1,ξ(i+2)]^1−di+ [f_i+1]^1−di[ωi+1,ξ(i+2)]^di[ω_i−1,ξ(i)].

It is now immediate that the assignment

xi−→[ωi,ξ(i+1)], fi−→f_i, x[αi]−→[ω_i,ξ(i+1)±2]^δi,i[ω(i, i+ 1)]^1−δi,i defines a homomorphism of ringsι:A(x, Qξ)→K0(Fξ).

1.9. The elementsι(x[α]),α∈Φ_>−1. — The formulas given in (1.2) and (1.3) can be rewritten as follows:

ι(x[αi,j]) = [ω(i, j+ 1)]^1−dj[ω_i,ξ(i+1)±2]^{δi•,j•dj}[ω(i, j•+ 1)]^{dj(1−δi•,j•)}, j >i.

(1.5)

We shall prove this reformulation by induction onj−i. Observe that induction begins when j =i by definition. For the inductive step applyι to both sides of the second equation in Proposition 1.5. We will show that the right hand side of this equation is the same as the right hand side of the equation in Proposition 1.6(ii), (iii). Hence the left hand sides must match up. The inductive step is immediate once we observe thatK0(Fξ)has no zero divisors.

To prove that the right hand sides are the same, suppose first thatj•6i(in particu- larj•=i•orj•=i). Applyingιto both sides of the second equation in Proposition 1.5 gives

[ω_j,ξ(j+1)]ι(x[α_i,j]) =f^d_j^j−1ι(x[α_i,j−1])[ω_j+1,ξ(j+2)]^1−dj

+f^1−d_j+1^jf^δ_i^i,j•[ω_j+1,ξ(j+2)]^dj[ω_i−1,ξ(i)]^1−δi,j•. The second term on the right hand side of the preceding equation is equal to the second term on the right hand side of the equation in Proposition 1.6(ii). To see that the first terms match up we use the inductive hypothesis forι(x[α_i,j−1])and see that it suffices to prove that

[ω_i,ξ(i+1)±2]^dj−1 = [ω_i,ξ(i+1)±2]^{δi•,(j−1)•}[ω(i,(j−1)•+ 1)]^{1−δi•,(j−1)•d}j−1

. Ifd_j−1= 0, then the preceding equality is obviously true. Since

d_j−1= 1 =⇒ (j−1) = (j−1)=j•=i =⇒ i•= (j−1)•

and the equality follows.

Ifi < j•, then the result follows if we prove that,

ι(x[αi,j−1]) = [ω(i, j)]^1−dj−1 [ωi,z]^δk,i•[ω(i, k+ 1)]^{1−δk,i•d}j−1

, ι(x[α_i,j•−1]) = [ω(i, j•)]^{1−dj• −1} [ω_i,z]^δi•,k•[ω(i, k•+ 1)]^{1−δi•,k•d}j• −1

,

where we recall thatk= (j•)•. Ifd_j−1= 0the first equality follows from the definition and the inductive hypothesis and ifd_j−1= 1 then(j−1) =j• and so(j−1)•=k.

The first equality again follows from the inductive hypothesis. The second equality is deduced in the same way from the inductive hypothesis.

1.10. — We prove now thatιis an isomorphism. Let{ω1, . . . , ωn}which are dual to the simple roots ofAnandP⁺be theirZ+-span. It is convenient to setω0=ωn+1= 0.

Let6be the usual partial order onP⁺ given byµ6λiffλ−µis in theZ+-span of {α₁, . . . , α_n}.

Define a morphism of monoids wt :P_ξ⁺ →P⁺ by settingwtωi,a=ωi. SinceFξ

is a tensor category it is well-known that the following holds in K0(Fξ); for ω = ω_i1,a1· · ·ω_ik,ak∈Pξ⁺:

(1.6) [ωi₁,a₁]· · ·[ωi_k,a_k] = [ω] + X

π∈P+ ξ

wtπ<wtω

r(ω,π)[π], for somer(ω,π)∈Z+.

A straightforward induction on wtω shows that K0(Fξ) is generated as a ring by the elements [ω_i,ξ(i)±1]. By Section 1.9 we see that ι({x[−αi], x[αi,i]}) = {[ω_i,ξ(i)+1],[ω_i,ξ(i)−1]}and hence it follows thatιis surjective.

We prove thatι is injective. Set

wt`xi=ω<sub>i,ξ(i+1)</sub>, wt`fi=f_i, wt`x[αi] =π, such that ι(x[αi]) = [π].

Extendwt_` in the obvious way to the basis ofA(x, ξ); if m=x^p₁¹· · ·x^p_nⁿx[α1]^m1· · ·x[αn]^mn

is a standard monomial inA(x, ξ)andf =f₁^r1· · ·f_n^rn∈Z[f₁^±1, . . . , f_n^±1]then wt`f m=

n

Y

i=1

f^r_iⁱω^p_{i, ξ(i+1)}ⁱ

ω^δξ(i),ξ(i+1)+1

i, ξ(i+1)+2 ω^δξ(i),ξ(i+1)−1

i,ξ(i+1)−2 ω^1−di+1, ξ(i+2)ⁱ

^mi

.

Lemma. — Let m,m⁰ be standard monomials in A(x, Qξ) andf,f⁰ be monomials in{fi:i∈[1, n]}. Then

wt`f m= wt`f⁰m⁰ ⇐⇒ f =f⁰ andm=m⁰. Proof. — Write

m=x^p₁¹· · ·x^p_nⁿx[α1]^m1· · ·x[αn]^mn, f =f₁^r1· · ·f_n^rn,

and letm⁰,f⁰ be defined similarly withpi replaced byp⁰_i etc. Ifp1>0 thenm1= 0 and using the fact thatPξ⁺ is a free abelian monoid we have

f^r₁¹ω^p_1,ξ(2)¹ =f^r

0 1

1 ω^p

0 1

1, ξ(2)ω^m

0

1δξ(1),ξ(2)+1

1, ξ(2)+2 ω^m

0

1δξ(1),ξ(2)−1

1, ξ(2)−2 . Sincef₁=ω_1,ξ(1)+1ω_i,ξ(1)−1, we get

r1+p1=r₁⁰ +p⁰₁, r1=m⁰₁+r₁⁰.

If m⁰₁ 6= 0 then p⁰₁ = 0 and we have r1 > r⁰₁ and r₁⁰ > r1 which is absurd. Hence m⁰₁ = 0 and so r⁰₁=r1 and p⁰₁ =p1. Writingm =x^p₁¹m1 andm⁰ =x^p₁¹m⁰₁ we see thatm1 andm⁰₁ are both standard monomials and

wt_{`</sub>f<sub>2</sub><sup>r2</sup>· · ·f<sub>n</sub><sup>rn</sup>m<sub>1</sub>= wt<sub>`}f^r

0 2

2 · · ·f^r

0

nnm⁰₁.

An obvious iteration of the preceding argument proves the lemma.

Suppose that

ι

X

r,s

cr,sf(s)mr

= 0,

wheremrvaries over standard monomials inA(x, Qξ), andf(s)varies over monomi- als inf_i,i∈[1, n]andc_r,s ∈Zwith only finitely many being non-zero. Assume for a contradiction thatcr,s6= 0for somer, sand letλbe a maximal element (with respect to the partial order onP⁺) of the set{wt(wt`f(s)mr) :cr,s6= 0}. Using (1.6) we get

0 = X

wt(wt`f(s)mr)=λ

cr,s[wt`f(s)mr] + X

wtω≯λ

nω[ω], nω∈Z.

Since the elements[ω],ω∈P_ξ⁺are linearly independent elements ofK0(Fξ)we get X

wt(wt_`f(s)m_r)=λ

cr,s[wt`f(s)mr] = 0.

By Lemma 1.10 the elements [wt`(f(s)mr)] are all distinct and hence also linearly independent. This forces cr,s = 0 contradicting our assumption and proves that ι is injective.

1.11. The elementsι(x[β1]x[β2]). — We now prove the final assertion of the theorem.

Write [ωs] = ι([x[βs]), s = 1,2 and let ω = ω1ω2. Assuming that [ω] 6= [ω1][ω2] we shall prove that x[α]x[β] is not a cluster monomial. By Proposition 1.7 we can write[ω₁][ω₂]as the non-trivial sum of elements which are images underι of cluster monomials. Since cluster monomials are linearly independent andιis an isomorphism we see that x[β1]x[β2]is not a cluster monomial and the proof of the main theorem is complete.

2. Proof of Proposition1.5and aq-character formula.

In this section we prove Proposition 1.5 which is a recursive formula for a cluster variable. We also solve this recursions and give a closed formula for the cluster variable in terms of the initial cluster and the frozen variables. In view of Section 1.9 this formula can also be viewed as giving theq-character of[ω],ω∈Pr_ξ in terms of the local Weyl modules and Kirillov-Reshetikhin modules.

2.1. — We briefly recall the definition (see [13]) of a cluster algebra. Let Q be a quiver with(n+m)-vertices labeled{1, . . . , n,1⁰, . . . , m⁰}and assume that the set of edges has no loops or2-cycles. A mutation ofQat a vertexiis the quiver obtained by performing the following three operations.

– reverse all edges ati,

– given edgesj→i→kadd a new edgej→k, – remove any two cycles that may have been created.

We shall assume that mutation is never allowed at the vertices labeled {1⁰, . . . , m⁰};

these are called the frozen vertices. Suppose that x={x1, . . . , xn, f1, . . . , fm} is an

algebraically independent set and let Q(x)be the field of rational functions in these variables. The setxis called the initial cluster and(x, Q)is called the initial seed.

Corresponding to a mutation ofQat a vertexidefine a new cluster x⁰={x⁰₁, . . . , x⁰_n, f1, . . . , fm}

by

x⁰_j=xj, j6=i, x⁰_ixi= Y

∃j→i inQ

fj

Y

∃j→i inQ

xj+ Y

∃i→k inQ

fk

Y

∃i→k inQ

xk.

The new cluster again consists of algebraically independent elements and we have a new seed (x⁰, Q⁰)where Q⁰ is the mutation of Q at i. Iterating this process defines a collection of new clusters and new seeds. An element of a given cluster is called a cluster variable. A cluster monomial is a product of cluster variables all belonging to the same cluster. The associated cluster algebra is the Z subring (of the field of rational functionQ(x)) generated by all the cluster variables.

2.2. The quiverQξ[i, j]. — Given 1 6i 6 n−1 set Qξ =Qξ[i, j] if j < i and let Q_ξ[i, i]be obtained by mutatingQ_ξ ati. Assume that we have definedQ_ξ[i, j−1]for j > iletQ_ξ[i, j]be the quiver defined by mutatingQ_ξ[i, j−1]at j.

Proposition 1.5 is a simple inspection when j = i and if j > i then it is a con- sequence of the discussion in Section 2.1, the following lemma and an induction on j−i.

Lemma. — Suppose that j > i and that we have an arrow (j−1) → j in Q_ξ. In Qξ[i, j−1]we have the following edges at the vertexj:

max{i−1, j•−1}

aj

(((j−1) joo

dj−1

1−dj

))

(j+ 1) dj

gg

max{i, j_•}⁰

b_j

44

j⁰ (j+ 1)⁰

1−d_j

gg

wherea_j = 1−δ_i,j• andb_j = min{1,(1−δ_j•,i•)d_j•−1+δ_j•,i}.

Proof. — We proceed by induction onj−i. To see that induction begins whenj=i+1 notice that

d_i= 1 =⇒ i= (i+ 1)• =⇒ a_i+1= 0, b_i+1= 1, d_i= 0 =⇒ i•= (i+ 1)• =⇒ a_i+1= 1, b_i+1= 0.

On the other hand, inQξ[i, i]which is the mutation of Qξ at i, an inspection shows that the edges ati+ 1are given as follows:

ioo (i+ 1)

1−d_i+1

**

(i+ 2) di+1

jj

i⁰

<<

(i+ 1)⁰ (i+ 2)⁰ 1−di+1

hh , di= 1,

(i−1)

!!

ioo (i+ 1)

1−di+1

**

(i+ 2) di+1

jj

(i+ 2)⁰ 1−di+1

hh , di= 0,

and it follows that induction begins. For the inductive step we assume that the result holds for the edges atj < n in Q_ξ[i, j−1]for and prove that it holds for the node j+ 1 inQξ[i, j].

Case 1. — If dj = 1 then j is a sink ofQξ by assumption and so we have an edge (j+ 1)→ j in Q_ξ. Hence by the inductive hypothesis the edges atj and (j+ 1)in Qξ[i, j−1]are

max{i−1, j•−1}

aj

''(j−1) joo

dj−1

(j+ 1)

oo

1−d_j+1 '' dj+1

))

(j+ 2) 1−dj+1

ii

max{i, j_•}⁰ bj

55

j⁰ (j+ 1)⁰

OO

(j+ 2)⁰. Mutating atj we see that the edges at (j+ 1)are

(j−1) //j //(j+ 1) dj−1

||

||

1−dj+1 ((

dj+1

**

(j+ 2) 1−dj+1

jj

j⁰ (j+ 1)⁰

OO

(j+ 2)⁰ .

The inductive step follows sincedj= 1 =⇒ (j+ 1)•=j and so

max{i−1,(j+1)•−1}=j−1, max{i,(j+1)•}⁰=j⁰, aj+1= 1 =dj, bj+1=d_j−1.

Case 2. — Ifdj= 0or equivalentlyj6=jthen inQξ we have an edge j→j+ 1. By the induction hypothesis, the edges atj and(j+ 1)in Qξ[i, j−1]are

max{i−1, j•−1}

aj

''(j−1) j

d_j−1

oo //(j+ 1)

1−dj+1

))

(j+ 2) d_j+1

ii

max{i, j•}⁰ b_j

66

j⁰ (j+ 1)⁰

__

(j+ 2)⁰ 1−dj+1

ff .

Mutating atj we obtain

max{i−1, j•−1}

aj

))

joo (j+ 1)

1−dj+1

**

(j+ 2) d_j+1

jj

max{i, j•}⁰

b_j 55

(j+ 2)⁰ 1−dj+1

hh

The inductive step follows from the fact thatdj= 0 =⇒ (j+ 1)•=j•< j and so max{i−1,(j+ 1)•−1}= max{i−1, j•−1}, max{i,(j+ 1)•}⁰= max{i, j•}⁰, and

aj+1=aj, bj+1=bj, dj= 0.

The proof of the lemma is complete.

2.3. The setΓi,j. — We continue to setdm=δm,m for16m6n. For i, j∈[1, n]

define sets Γ_i,j as follows: Γ_i,j ={0} ifj < i and if i 6j then Γ_i,j is the subset of Z^j−i+2+ consisting of elements ε = (ε_i, . . . , ε_j+1) satisfying the following conditions:

forr, m∈[i, j]withr6mandσr,m(ε) =εr+· · ·+εm, we have εj+1= 1 + (dj−1)σ_{max{i,j•+1},j}(ε), (2.1)

σ_{max{i,j•+1},j}(ε)61, (2.2)

σ_i,i(ε)616σ_i,i+1(ε) if i6j, (2.3)

σm+1,(m+1)(ε)616σm+1,(m+1)+1(ε), if i6m=m< j•. (2.4)

Clearly,εm∈ {0,1}fori6m6j+ 1. Fori6j let

Γ¹_i,j ={ε∈Γi,j :σ_{max{i,j•+1},j}(ε) = 1}, Γ⁰_i,j={ε∈Γi,j:σ_{max{i,j•+1},j}(ε) = 0}.

The condition in (2.2) shows that

Γi,j = Γ¹_i,jtΓ⁰_i,j. We shall use the following freely:

(2.5) dm−1= 0 ⇐⇒ (m−1)•=m•, dm−1= 1 ⇐⇒ m•=m−1.

Lemma. — Forj > ithe assignments

(εi, . . . , εj)−→(εi, . . . , εj, dj),

(ε_i, . . . , ε_j•)−→(ε_i+δ_i,j•, . . . , ε_j•,0, . . . ,0,1),

define bijections ι_j−1: Γ_i,j−1→Γ¹_i,j andιj•−1: Γi,j•−1→Γ⁰_i,j respectively.

Proof. — For the first assertion of the lemma we must prove that

eε= (εi, . . . , εj)∈Γi,j−1 ⇐⇒ ε= (εi, . . . , εj, dj)∈Γ¹_i,j.

Clearly we haveσm,r(ε) =σm,r(eε)for alli6m6r6j. Using (2.5) we see that (∗) εj = 1 + (dj−1−1)σmax{i,(j−1)•+1},j−1(eε) ⇐⇒ σ_{max{i,j•+1},j}(ε) = 1.

It follows thateεsatisfies (2.1) ifε∈Γ¹_i,j. It also proves thatεsatisfies (2.1) and (2.2) if eε ∈ Γ_i,j−1. To see that eε satisfies (2.2) if ε ∈ Γ¹_i,j we note that this is clear if (j−1)• = j• and if j• = j−1 it follows from the fact that ε satisfies (2.4) with m= (j−1)•.

It is obvious thateεsatisfies (2.3) (resp. (2.4)) ifε∈Γ¹_i,j; it is also obvious thatε satisfies these inequalities ifeε∈Γ_i,j−1as long asi6j−1(resp.i6m <(j−1)•)).

If i =j thendj = 1 and j• =i•< i. Using (∗) and the fact that we have already proved thatεsatisfies (2.1) we get

σ_{max{i,j•+1},j}(ε) = 16maxσ_{max{i,j•+1},j+1}(ε) = 2,

proving that (2.3) holds for ε. If (j −1)• 6 m = m < j• then we must have m= (j−1)•, andj•=j−1 = (m+ 1). It follows that d_j−1= 1, ε_j= 1and so we have

σ_{(j−1)•+1,j−1}(eε) =σ_{(j−1)•+1,j−1}(ε)61 =εj 6σ_{(j−1)•+1,j}(ε), proving thatεsatisfies (2.4). The proof of the first assertion is complete.

We prove the second assertion of the lemma; note that if ε∈ Γ⁰_i,j then we must have ε_m= 0forj•+ 16m6j and hence by (2.1) we also haveε_j+1= 1. Since

j•6i =⇒ Γ_i,j•−1={0} and Γ⁰_i,j={(δ_i,j•, . . . ,0,1)},

the result is trivially true in this case. We assume from now on that j• > i (in particularj•>i) and let

eε= (εi, . . . , εj•) ε= (εi, . . . , εj•,0, . . . ,0,1).

Suppose that eε∈Γi,j•−1. It is obvious that εsatisfies (2.1) and (2.2) and (2.3) and fori6m <(j•−1)•thatεsatisfies (2.4). If(j•−1)•6m=m6j•−1then either m= (j•−1)•orm=j•−1. In the first case the first inequality in (2.4) forεis just (2.2) foreεwhile the second inequality follows from (2.1) foreε. Ifm=m=j•−1, then (2.1) forcesεj• = 1and hence we have εj•616εj•+εj•+1. This proves that (2.4) holds forεand soε∈Γ⁰_i,j.

Next we assume thatε∈Γ⁰_i,jand prove thateε∈Γi,j_•−1. To prove that (2.1) holds for eε it suffices to observe that if j• = i (resp.(j•)• > i) then (2.3) (resp. (2.4)) forεgives

σ_{max{i,(j•)•+1},j•}(eε) =σ_{max{i,(j•)•+1},j•}(ε) = 1.

Ifd_j•−1= 1then(j•)•=j•−1and so the preceding equality isε_j•= 1as needed. If d_j•−1 = 0then(j•−1)•= (j•)•and again the preceding equality is a reformulation of (2.1) for eε. The fact thateεsatisfies (2.2) follows by using (2.3) forε if(j•)•< i and using (2.4) forεotherwise. It is clear that (2.3) and (2.4) hold foreεsince they are the same as the corresponding ones forεand the proof of the lemma is complete.

2.4. The setsΓ⁰_i,j. — Fori6j define a map

p_ij : Γ_i,j−→Z^(j−i+2), p_i,j(ε_i, . . . , ε_j+1) = (ε⁰_i, . . . , ε⁰_j+1), as follows:

– ε⁰_j+1= (1−d_j)σ_{max{i,j•+1},j}(ε) +d_j(1−σ_{max{i,j•+1},j}(ε)), – ifi•=m•or σ_{max{i,(m•)•+1},m•}= 1then,

ε⁰_m=

((dm−1)εm+1−dm, σ_{max{i,m•+1},m}(ε) = 0, dm−(εm+εm+1), σ_{max{i,m•+1},m}(ε) = 1, – ifm•>iandσ_{max{i,(m•)•+1,m•}}(ε) = 0thenε⁰_m=dm(1−εm+1).

It is easily seen thatε⁰_m∈ {−1,0,1}fori6m6j. Fori6j letΓ⁰_i,j be the image ofpij and set Γ⁰_i,j={0}ifi > j.

Lemma. — Let16i6j6n.

(i) If eε= (εi, . . . , εj)∈Γi,j−1 then

p_ij−1(eε) = (ε⁰_i, . . . , ε⁰_j) =⇒ pij(ι_j−1(eε)) = (ε⁰_i, . . . , ε⁰_j−1,−1 +ε⁰_j,1−dj).

(ii) If eε= (εi, . . . , εj_•)∈Γi,j_•−1 then

p_ij•−1(eε) = (ε⁰_i, . . . , ε⁰_j•) =⇒ p_ij(ι_j•−1(eε)) = (ε⁰_i, . . . , ε⁰_j•,0· · · ,0,−1, d_j).

Proof. — Letε= (ε_i, . . . , ε_j, d_j) =ι_i,j−1(eε)and letp_ij(ε) = (ε⁰⁰_i, . . . , ε⁰⁰_j+1). Since σm,r(eε) =σm,r(ε), m6r6j,

it is clear from the definition thatε⁰_m=ε⁰⁰_mifm6j−1.

By Lemma 2.3 we haveε∈Γ¹_i,j and hence σ_{max{i,j•+1},j}(ε) = 1. It is immediate from the definition ofp_ij thatε⁰⁰_j+1= 1−d_j. We now prove thatε⁰⁰_j =−1 +ε⁰_j; using the definition ofε⁰_j this is equivalent to proving

ε⁰⁰_j =−1 + (1−d_j−1)σmax{i,(j−1)•+1},j−1(eε)

+d_j−1(1−σmax{i,(j−1)•+1,},j−1(eε))

=−1 + (1−d_j−1)σ_{max{i,j•+1},j−1}(ε) +d_j−1(1−σ_{max{i,(j•)•+1},j•}(ε)).

(2.6)

If j• > i and σ_{max{i,(j•)•+1},j•}(ε) = 0 then by (2.4) we have εj_•+1 = 1 and so σj•+1,j−1(ε) = 1. This means that the right hand side of (2.6) is zero. Since by definitionε⁰⁰_j =dj(1−dj) = 0the result is proved in this case.