Friezes, Strings and Cluster Variables
aa r X i v : . [ m a t h . A C ] M a r FRIEZES, STRINGS AND CLUSTER VARIABLES
IBRAHIM ASSEM, GRÉGOIRE DUPONT, RALF SCHIFFLER AND DAVID SMITH
Abstract.
To any walk in a quiver, we associate a Laurent polynomial. Whenthe walk is the string of a string module over a 2-Calabi-Yau tilted algebra, weprove that this Laurent polynomial coincides with the corresponding clustercharacter of the string module, up to an explicit normalising monomial factor.
Introduction
In the early 2000, S. Fomin and A. Zelevinsky introduced the class of clusteralgebras with the purpose of building a combinatorial framework for studying totalpositivity in algebraic groups and canonical bases in quantum groups, see [27]. Sincethen, the study of cluster algebras was shown to be connected to several areas ofmathematics, notably combinatorics, Lie theory, Poisson geometry, Teichmüllertheory, mathematical physics and representation theory of algebras.A cluster algebra is a commutative algebra generated by a set of variables, called cluster variables , obtained recursively by a combinatorial process known as muta-tion , starting from an initial set of cluster variables, the initial cluster , and a quiverwithout cycles of length at most two. One of the most remarkable facts about clus-ter variables is that they can be expressed as Laurent polynomials in terms of theinitial cluster variables [27]; this is the so-called
Laurent phenomenon . Also, it isconjectured that the coefficients in this expression are always non-negative; this isthe positivity conjecture . The problem of computing explicitly the cluster variablesis a difficult one and has been extensively studied. The most general results knownat the present time are in [30, 45, 42].In order to compute cluster variables, one may use friezes . Friezes, which go backto works of Coxeter and Coxeter-Conway [21, 19, 20], are an efficient combinatorialtool which mimics the application of mutations on sinks or sources of the givenquiver, hence an obvious combinatorial connection with cluster algebras [17, 14, 46,40, 5]. It is now known that connections between friezes and cluster algebras aredeeper than just combinatorics, see for instance [24, 4, 38, 7, 29]. Our starting pointfor the present paper was the result in [5] giving an explicit formula as a productof × matrices for all cluster variables in coefficient-free cluster algebras of type A and all but finitely many cluster variables in coefficient-free cluster algebras oftype e A , thus explaining at the same time the Laurent phenomenon and positivity.Our objective here is to show that the same technique can be used for computingthe cluster variables associated with the string modules over a 2-Calabi-Yau tiltedalgebra (in the sense of [48]).Besides friezes, our second main tool is the notion of a cluster character . In [14],Caldero and Chapoton noticed that cluster variables in simply-laced coefficient-free Date : October 12, 2018.2010
Mathematics Subject Classification. cluster algebras of finite type can be expressed as generating series of Euler-Poincarécharacteristics of Grassmannians of submodules. Generalising this work, Caldero-Keller [16], Palu [44] and Fu-Keller [30] introduced the notion of a cluster characterassociating to each module M over a 2-Calabi-Yau tilted algebra B T a certainLaurent polynomial X TM allowing one to compute a corresponding cluster variable.In general, cluster characters are hard to compute because one first needs to findthe Euler characteristics of Grassmannians of submodules, and then the dimensionsof certain Hom-spaces in the corresponding 2-Calabi-Yau category.One class of algebras, however, whose representation theory is reasonably well-understood is the class of string algebras, introduced by Butler and Ringel in [13](see also [53]). In particular, indecomposable modules over string algebras arepartitioned into two sets: string and band modules, and only string modules canbe associated with cluster variables. The Euler characteristics of Grassmanniansof submodules of string modules were computed by Cerulli and Haupt [18, 33].Nevertheless, their methods do not allow one to compute explicitly the associatedcluster character.The main result of this paper gives an explicit formula for the cluster characterassociated with a string module over a 2-Calabi-Yau tilted algebra. This can bestated as follows. To any walk c in a locally finite quiver Q , we associate a Laurentpolynomial L c in the ring of Laurent polynomials in the indeterminates x i indexedby the set Q of points of Q , which can be expressed as a product of × matricesas in [5] (see Section 1.3 below). Now, we let T be a tilting object in a Hom-finitetriangulated 2-Calabi-Yau category C and B T = End C ( T ) be the corresponding2-Calabi-Yau tilted algebra whose ordinary quiver is denoted by Q . Moreover, toany string B T -module M , we associate a tuple of integers n M = ( n i ) i ∈ Q whichwe call the normalisation of M (see Section 5.2), and the Laurent polynomial L M which is just the Laurent polynomial L c attached to the string c of M in the quiver Q . Using the notation x n M = Q i ∈ Q x n i i , our main result (Theorem 5.11 below)can be stated as saying that : X TM = 1 x n M L M . This result entails several interesting consequences. We first obtain the positivityof the Laurent polynomial X TM for any string B T -module M (see Corollary 6.4), thusreproving a result of Cerulli and Haupt [18, 33]. Our results also apply to the casesof string modules over cluster-tilted algebras, string modules over gentle algebrasarising from unpunctured surfaces (see [2]) and, more generally, string modules overfinite dimensional Jacobian algebras associated with quivers with potentials in thesense of [22]. We also obtain a new proof of the positivity conjecture for clusteralgebras arising from surfaces without punctures, see [50, 49, 42].The paper is organised as follows. Section 1 introduces the basic definitions andpresents our combinatorial formula. Section 2 introduces the concept of realisablequadruples which is the context in which our formula will actually compute clus-ter characters. This is closely related to the notion of triangulated 2-Calabi-Yaurealisation in the sense of Fu and Keller [30]. Section 3 recalls all the necessarybackground from [44, 30] concerning cluster characters. Sections 4 and 5 containthe proof of our main result. In Section 4, we actually prove a weaker version ofour theorem which will be used in order to prove the general case in Section 5. InSection 6 we present possible applications of the results to the study of positivity in RIEZES, STRINGS AND CLUSTER VARIABLES 3 cluster algebras. In Section 7, we investigate the normalising factor and explicitlycompute it for several cases of string modules over cluster-tilted algebras. The lastsection presents some detailed examples.1.
The matrix formula
Notations.
Throughout the article, k denotes an algebraically closed field.Given a quiver Q , we denote by Q its set of points and by Q its set of arrows.For any arrow α ∈ Q , we denote by s ( α ) its source and by t ( α ) its target . Wesometimes simply write α : s ( α ) → t ( α ) or s ( α ) α −→ t ( α ) . For any point i ∈ Q , weset Q ( i, − ) = { α ∈ Q | s ( α ) = i } , Q ( − , i ) = { α ∈ Q | t ( α ) = i } and if F is a subset of points in Q , we set Q ( i, F ) = { α ∈ Q | s ( α ) = i, t ( α ) ∈ F } , Q ( F, i ) = { α ∈ Q | s ( α ) ∈ F, t ( α ) = i } and finally, for any i, j ∈ Q , we set Q ( i, j ) = Q ( i, − ) ∩ Q ( − , j ) .To any quiver Q , we associate a family x Q = { x i | i ∈ Q } of indeterminates over Z . We let L ( x Q ) = Z [ x ± i | i ∈ Q ] be the ring of Laurent polynomials in thevariables x i , with i ∈ Q and F ( x Q ) = Q ( x i | i ∈ Q ) be the field of rationalfunctions in the variables x i , with i ∈ Q . For any d = ( d i ) i ∈ Q ∈ Z Q , we set x d Q = Q i ∈ Q x d i i .A bound quiver is a pair ( Q, I ) such that Q is a finite quiver (that is, Q and Q are finite sets) and I is an admissible ideal in the path algebra k Q of Q . Given afinite dimensional basic k -algebra B , there exists a bound quiver ( Q, I ) such that B ≃ k Q/I and the quiver Q is called the ordinary quiver of B (see for instance[6]). We always identify the category mod- B of finitely generated right B -moduleswith the category rep( Q, I ) of k -representations of Q satisfying the relations in I .For a B -module M , we denote by M ( i ) the k -vector space at the point i ∈ Q and M ( α ) the k -linear map at the arrow α ∈ Q .1.2. Walks and strings.
Let Q be a quiver. For any arrow β ∈ Q , we denoteby β − a formal inverse for β , with s ( β − ) = t ( β ) , t ( β − ) = s ( β ) and we set ( β − ) − = β .A walk of length n ≥ in Q is a sequence c = c · · · c n where each c i is an arrow ora formal inverse of an arrow and such that t ( c i ) = s ( c i +1 ) for any i ∈ { , . . . , n − } .The source of the walk c is s ( c ) = s ( c ) and the target of the walk c is t ( c ) = t ( c n ) .With any point i ∈ Q , we associate a walk e i of length zero which is given by thestationary path at point i . For any walk c on Q , we denote by c the walk of lengthzero c = e s ( c ) .If ( Q, I ) is a bound quiver, a string in ( Q, I ) is either a walk of length zero ora walk c = c · · · c n of length n ≥ such that c i = c − i +1 for any i ∈ { , . . . , n − } and such that no walk of the form c i c i +1 · · · c t nor its inverse belongs to I for ≤ i and t ≤ n . If Q is a quiver, a string in the quiver Q is a string in the bound quiver ( Q, (0)) . If B ≃ k Q/I is a finite dimensional k -algebra and c is a string in ( Q, I ) ,then we also say that c is a string in B .To any string in a finite dimensional k -algebra B , we can naturally associatean indecomposable finite dimensional right B -module M c , called string module asin [13, §3]. Namely, if B has ordinary quiver Q and c is a string, we define M c as follows. If c = e v has length zero, then M c is the simple representation S v IBRAHIM ASSEM, GRÉGOIRE DUPONT, RALF SCHIFFLER AND DAVID SMITH at the point v ∈ Q . Otherwise, c is a string of length n ≥ and we write c = v c · · · c n v n +1 . For any v ∈ Q , we set I v = { i ∈ [1 , n + 1] | v i = v } and we define M c ( v ) as the | I v | -dimensional k -vector space with basis z i , with i ∈ I v . For any ≤ i ≤ n , if c i = β ∈ Q , we set M c ( β )( z i − ) = z i and if c i = β ∈ Q − , we set M c ( β )( z i ) = z i − . Finally, if α : v −→ v ′ is an arrow in Q and if z j is one of the basis vector of M c ( v ) such that M c ( α )( z j ) is not yet defined,we set M c ( α )( z j ) = 0 .A string module is also called a string representation of the corresponding boundquiver. For any string module M , we denote by s ( M ) the corresponding string.1.3. A formula for walks.
For any locally finite quiver Q , we define a family ofmatrices with coefficients in Z [ x Q ] = Z [ x i | i ∈ Q ] as follows.For any arrow β ∈ Q , we set A ( β ) = (cid:20) x t ( β ) x s ( β ) (cid:21) and A ( β − ) = (cid:20) x t ( β ) x s ( β ) (cid:21) . Let c = c · · · c n be a walk of length n ≥ in Q . For any i ∈ { , . . . , n } we set v i +1 = t ( c i ) (still with the notation c = e s ( c ) ) and V c ( i ) = Y α ∈ Q ( v i , − ) α = c ± i ,c ± i − x t ( α ) Y α ∈ Q ( − ,v i ) α = c ± i ,c ± i − x s ( α ) . We then set L c = 1 x v · · · x v n +1 (cid:2) , (cid:3) V c (1) n Y i =1 A ( c i ) V c ( i + 1) ! (cid:20) (cid:21) ∈ L ( x Q ) . If c = e i is a walk of length 0 at a point i , we similarly set V e i (1) = Y α ∈ Q ( i, − ) x t ( α ) Y α ∈ Q ( − ,i ) x s ( α ) . and L e i = 1 x i (cid:2) , (cid:3) V e i (1) (cid:20) (cid:21) ∈ L ( x Q ) . In other words, if c is any walk, either of length zero, or of the form c = c · · · c n ,we have(1.1) L c = 1 Q ni =0 x t ( c i ) (cid:2) , (cid:3) n Y i =0 A ( c i ) V c ( i + 1) ! (cid:20) (cid:21) ∈ L ( x Q ) with the convention that A ( c ) is the identity matrix. RIEZES, STRINGS AND CLUSTER VARIABLES 5
Example 1.1.
Consider the quiver γ (cid:30) (cid:30) ======== Q : 1 α / / β @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) δ / / ǫ / / and consider the path c = δ − βγ in Q . Then, L c = 1 x x x (cid:18)(cid:2) , (cid:3) (cid:20) x x (cid:21) (cid:20) x x (cid:21) (cid:20) x (cid:21)(cid:20) x x (cid:21) (cid:20) (cid:21) (cid:20) x x (cid:21) (cid:20) x x (cid:21) (cid:20) (cid:21)(cid:19) = x x x + 2 x x x x + x x x x x + x x x + x x x + x x x x x x A formula for string modules.Definition 1.2.
Let B be a finite dimensional k -algebra with bound quiver ( Q, I ) and let M be a string B -module with corresponding string s ( M ) . We set L M = L s ( M ) ∈ L ( x Q ) . If Q is a quiver, a subquiver R of Q is a quiver R such that R ⊂ Q and suchthat for any i, j ∈ R , the set of arrows from i to j in R is a subset of the set ofarrows from i to j in Q . If R is a subquiver of Q , we naturally identify L ( x R ) with a subring of L ( x Q ) .A subquiver R of Q is called a full subquiver if for all i, j ∈ R the set of arrowsfrom i to j in R equals the set of arrows from i to j in Q . If B is a finitedimensional k -algebra with bound quiver ( Q, I ) and M is a B -module, the support of M is the full subquiver supp( M ) of Q consisting of the points i ∈ Q such that M ( i ) = 0 . The closure of the support of M is the full subquiver supp( M ) of Q formed by the points i ∈ Q which are in the support of M or such that thereexists an arrow α such that s ( α ) ∈ supp( M ) and t ( α ) = i or t ( α ) ∈ supp( M ) and s ( α ) = i .With these identifications, if B is a finite dimensional k -algebra and M is a string B -module, then L M ∈ L ( x supp( M ) ) . Realisable quadruples
In the previous section we associated with any string module M over a finitedimensional algebra B a certain Laurent polynomial L M . In this section we providea context in which the algebra B arises in connection with some cluster algebras sothat we can compare the Laurent polynomials L M with cluster variables or, moregenerally, with cluster characters. The context in which we work is the context oftriangulated 2-Calabi-Yau realisations introduced in [30].2.1. Definitions. An ice quiver is a pair ( Q , F ) such that Q is a finite connectedquiver without loops and 2-cycles and F is a (possibly empty) subset of points of Q , called frozen points , such that there are no arrows between points in F . The unfrozen part of ( Q , F ) is the full subquiver of Q obtained by deleting the pointsin F . IBRAHIM ASSEM, GRÉGOIRE DUPONT, RALF SCHIFFLER AND DAVID SMITH
For any quiver Q , we denote by B ( Q ) = ( b ij ) ∈ M Q ( Z ) the skew-symmetricmatrix defined for any i, j ∈ Q by b ij = | Q ( i, j ) | − | Q ( j, i ) | . If ( Q , F ) is an ice quiver with unfrozen part Q , then we can fix an ordering ofthe points in Q such that B ( Q ) = (cid:20) B ( Q ) − C t C F × F (cid:21) where C is a matrix in M F × Q ( Z ) .For any ice quiver ( Q , F ) with unfrozen part Q , we denote by A ( Q , F ) thecluster algebra of geometric type with initial seed ( e B ( Q ) , x , y ) where e B ( Q ) = (cid:20) B ( Q ) C (cid:21) , x = ( x i , i ∈ Q ) and y = ( x i , i ∈ F ) (see [28]).A k -linear category C is called Hom-finite if Hom C ( M, N ) is a finite dimensional k -vector space for any two objects M, N in C . A k -linear triangulated category C is called -Calabi-Yau if there is a bifunctorial isomorphism Hom C ( X, Y ) ≃ D Hom C ( Y, X [2]) where D = Hom k ( − , k ) is the standard duality and [1] denotes the suspensionfunctor.For any objects M, N in C , we denote by Ext C ( M, N ) the space Hom C ( M, N [1]) .An object M in the category C is called rigid if Ext C ( M, M ) = 0 . An object T in C is called a tilting object in C if for any object X in C , the equality Ext C ( T, X ) = 0 isequivalent to the fact that X belongs to the additive subcategory add T of C . Notethat these objects are sometimes called cluster-tilting in the literature. It is knownthat the combinatorics of cluster algebras are closely related to the combinatoricsof tilting objects in triangulated 2-Calabi-Yau categories [8]. Definition 2.1. A realisable quadruple is a quadruple ( Q , F, C , T ) such that :(a) ( Q , F ) is an ice quiver ;(b) C is a Hom-finite triangulated 2-Calabi-Yau category whose tilting objects forma cluster structure in the sense of [8] ;(c) T is a tilting object in C ;(d) the ordinary quiver of B T = End C ( T ) is Q .Following Reiten [48], every algebra of the form End C ( T ) as above is called a . In order to simplify terminology, a category C satisfying(b) is simply called a triangulated 2-Calabi-Yau category . Definition 2.2.
Given a realisable quadruple ( Q , F, C , T ) , a B T -module M iscalled :a) unfrozen if M ( i ) = 0 implies i F ;b) unfrozen indecomposable if it is unfrozen and indecomposable as a B T -module ;c) unfrozen sincere if M ( i ) = 0 if and only if i F . RIEZES, STRINGS AND CLUSTER VARIABLES 7
Examples.
The notion of realisable quadruple covers a lot of situations in thecontext of cluster algebras. We now list some examples of such situations.
Example 2.3.
Let Q be an acyclic quiver and let C be the cluster category of Q ,first defined in [10] (see [15] for an alternative description in Dynkin type A ). Itis canonically equipped with a structure of triangulated 2-Calabi-Yau category [34]and for any tilting object T in C , an algebra of the form End C ( T ) is called a cluster-tilted algebra , as defined in [11] (see also [15]). Thus, if Q T is the ordinary quiverof the cluster-tilted algebra End C ( T ) , the quadruple ( Q T , ∅ , C , T ) is realisable andevery B T -module is unfrozen. Example 2.4.
Let ( Q, W ) be a Jacobi-finite quiver with potential, that is, a quiverwith potential in the sense of [22] such that the Jacobian algebra J ( Q,W ) is finitedimensional. Let C ( Q,W ) be the generalised cluster category constructed by Amiot[1, §3]. Then C ( Q,W ) is a triangulated 2-Calabi-Yau category and there exists atilting object T in C ( Q,W ) such that ( Q, ∅ , C ( Q,W ) , T ) is a realisable quadruple [1,Theorem 3.6].This example is of particular interest for the study of cluster algebras arisingfrom surfaces in the sense of [26] (see also Section 6.2 for more details). Indeed,Labardini associated a non-degenerate Jacobi-finite quiver with potential ( Q, W ) to any marked surface ( S, M ) with non-empty boundary [39]. Thus, the generalisedcluster category C ( Q,W ) provides a categorification for the cluster algebra A ( S, M ) associated with the surface. Moreover, if the surface is unpunctured , that is, if thereare no marked points in the interior of the surface, it is known that the Jacobianalgebra J ( Q,W ) is a string algebra (it is in fact a gentle algebra, see [2]). The J ( Q,W ) -modules without self-extension are thus string modules and it follows from[30] that the cluster variables in A ( S, M ) can be studied via cluster charactersassociated with string J ( Q,W ) -modules (see Section 3 for details). Example 2.5.
Let A be a finite dimensional k -algebra of global dimension 2 andlet e Q be the ordinary quiver of the relation extension of A , that is, the trivialextension of A by the A -bimodule Ext A ( DA, A ) (see for instance [3]). Then Amiotalso associated with A a generalised cluster category C A and proved that thereexists a tilting object T in C A such that ( e Q, ∅ , C A , T ) is a realisable quadruple [1,Theorem 4.10]. Example 2.6.
We now give a fundamental example with a non-empty set of frozenpoints. For any quiver Q , we denote by ( Q pp , Q ′ ) its principal extension . It isdefined as follows. We fix a copy Q ′ = { i ′ | i ∈ Q } of Q and we set Q pp0 = Q ⊔ Q ′ .The arrows in Q pp1 between two points in Q are the same as in Q and for any i ∈ Q we add an extra arrow i ′ −→ i in Q pp1 . Thus, the matrix of Q pp is given by B ( Q pp ) = (cid:20) B ( Q ) − I Q ′ I Q ′ (cid:21) where I Q ′ denotes the identity matrix in M Q ′ ( Z ) .Now, if Q is the ordinary quiver of a cluster-tilted algebra B T , it is knownthat there exists a 2-Calabi-Yau category C pp , endowed with a tilting object T pp ,obtained via a process of principal gluing , such that ( Q pp , Q ′ , C pp , T pp ) is a realis-able quadruple for which every B T -module can naturally be viewed as an unfrozenmodule (see [30, §6.3] or Corollary 2.12). IBRAHIM ASSEM, GRÉGOIRE DUPONT, RALF SCHIFFLER AND DAVID SMITH
Blown-up ice quivers and their realisations.
In the proof of the maintheorem of this article, we are interested in a particular family of ice quivers, called blown-up . We now give some details concerning these quivers.
Definition 2.7.
We say that an ice quiver ( Q , F ) is blown-up if | Q ( i, − ) ∪ Q ( − , i ) | ≤ for every point i ∈ F , that is, if there exists at most one arrowstarting or ending at any frozen point. Example 2.8. a) Any ice quiver with an empty set of frozen points is blown-up.b) If Q is any quiver, then its principal extension ( Q pp , Q ′ ) defined in Example 2.6is blown-up. Remark 2.9.
Any blown-up ice quiver ( Q , F ) whose unfrozen part is acyclic canbe embedded in a realisable quadruple ( Q , F, C , T ) . Indeed, since ( Q , F ) is blown-up with an acyclic unfrozen part, Q is also acyclic. Let thus C be the clustercategory of the quiver Q . Then the path algebra k Q is identified with a tiltingobject in C and the corresponding cluster-tilted algebra is isomorphic to k Q sothat its ordinary quiver is Q . Thus ( Q , F, C , k Q ) is a realisable quadruple.We can construct a wide class of examples of realisable quadruples with thefollowing proposition, due to Amiot : Proposition 2.10 ([1]) . If C and C are two generalised cluster categories associ-ated with Jacobi-finite quivers with potentials ( Q , W ) and ( Q , W ) , then for anymatrix C with non-negative integer entries, there exists a Jacobi-finite quiver withpotential ( Q ′ , W ′ ) whose generalised cluster category C ( Q ′ ,W ′ ) has a tilting object T ′ such that the matrix associated with the ordinary quiver of the 2-Calabi-Yau tiltedalgebra End C ( Q ′ ,W ′ ) ( T ′ ) is B ′ = (cid:20) B ( Q ) − C t C B ( Q ) (cid:21) . Corollary 2.11.
Let ( Q , F ) be a blown-up ice quiver with unfrozen part Q . Assumethat there exists a potential W on Q such that the Jacobian algebra J ( Q,W ) is finitedimensional. Then there exists a triangulated 2-Calabi-Yau category C and a tiltingobject T in C such that ( Q , F, C , T ) is a realisable quadruple.Proof. We let C (0) be the generalised cluster category associated with the quiverwith potential ( Q, W ) . According to Example 2.4, there exists a tilting object T (0) in C such that the quadruple ( Q, ∅ , C (0) , T (0) ) is realisable.In order to simplify notations, we write F = { , . . . , m } . For any i ∈ F , wedenote by C A i a copy of the cluster category of type A , which is a particularcase of generalised cluster category in the sense of Amiot. We now construct, byinduction on i ∈ { , . . . , m } , a category C ( i ) with a tilting object T ( i ) such that theordinary quiver of the 2-Calabi-Yau tilted algebra End C ( i ) ( T ( i ) ) is the full subquiver ( Q , F ) formed by points in Q and frozen points in { , . . . , i } .Since ( Q , F ) is blown-up, for every i ∈ { , . . . , m } , there exists a unique arrow α i ∈ Q ( i, − ) ⊔ Q ( − , i ) . If α i ∈ Q ( i, − ) , we construct the category C ( i ) gluing C ( i − and C A i as provided by Proposition 2.10 with C = C ( i − and C = C A i and we denote by T ( i ) the canonical tilting object in this gluing. If α i ∈ Q ( − , i ) ,we construct a gluing C ( i ) of C A i and C ( i − as provided by Proposition 2.10 with C = C A i and C = C ( i − and we denote by T ( i ) the canonical tilting object inthis gluing. We finally set e C = C ( m ) and e T = T ( m ) . (cid:3) RIEZES, STRINGS AND CLUSTER VARIABLES 9
Corollary 2.12.
Let ( Q , F ) be a blown-up ice quiver whose unfrozen part is theordinary quiver of a cluster-tilted algebra. Then ( Q , F ) can be embedded in a real-isable quadruple ( Q , F, C , T ) .Proof. It is proved in [35] and [9] that any cluster-tilted algebra is the Jacobian alge-bra of a Jacobi-finite quiver with potential. The result thus follows from Corollary2.11. (cid:3) Cluster characters with coefficients
In this section, we collect some background concerning Fu-Keller’s cluster char-acters [30]. These characters allow one to give an explicit realisation of the ele-ments in a cluster algebra in terms of the geometry and the homology underlyinga 2-Calabi-Yau category.We fix a triangulated 2-Calabi-Yau category C with suspension functor [1] andwe fix a tilting object T in C .We let Q be the ordinary quiver of the 2-Calabi-Yau tilted algebra B T =End C ( T ) . Indecomposable direct summands of T label the points in Q . Let F be a set of frozen points in Q . We set T F = L i ∈ F T i . Consider the full subcat-egory U of C formed by the objects X such that Hom C ( T F , X ) = 0 . Let (add T [1]) be the ideal consisting of those morphisms factoring through objects of add T [1] . Theorem 3.1 ([11, 36]) . The functor
Hom C ( T, − ) induces an equivalence Hom C ( T, − ) : C / (add T [1]) ∼ −→ mod- B T . The equivalence
Hom C ( T, − ) induces an equivalence between U / (add T [1]) andthe subcategory of mod- B T consisting of B T -modules supported on the unfrozenpart of ( Q , F ) , which is denoted by Q . Slightly abusing notations, an object M in U / (add T [1]) is identified with its image Hom C ( T, M ) in mod- B T . Conversely,any B T -module supported on Q is viewed as an object in U / (add T [1]) .For any i ∈ Q , we denote by S i the simple B T -module corresponding to thepoint i . We denote by h− , −i the truncated Euler form on mod- B T defined by h M, N i = dim Hom B T ( M, N ) − dim Ext B T ( M, N ) for any B T -modules M and N . We denote by h− , −i a the anti-symmetrised Eulerform on mod- B T defined by h M, N i a = h M, N i − h
N, M i for any B T -modules M and N . Lemma 3.2 ([44]) . For any i ∈ Q , the form M
7→ h S i , M i on mod- B T onlydepends on the class [ M ] of M in the Grothendieck group K ( mod- B T ) of mod- B T . For any B T -module M and any e ∈ K ( mod- B T ) , we let Gr e ( M ) denote theset of submodules N of M whose class [ N ] in K ( mod- B T ) equals e . This setis called the Grassmannian of submodules of M of dimension e . It is a projectivevariety and we denote by χ (Gr e ( M )) its Euler-Poincaré characteristic (with respectto the singular cohomology if k is the field of complex numbers, and to the étalecohomology with compact support if k is arbitrary). Definition 3.3 ([44]) . The cluster character associated with ( C , T ) is the uniquemap X T ? : Ob( C ) −→ L ( x Q ) such thata) X TT i [1] = x i for any i ∈ Q ;b) If M is indecomposable and not isomorphic to any T i [1] , then X TM = X e ∈ N Q χ (Gr e (Hom C ( T, M ))) Y i ∈ Q x h S i , e i a −h S i , Hom C ( T,M ) i i ; c) For any two objects M, N in C , X TM ⊕ N = X TM X TN . We recall that an indecomposable object X in C is called reachable from T if itis a direct summand of a tilting object which can be obtained from T by a finitenumber of mutations, see [8]. In particular, any reachable object is rigid. Theorem 3.4 ([30]) . The map X T ? induces a surjection from the set of indecom-posable objects in C which are reachable from T to the set of cluster variables inthe cluster algebra A ( Q , F ) . Note that identifying C / (add T [1]) with mod- B T using the functor Hom C ( T, − ) ,we associate to any indecomposable B T -module M the cluster character of an in-decomposable lifting M of M in C / (add T [1]) . Thus, we set X TM = X TM = X e ∈ N Q χ (Gr e ( M )) Y i ∈ Q x h S i , e i a −h S i ,M i i and this way, we may view X T ? as a map on the set of objects in mod- B T .4. A formula for Dynkin type A with coefficients We now start the proof of our main result, which will be stated in Theorem5.11. This section is devoted to the first step of the proof in which we establishthe theorem for specific modules in the particular case of blown-up quivers withunfrozen part of Dynkin type A . The result we prove in this section is the following : Theorem 4.1.
Let ( Q , F ) be a blown-up ice quiver with unfrozen part Q of Dynkintype A and let ( Q , F, C , T ) be a realisable quadruple. Let B T = End C ( T ) . Thenfor any indecomposable B T -module M , the following hold :a) If M is unfrozen, then it is a string module ;b) If M is a submodule of the unique unfrozen sincere module, then the correspond-ing cluster character is given by X TM = L M . Every B T -module M supported on Q has the structure of a module over thepath algebra H = k Q of the quiver Q . Since Q is of Dynkin type A every suchmodule M is a string module if it is indecomposable. This proves the first point ofthe theorem.To prove the second point, we need to collect the necessary background concern-ing the matrix product formula from [5], which we do in Section 4.1. Section 4.2 isdevoted to the proof when ( Q , F ) is the principal extension of a Dynkin quiver oftype A . In Section 4.3, we use the Fomin-Zelevinsky separation formula in order todeduce the general case. RIEZES, STRINGS AND CLUSTER VARIABLES 11
Background on the matrix product formula.
Let Q be a Dynkin quiver oftype A n with n ≥ . Attach to each point i of Q a cluster variable x i . Because Q isof Dynkin type A n , it is well-known that the corresponding coefficient-free clusteralgebra A ( Q, ∅ ) is generated by n ( n +1)2 + n cluster variables. If C Q denotes thecluster category of Q , the set of indecomposable objects in C Q can be identified withthe disjoint union of the set of indecomposable k Q -modules and { P i [1] | i ∈ Q } where P i denotes the indecomposable k Q -module associated with the point i ∈ Q . Identifying the path algebra k Q with a tilting object in C Q , the quadruple ( Q, ∅ , C Q , k Q ) is realisable and the associated cluster character M X M = X k QM induces a bijection from the set of indecomposable objects in C Q to the set of clustervariables in A ( Q, ∅ ) . Moreover, this bijection sends each object of the form P i [1] onto x i , and each indecomposable k Q -module M onto the unique cluster variablehaving, in its reduced form, x dim M as denominator, where dim M = (dim k M ( i )) ∈ N Q , see [16].Under this identification, one can position the cluster variables of A ( Q, ∅ ) intoa grid underlying the Auslander-Reiten quiver of C Q . This positioning of variableson the grid corresponds to the frieze on the repetition quiver Z Q op associated withthe opposite quiver of Q , as constructed in [5]. We illustrate this on an example.Let Q be a Dynkin quiver of type A n , say α / / β / / γ o o δ / / ǫ o o ζ o o η o o θ / / ι / /
10 11 . κ o o (here n = 11 ). The corresponding grid, in which we illustrated the cluster variablesassociated with the indecomposable projective k Q -module P i , and those associatedwith the indecomposable injective k Q -module I i , is as follows (the variable X M ( u,v ) will be explained thereafter). x x x X P X P x x x x X P ∗ . . . x x X P X P X P X P ∗ . . . x X P X P X M ( u,v ) ∗ . . . x X P ∗ . . . X P ∗ ∗ ∗ ∗ ∗ ∗ ∗ X I X I X I . . . ∗ X I X I x x x . . . ∗ X I x x . . . ∗ X I x . . . X I X I X I x X I x x x x In [5, §8.2], the authors defined, for each cell ( u, v ) in the grid, a Laurent poly-nomial t Q op ( u, v ) , whose definition depends on the region of the grid in which thecell is located.For the purpose of our paper, it suffices to consider only the north-west compo-nent of the grid together with its south-east frontier of asterisks. It is important to observe that in mod- k Q , this part of the grid contains exactly all the indecompos-able submodules of the unique indecomposable sincere k Q -module, which is exactlypositioned at the intersection of the vertical and the horizontal lines of asterisks(see for instance [32]). This observation will be crucial in Lemma 4.2 below and inSection 5.We give more details on this region. For any cell ( u, v ) located in the north-west component of the grid (and not on its south-east frontier of asterisks), itshorizontal and vertical projections onto the initial variables x i determine a word x k x k +1 . . . x k + l +1 . Following [5], for each j = 1 , , . . . , l − , let M ( x j , x j +1 ) = (cid:20) x j x j +1 (cid:21) if x j is to the left of x j +1 , (cid:20) x j +1 x j (cid:21) if x j is below x j +1 . Then t Q op ( u, v ) = 1 x k +1 · · · x k + l [1 , x k ] l − Y j =1 M ( x k + j , x k + j +1 ) (cid:20) x k + l +1 (cid:21) . To determine the corresponding indecomposable k Q -module M ( u,v ) for which t Q op ( u, v ) = X M ( u,v ) , the horizontal and vertical projections onto the cluster vari-ables X P i gives the word X P k +1 X P k +2 . . . X P k + l , meaning that M ( u,v ) is the stringmodule corresponding to the unique string from the point k + 1 to the point k + l in Q . In our example, the cell ( u, v ) gives rise to the word x x . . . x , leading to t Q op ( u, v ) = 1 x x x x [1 , x ] (cid:20) x x (cid:21) (cid:20) x x (cid:21) (cid:20) x x (cid:21) (cid:20) x (cid:21) and M ( u,v ) corresponds to the string module whose corresponding string is γ − δǫ − in Q .Now, a close inspection of the formulae (in the coefficient-free situation) gives t Q op ( u, v ) = L M ( u,v ) whenever ( u, v ) lies in the north-west region.As mentioned above, we also need to consider the south-east frontier of asterisksof the north-west region. Observe that on this frontier, L M ( u,v ) is defined, while t Q op ( u, v ) is not. To fix this, one can augment Q op with two new sinks, labeled and n + 1 , in order to obtain a Dynkin quiver Q op of type A n +2 in such a waythe cells which were on the frontier of asterisks now lie in the north-west region inthe grid corresponding to Q op ; thus t Q op ( u, v ) can be defined. In our example, itsuffices to let Q op be given by o o α o o β o o γ / / δ o o ǫ / / ζ / / η / / θ o o ι o o κ / / / / . It is then easily checked that L M ( u,v ) = t Q op ( u, v ) | x =1 x n +1 =1 , that is, L M ( u,v ) is obtainedfrom t Q op ( u, v ) by specialising the initial cluster variables x and x n +1 to . Observethat this relation also holds true for any cell located in the north-west region. So,in general, one can write L M ( u,v ) = t Q op ( u, v ) | x =1 x n +1 =1 whenever ( u, v ) lies in the north-west region or on its south-east frontier of asterisks. RIEZES, STRINGS AND CLUSTER VARIABLES 13
In the situation where we deal with arbitrary coefficients, let ( Q , F ) be an icequiver with unfrozen part Q of Dynkin type A . Then, generalising the coefficient-free situation, we let t Q op ( u, v ) = L M ( u,v ) . For each i ∈ Q , let y i = Y α ∈ Q ( F,i ) x s ( α ) and z i = Y α ∈ Q ( i,F ) x t ( α ) . and for any d = ( d i ) i ∈ Q ∈ N Q , we set y d = Q i ∈ Q y d i i and z d = Q i ∈ Q z d i i .Then, a tedious and combinatorial adaptation of Lemmata 5, 6 and Theorem 4in [5], in which one needs to embed Q op in Q op as above, allows us to obtain thefollowing recurrence relations, whose verification is left to the reader. Lemma 4.2. a) For any i ∈ Q , we have (4.2) x i L P i − y i Y α ∈ Q ( i, − ) L P t ( α ) Y α ∈ Q ( − ,i ) x s ( α ) = z dim P i . b) For any non-projective indecomposable submodule M of the unique indecompos-able unfrozen sincere k Q -module, we have (4.3) L τM L M − L E = y dim τM z dim M where E is the middle term of the almost split exact sequence −→ τ M i −→ E p −→ M −→ in mod- k Q . Proof of Theorem 4.1 - first step : Principal coefficients.
We start withtwo lemmata concerning cluster characters associated with principal extensions ofacyclic quivers. These are analogues to [14, Lemma 3.9 and Proposition 3.10](see also [25, Lemma 2.2]) which provide representation-theoretical interpretationsof certain exchange relations in a cluster algebra with principal coefficients at anacyclic seed.Let Q be an acyclic quiver and let H = k Q be the path algebra of Q , whichis finite-dimensional and hereditary. The map sending an H -module module to itsdimension vector allows us to identify the Grothendieck group K ( mod- H ) with Z Q . If ( Q , F ) is the principal extension ( Q pp , Q ′ ) of Q defined in Example 2.6, wedenote by X pp? the corresponding cluster character and by L pp the matrix formulaof equation (1.1). For every i ′ ∈ Q ′ , we set y i = x i ′ . It is known that X pp M = X e ∈ N Q χ (Gr e ( M )) Y i ∈ Q x −h e , [ S i ] i−h [ S i ] , [ M ] − e i i y m i − e i i where the Euler forms and the Grassmannian are considered in mod- H and where [ M ] = ( m i ) i ∈ Q (see for instance [25, Remark 2.4]). Lemma 4.3.
Let Q be an acyclic quiver. Then for any i ∈ Q , we have (4.4) x i X pp P i − y i Y α ∈ Q ( i, − ) X pp P t ( α ) Y α ∈ Q ( − ,i ) x s ( α ) = 1 . where P j is the indecomposable projective k Q -module associated with the point j ∈ Q . Proof.
The proof is a straightforward adaptation of the proof of [14, Lemma 3.9],we give it for completeness. We recall that for any i ∈ Q , we have Rad P i = M α ∈ Q ( i, − ) P t ( α ) and thus X ppRad P i = Y α ∈ Q ( i, − ) X pp P t ( α ) . We also recall that P i / Rad P i ≃ S i and that a submodule M of P i either equals P i or is a submodule of Rad P i . We set dim P i = m = ( m j ) j ∈ Q and let δ be suchthat δ ij = 1 if i = j and δ ij = 0 otherwise. Thus X ppRad P i = X e χ (Gr e (Rad P i )) Y l ∈ Q x −h e , [ S l ] i−h [ S l ] , m − [ S i ] − e i l y m l − δ il − e l l = X e χ (Gr e (Rad P i )) Y l ∈ Q (cid:16) x −h e , [ S l ] i−h [ S l ] , m − e i l y m l − e l l (cid:17) x h [ S l ] , [ S i ] i l y − δ il l = y − i Y α ∈ Q ( − ,i ) x − s ( α ) x i X e χ (Gr e (Rad P i )) Y l ∈ Q x −h e , [ S l ] i−h [ S l ] , m − e i l y m l − e l l but X pp P i = x − i + X e χ (Gr e (Rad P i )) Y l ∈ Q x −h e , [ S l ] i−h [ S l ] , m − e i l y m l − e l l . Thus, x i X pp P i = y i Y α ∈ Q ( − ,i ) x s ( α ) X ppRad P i + 1 from which we deduce (4.4). (cid:3) Lemma 4.4.
Let Q be an acyclic quiver. Then for any non-projective k Q -module M , we have (4.5) X pp τM X pp M − X pp E = y dim τM where E is the central term of the almost split exact sequence −→ τ M i −→ E p −→ M −→ .Proof. This result is an analogue of [14, Proposition 3.10]. We sketch the proof forthe convenience of the reader.We write N = τ M , dim N = n = ( n l ) l ∈ Q and dim M = m = ( m l ) l ∈ Q . Itfollows from the definition of the character that we have X pp N X pp M = X pp N ⊕ M = X e ∈ N Q χ (Gr e ( N ⊕ M )) Y l ∈ Q x −h e , [ S l ] i−h [ S l ] , n + m − e i l y m l + n l − e l .l but there is a surjective map with affine fibres Gr e ( N ⊕ M ) −→ G f + g = e Gr f ( N ) × Gr g ( M ) so that χ (Gr e ( N ⊕ M )) = P f + g = e χ (Gr f ( N )) χ (Gr g ( M )) and thus X pp N ⊕ M = X f , g χ (Gr f ( N )) χ (Gr g ( M )) Y l ∈ Q x −h f + g , [ S l ] i−h [ S l ] , n + m − f − g i l y m l + n l − f l − g l .l RIEZES, STRINGS AND CLUSTER VARIABLES 15
Now, it follows from [14, Lemma 3.11] that every fibre of the map ζ : Gr e ( N ⊕ M ) −→ G f + g = e Gr f ( N ) × Gr g ( M ) U ( i − ( U ) , p ( U )) is an affine space except over the point (0 , M ) where it is empty. It thus followsthat X pp N ⊕ M = X pp E + Y l ∈ Q y n l l which establishes (4.5). (cid:3) We now prove the second point of Theorem 4.1 for principal coefficients :
Proposition 4.5.
Let Q be a Dynkin quiver of type A . Then for any indecompos-able submodule M of the unique unfrozen sincere k Q -module, we have X pp M = L pp M . Proof.
The proof directly follows from Lemmata 4.2, 4.3 and 4.4, keeping in mindthat for principal coefficients we have z i = 1 for each i in Lemma 4.2. (cid:3) Proof of Theorem 4.1 - second step.
We now finish the proof of Theorem4.1. For this, the strategy is to apply the Fomin-Zelevinsky separation formulato the equality established in Proposition 4.5. We let ( Q , F ) be a blown-up icequiver with unfrozen part Q of Dynkin type A and we fix a realisable quadruple ( Q , F, C , T ) and an unfrozen indecomposable B T -module M which has a naturalstructure of H -module where H is the path algebra of Q . We denote by ( Q pp , Q ′ ) the principal extension of Q . Every H -module can naturally be viewed as a B T -module or as a k Q pp -module. According to Proposition 4.5, we know that L pp M = X pp M . We now want to prove that L M = X TM where L ? denotes the matrix formulaassociated with ( Q , F ) in equation (1.1).In order to simplify the notations, we identify Q with { , . . . , n } and for every i ∈ Q , we denote i ′ by n + i . Let P be the tropical semifield generated by the x i ,with i ∈ F , and endowed with the auxiliary addition Y i x a i i ⊕ Y i x b i i = Y i x min { a i ,b i } i . For any i ∈ Q , we set w i = Y α ∈ Q ( F,i ) x s ( α ) Y α ∈ Q ( i,F ) x − t ( α ) . With the notations of Section 4.1, we can thus write w i = y i z − i . Following [28], forevery subtraction-free rational expression f in the variables x , . . . , x n , we definethe separation of f as σ ( f ) = f ( x , . . . , x n , w , . . . , w n ) f | P (1 , . . . , , w , . . . , w n ) where the f | P means that we have replaced the ordinary addition in F ( x F ) by theauxiliary addition ⊕ of the semifield P . This can be done since f is subtraction-free.Now, we note that, for every H -module, L pp M is a subtraction-free rational ex-pression by definition. Also, for every indecomposable H -module M , X pp M is a cluster variable so that it is defined with a finite number of mutations, which areall subtraction-free. Thus, X pp M is also a subtraction-free rational expression (see[28, §3]) and we can apply σ to both X pp M and L pp M . Since L pp M = X pp M by Proposition4.5, we have σ ( L pp M ) = σ ( X pp M ) . Thus, we only need to prove that σ ( X pp M ) = X TM and σ ( L pp M ) = L M .The equality σ ( X pp M ) = X TM follows directly from [28, Theorem 3.7] since X pp? and X T ? induce bijections from the set of indecomposable H -modules to the set ofcluster variables in the cluster algebras A ( Q pp , Q ′ ) and A ( Q , F ) respectively andthese bijections respect denominator vectors.The equality σ ( L pp M ) = L M is obtained from the following observation. We write c = s ( M ) . For each j ∈ F , x j appears in exactly one w i , since there is exactlyone i ∈ Q which is adjacent to j in Q . On the other hand, in the product ofmatrices L pp M , x n + i appears in at most one matrix, namely in the matrix V pp c ( i ) (where V pp c ( i ) denotes the matrices arising in the matrix product L pp c ). Thus, usingthe definition of the addition in P and the fact that L pp M (1 , . . . , , x n +1 , . . . , x n ) hasconstant term , it follows that L pp M | P (1 , . . . , , w , . . . , w n ) = Y i ∈ (supp( M )) Y α ∈ Q pp1 ( i,F ) x − t ( α ) . Now, since Y α ∈ Q ( i,F ) x t ( α ) ( V pp c ( i )( x , . . . , x n , w , . . . , w n )) = V c ( i ) , we get σ ( L pp M ) = L M . This ends the proof of Theorem 4.1. (cid:3) The general case
We now deduce the main theorem from Theorem 4.1. For this, we use some“blow-up” techniques which are described in Section 5.1. These techniques introducesome “error” in the computation of the characters but this can be controlled witha normalising factor that we introduce in Section 5.2. The main result (Theorem5.11) is stated in Section 5.4.5.1.
Blowing up quivers along string modules.
In this section, we fix a quiver Q , we denote by M a string representation of Q and we write c = s ( M ) . If c is ofpositive length n ≥ , we write c = c · · · c n .The blow-up g Q M of Q along M is the quiver constructed as follows. Let { v , . . . , v n +1 } be a set. For any i ∈ { , . . . , n } , we set β i from v i to v i +1 which isan arrow (or a formal inverse of an arrow, respectively) in ( g Q M ) if c i is an arrow(or a formal inverse of an arrow, respectively) in Q . For any i ∈ { , . . . , n } andfor any arrow α ∈ Q such that α = c ± i , c ± i − , if s ( α ) = t ( c i ) (or t ( α ) = s ( c i ) ,respectively), we create a new point, denoted by t ( α ) α ; i (or s ( α ) α ; i , respectively)and an arrow α v i : v i −→ t ( α ) α ; i (or α v i : s ( α ) α ; i −→ v i , respectively). Example 5.1.
Consider the quiver Q : 1 α / / γ / / ǫ / / δ / / RIEZES, STRINGS AND CLUSTER VARIABLES 17 and let M be the string representation corresponding to the walk c = ǫ − γ , that is M : 0 / / k [0 , t / / [1 , t / / k / / . Then, the quiver g Q M is g Q M γ ;1 γ v / / v δ v (cid:15) (cid:15) v β o o β / / v δ v (cid:15) (cid:15) ǫ ;3 . ǫ v o o δ ;1 α ;2 α v O O δ ;3 We recall the following definition from [33] :
Definition 5.2.
Let Q and S be two quivers. A winding of quivers Φ : Q −→ S isa pair Φ = (Φ , Φ ) where Φ : Q −→ S and Φ : Q −→ S are such that :a) Φ is a morphism of quivers, that is s ◦ Φ = Φ ◦ s and t ◦ Φ = Φ ◦ t ;b) If α, α ′ ∈ Q with α = α ′ and s ( α ) = s ( α ′ ) , then Φ ( α ) = Φ ( α ′ ) ;c) If α, α ′ ∈ Q with α = α ′ and t ( α ) = t ( α ′ ) , then Φ ( α ) = Φ ( α ′ ) ;With the above notations, the maps Φ : v i t ( c i − ) for any i ∈ { , . . . , n + 1 } v α ; i v for any v ∈ supp( M ) and any α, i and Φ : (cid:26) β i c i for any i ∈ { , . . . , n } α v i α for any arrow of the form α v i induce a winding of quivers Φ : g Q M −→ supp( M ) .Let Φ ∗ be the map from the set of objects in rep( g Q M ) to the set of objects in rep(supp( M )) which associates to a representation e V of g Q M the representation V given by V ( i ) = M j ∈ Φ − ( i ) e V ( j ) and V ( α ) = M β ∈ Φ − ( α ) e V ( β ) for any i ∈ (supp( M )) and any α ∈ (supp( M )) . At the level of dimension vectors, Φ ∗ induces a natural map φ : N ( g Q M ) −→ N (supp( M )) .We define a representation f M of g Q M by setting for any point v ∈ ( g Q M ) f M ( v ) = (cid:26) k if v = v i for some i ∈ { , . . . , n } , otherwiseand for any arrow α ∈ ( g Q M ) f M ( α ) = (cid:26) k if α = β ± i for some i ∈ { , . . . , n } , otherwise. Lemma 5.3.
Let B be a finite dimensional k -algebra. Then for any string B -module M , we have Φ ∗ ( f M ) ≃ M .Proof. This follows from the construction. (cid:3)
Definition 5.4. If B is a finite dimensional k -algebra with bound quiver ( Q, I ) and M is a B -module, the border ∂M of M is the set of points in the closure supp( M ) of the support of M in Q and which do not lie in the support supp( M ) of M .Thus, with the above notations, the border ∂ f M consists of all the points in ( g Q M ) which do not lie in the support of f M . Lemma 5.5.
Let B be a finite dimensional k -algebra. Then for any string B -module M , the pair ( g Q M , ∂ f M ) is a blown-up ice quiver whose unfrozen part isof Dynkin type A . Moreover, the representation f M is a sincere unfrozen stringrepresentation of g Q M .Proof. The first assertion follows from the construction of g Q M and f M . For thesecond assertion, we observe that f M is supported on the unfrozen part of g Q M whichis of Dynkin type A . It is unfrozen sincere and indecomposable by construction sothat it is a string representation. (cid:3) Example 5.6.
Let B be the path algebra of the quiver Q considered in Example5.1 and let M be the string module considered in that same example. Then, therepresentation f M of the quiver g Q M is f M : 0 / / k (cid:15) (cid:15) k k o o k / / k (cid:15) (cid:15) o o O O so that ( g Q M , ∂ f M ) is indeed a blown-up ice quiver with unfrozen part of Dynkintype A .5.2. Normalisation.
In this section M denotes a string module over a finite di-mensional algebra B with bound quiver ( Q, I ) . Definition 5.7.
The normalising vector of M is n M = ( n i ) i ∈ supp( M ) ∈ N supp( M ) given by n i = h S i , M i − X j ∈ Φ − ( i ) D S j , f M E for any i ∈ supp( M ) where the first truncated Euler form is considered in mod- B and the second truncated Euler form is considered in mod- k g Q M .The normalising factor of M is x n M = Y i ∈ supp( M ) x n i i . This normalisation is actually easy to compute in several usual situations as itis explained in Section 7.
Example 5.8.
Consider the finite dimensional algebra B whose ordinary quiver is γ (cid:0) (cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Q : 1 α / / β ^ ^ ======== RIEZES, STRINGS AND CLUSTER VARIABLES 19 and whose relations are given by the vanishing of all paths of length two, that is αβ = βγ = γα = 0 . It is a cluster-tilted algebra of type A . Consider the projectivemodule M associated with the point . It is a string B -module with string α . Theassociated blown-up quiver g Q M is γ ;1 −→ v −→ v −→ β ;2 and the representation f M is −→ k k −→ k −→ . Then, one has h S , M i = D S v , f M E = 0 , h S , M i = D S v , f M E = 1 and h S , M i = 0 whereas D S β ;2 , f M E + D S γ ;1 , f M E = 0 − − .Thus, the normalisation is n M = (0 , , .5.3. Blow-ups and cluster characters.
We keep the notations of Section 5.1.As usual, we naturally identify L ( x supp( M ) ) to a subring of L ( x Q ) . We considerthe following surjective morphism of Z -algebras : π : L ( x g Q M ) −→ L ( x supp( M ) ) defined by π ( x v i ) = x t ( c i − ) for any ≤ i ≤ n + 1 and π ( x v α ; i ) = x v for any v ∈ supp( M ) and any α, i .We now observe that for any unfrozen string module M with respect to a real-isable quadruple ( Q , F, C , T ) , the Laurent polynomial X TM is in the image of thefunction π . Lemma 5.9.
Let ( Q , F, C , T ) be a realisable quadruple and let M be an unfrozenstring module. Then X TM ∈ L ( x supp( M ) ) . Proof.
We recall that unfrozen B T -modules are identified via Hom C ( T, − ) with theobjects M in U / (add T [1]) . We have X TM = X e ∈ N Q χ (Gr e ( M )) Y i ∈ Q x h S i , e i a −h S i ,M i i . Any dimension vector e such that Gr e ( M ) = ∅ is supported on supp( M ) . Thus, if i ∈ Q is not in supp( M ) , then h S i , e i a = 0 and h S i , M i = 0 . In particular, X TM = X e ∈ N Q χ (Gr e ( M )) Y i ∈ supp( M ) x h S i , e i a −h S i ,M i i ∈ L ( x supp( M ) ) . (cid:3) Let e C be the cluster category of the quiver g Q M and let e T = k g Q M be the pathalgebra of g Q M , which is identified with a tilting object in e C . We denote by e X ? thecorresponding cluster character with values in L ( x g Q M ) . Proposition 5.10.
Let ( Q , F, C , T ) be a realisable quadruple and let M be anunfrozen string module. Then, π ( e X f M ) = x n M X TM . Proof.
We have X TM = X e ∈ N Q χ (Gr e ( M )) Y i ∈ Q x h S i , e i a −h S i ,M i i = X e ∈ N supp( M )0 χ (Gr e ( M )) Y i ∈ (supp( M )) x h S i , e i a −h S i ,M i i Now, for any e ∈ N supp( M ) , since Φ is a winding of quivers and Φ ∗ ( f M ) = M , itfollows from [33, Theorem 1.2 (a)] that χ (Gr e ( M )) = X f ∈ φ − ( e ) χ (Gr f ( f M )) . Fix e ∈ N supp( M ) and f ∈ φ − ( e ) . We now prove that for any i ∈ supp( M ) , wehave(5.6) x h S i , e i a i = π Y j ∈ Φ − ( i ) x h S j , f i a j . where the anti-symmetrised Euler form in the left-hand side is taken in mod- k g Q M and the anti-symmetrised Euler forms in the right-hand side are taken in mod- B .First note that π Y j ∈ Φ − ( i ) x h S j , f i a j = x P j ∈ Φ −
10 ( i ) h S j , f i a i so that it is enough to prove that h S i , e i a = X j ∈ Φ − ( i ) h S j , f i a . For any module U and any integer n ≥ , we denote by nU the direct sumof n copies of U . Now, since h S i , −i a is well-defined on the Grothendieck group K ( mod- B T ) (Lemma 3.2), we have h S i , e i a = X k ∈ supp( M ) h S i , e k S k i a = X k ∈ supp( M ) * S i , X l ∈ Φ − ( k ) f l S k + a = X k ∈ supp( M ) X l ∈ Φ − ( k ) f l h S i , S k i a RIEZES, STRINGS AND CLUSTER VARIABLES 21 and X j ∈ Φ − ( i ) h S j , f i a = X k ∈ supp( M ) X l ∈ Φ − ( k ) X j ∈ Φ − ( i ) f l h S j , S l i a = X k ∈ supp( M ) X l ∈ Φ − ( k ) f l X j ∈ Φ − ( i ) h S j , S l i a . But for any k ∈ supp( M ) and any l ∈ Φ − ( k ) , we have X j ∈ Φ − ( i ) h S j , S l i a = h S i , S k i a which proves (5.6).Now, e X f M = X f ∈ N ( g QM )0 χ (Gr f ( f M )) Y i ∈ ( g Q M ) x h S i , f i a − h S i , f M i i so that, π ( e X f M ) = X e ∈ N supp( M )0 X f ∈ φ − ( e ) χ (Gr f ( f M )) Y i ∈ supp( M ) π Y j ∈ Φ − ( i ) x h S j , f i a − h S j , f M i j = X e ∈ N supp( M )0 X f ∈ φ − ( e ) χ (Gr f ( f M )) Y i ∈ supp( M ) x h S i , e i a i π Y j ∈ Φ − ( i ) x − h S j , f M i j = X e ∈ N supp( M )0 χ (Gr e ( M )) Y i ∈ supp( M ) x h S i , e i a − P j ∈ Φ −
10 ( i ) h S j , f M i i = x n M X TM . This finishes the proof. (cid:3)
The main theorem.
We can now prove the main theorem of the article :
Theorem 5.11.
Let ( Q , F, C , T ) be a realisable quadruple and let M be an unfrozenstring module with respect to this quadruple. Then X TM = 1 x n M L M . Proof.
We first notice that with the above notations, it follows directly from thedefinitions that π ( e L f M ) = L M . Thus, X TM = 1 x n M π ( e X f M ) = 1 x n M π ( e L f M ) = 1 x n M L M where the first equality follows from Proposition 5.10, the second equality fromTheorem 4.1 and Lemma 5.5 and the third from the above observation. (cid:3) Applications
In this section, we give several applications of Theorem 5.11.
Computing cluster variables.
We now prove that the formula given inequation (1.1) allows one to compute the cluster variables in several situations.
Corollary 6.1.
Let ( Q , F ) be an ice quiver. Assume that there exists a realis-able quadruple ( Q , F, C , T ) such that every rigid object in C is reachable from T .Then for any unfrozen End C ( T ) -module M which is rigid and indecomposable, theLaurent polynomial x n M L M is a cluster variable in A ( Q , F ) .Proof. We identify M with an indecomposable lifting in C for the functor Hom C ( T, − ) , which is also rigid in C by [52, 31]. It follows from Theorem 3.4that the map M X TM induces a surjection from the set of reachable rigid ob-jects in to the set of cluster variables in A ( Q , F ) . It is also rigid in C , hence, byhypothesis, it is reachable and thus X TM is a cluster variable in A ( Q , F ) . (cid:3) Corollary 6.2.
Let B T be a cluster-tilted algebra. Then for any rigid string B T -module M , the Laurent polynomial x n M L M is a cluster variable in the coefficient-free cluster algebra associated with the ordinary quiver of B T .Proof. Let C be a cluster category, T be a tilting object in C and Q T be theordinary quiver of B T . Then the quadruple ( Q T , ∅ , C , T ) is realisable (see Example2.3). Moreover, it follows from [10] that every indecomposable rigid object in C isreachable from T . The result thus follows from Corollary 6.1. (cid:3) Corollary 6.3.
Let B T be a cluster-tilted algebra of Dynkin type A or euclideantype e A . Then, the map M x n M L M induces a bijection from the set of indecomposable rigid B T -modules to the set ofcluster variables in the coefficient-free cluster algebra associated with the ordinaryquiver of B T which do not belong to the initial cluster.Proof. Cluster-tilted algebras of Dynkin type A or euclidean type e A are stringalgebras [2]. Thus, each indecomposable rigid B T -module is a string module. Nowif C is a cluster category and T is a tilting object in C such that B T = End C ( T ) ,it follows in particular from Theorem 5.11 that for every indecomposable rigid B T -module M , we have X TM = x n M L M . But then it follows from [44] that M x n M L M is a bijection from the set of indecomposable rigid B T -modules to the setof cluster variables in the coefficient-free cluster algebra A ( Q T , ∅ ) which do notbelong to the initial cluster x Q T where Q T denotes the ordinary quiver of B T . (cid:3) Positivity in cluster algebras.
We now prove positivity results in clusteralgebras using formula (1.1). We first recall the basic notions concerning positivityin cluster algebras.If P is an abelian group, we denote by ZP its group ring. Let n ≥ be an integerand x = ( x , . . . , x n ) be an n -tuple of indeterminates. We denote by Z ≥ P [ x ± ] the semiring of subtraction-free Laurent polynomials in the variables in x withcoefficients in ZP , that is, the set of elements of the form f ( x , . . . , x n ) x d · · · x d n n where f is a polynomial in n variables whose coefficients are non-negative linearcombinations of elements of P and d , . . . , d n ∈ Z . In particular, in the case when RIEZES, STRINGS AND CLUSTER VARIABLES 23 P = { } , the notation Z ≥ [ x ± ] denotes the semiring of Laurent polynomials invariables x , . . . , x n with non-negative coefficients.If Q is a quiver without loops and 2-cycles, if P is any semifield and if y isa Q -tuple of elements of P , it follows from the Laurent phenomenon that everycluster variable in the cluster algebra A ( Q, x Q , y ) is an element of ZP [ c ± ] whenexpressed in any cluster c of A ( Q, x Q , y ) [27]. The Fomin and Zelevinsky positivityconjecture , stated in [27], asserts that every cluster variable is actually an elementof Z ≥ P [ c ± ] . This conjecture was proved for rank two cluster algebras [51, 41, 23],cluster algebras arising from surfaces with or without punctures [50, 49, 42], clusteralgebras with bipartite seeds [43] and weaker versions were proved for acyclic clusteralgebras [5, 47, 43]. Corollary 6.4.
Let ( Q , F, C , T ) be a realisable quadruple and let M be an unfrozenstring module with respect to this quadruple. Then X TM ∈ Z ≥ [ x supp( M ) ± ] . Proof.
The set Z ≥ [ x supp( M ) ] is a semiring for the usual sum and product. Thus,the set M ( Z ≥ [ x supp( M ) ]) is also a semiring for the usual sum and product ofmatrices. It follows that L M ∈ Z ≥ [ x supp( M ) ± ] and by Theorem 5.11, X TM = L M .This proves the corollary. (cid:3) This corollary was also obtained by Cerulli and Haupt [33, 18].
Corollary 6.5.
Let B T be a cluster-tilted algebra. Then X TM ∈ Z ≥ [ x supp( M ) ± ] for any string B T -module M . (cid:3) We now provide a new proof of the Fomin and Zelevinsky positivity conjecturefor cluster algebras arising from unpunctured surfaces. We first set some notations.If ( S, M ) is an oriented bordered marked surface, Fomin, Shapiro and Thurstonassociated with any triangulation Γ of ( S, M ) a quiver Q Γ [26]. For any semifield P and any ( Q Γ ) -tuple y of elements of P , we denote by A ( Q Γ , x Q Γ , y ) the clusteralgebra with initial seed ( Q Γ , x Q Γ , y ) (see [28]). Such a cluster algebra does notdepend on the choice of the triangulation Γ of ( S, M ) , up to an isomorphism of ZP -algebras [26]. It is called the cluster algebra arising from the surface ( S, M ) and is denoted by A ( S, M ) .For this class of cluster algebras, Fomin and Zelevinsky positivity conjectureamounts to saying that any cluster variable z in A ( S, M ) belongs to Z ≥ P [ x ± Q Γ ] for any choice of triangulation Γ [27, §3]. This conjecture was proved in [50, 49]for unpunctured surfaces and in [42] for arbitrary surfaces. We now provide a newindependent representation-theoretical proof of the positivity conjecture for clusteralgebras arising from unpunctured surfaces. Corollary 6.6.
Let ( S, M ) be an unpunctured surface. Then the positivity conjec-ture holds for A ( S, M ) equipped with an arbitrary choice of coefficients.Proof. Using the Fomin-Zelevinsky separation formula [28, Theorem 3.7], it isenough to prove the result for principal coefficients. Let Γ be any triangulationof ( S, M ) and let ( Q Γ , W Γ ) be the quiver with potential associated with this tri-angulation as in [2] (see also [39]). Let J Γ = J ( Q Γ ,W Γ ) be the correspondingJacobian algebra which is finite dimensional [2]. It follows from Corollary 2.11 thatthe principal extension ( Q ppΓ , ( Q Γ ) ′ ) can be embedded in a realisable quadruple ( Q ppΓ , ( Q Γ ) ′ , C , T ) . Now, let z be a cluster variable in A ( S, M ) , it follows fromTheorem 3.4 that either z belongs to the initial cluster x Q Γ or z = X TM for someindecomposable rigid J Γ -module M . Since the algebra J Γ is a string algebra, anyindecomposable rigid J Γ -module is a string module. Thus, if we set y i = x i ′ forany i ∈ ( Q T ) , it follows from Corollary 6.4 that z = X TM ∈ Z ≥ [ x supp( M ) ± ] ⊂ Z ≥ [ y ± ][ x ± Q Γ ] = Z ≥ P [ x ± Q Γ ] . (cid:3) SL (2 , Z ) and total Grassmannians of submodules.Definition 6.7. Given a finite dimensional k -algebra B and a finitely generated B -module M , the total Grassmannian of submodules of M is the set Gr( M ) = G e ∈ K ( mod- B ) Gr e ( M ) . Since M is finite dimensional, all but finitely many Gr e ( M ) are empty so that Gr( M ) is a finite disjoint union of projective varieties. It is thus endowed with astructure of projective variety and we can consider its Euler-Poincaré characteristic χ (Gr( M )) = P e χ (Gr e ( M )) ∈ Z .For modules over hereditary algebras, these numbers can be computed withfriezes, as shown in [4, §4]. Here we prove that, for string modules over 2-Calabi-Yau tilted algebras, these numbers can be computed with products of matrices in SL (2 , Z ) .If Q is any quiver, for any β ∈ Q , we define two matrices in SL (2 , Z ) : a ( β ) = (cid:20) (cid:21) and a ( β − ) = (cid:20) (cid:21) and for any walk c = c · · · c n in Q , we set l c = (cid:2) , (cid:3) n Y i =1 a ( c i ) ! (cid:20) (cid:21) ∈ N . Finally, if c is a walk of length 0, we set l c = 2 . Proposition 6.8.
For any string module M over a finite dimensional algebra, wehave χ (Gr( M )) = l s ( M ) . Proof.
Let B be a finite-dimensional algebra with ordinary quiver Q and let M be astring B -module. We use the notations of Section 5.1. Let f M be the representationof the blow-up g Q M of Q along M . Then f M is a string representation of g Q M andwe have l s ( M ) = l s ( f M ) . Moreover, it follows from [33, Theorem 1.2 (a)] that X e ∈ N Q χ (Gr e ( M )) = X e ∈ N Q X f ∈ φ − ( e ) χ (Gr f ( f M )) = X f ∈ N ( g QM )0 χ (Gr f ( f M )) so that the characteristic of the total Grassmannian of M equals the characteristicof the total Grassmannian of f M . Since Grassmannians of submodules of f M onlydepends on the support of f M , it is enough to prove the proposition for indecom-posable modules over path algebras of Dynkin quivers of type A . RIEZES, STRINGS AND CLUSTER VARIABLES 25
Let thus B be the path algebra of a Dynkin quiver Q of type A , let C be thecluster category of Q and let T = k Q . Then ( Q, ∅ , C , T ) is a realisable quadruplesuch that B = End C ( T ) . According to Theorem 5.11, we have X TM = 1 x n M Q L M . Consider the surjective morphism of Z -algebras. p : (cid:26) L ( x Q ) −→ Z x i for any i ∈ Q . Then it follows from the definition of the cluster character that p ( X TM ) = χ (Gr( M )) .We now prove that p ( L M ) = l s ( M ) . If the string s ( M ) of M is of length zero, M is simple and thus χ (Gr( M )) = χ (Gr [0] ( M )) + χ (Gr [ M ] ( M )) = 1 + 1 = 2 = l s ( M ) . Otherwise, we write s ( M ) = c · · · c n with n ≥ . Then, for any i ∈ { , . . . , n } , wehave p ( A ( c i )) = a ( c i ) and p ( V c ( s ( c i ))) = p ( V c ( t ( c i ))) is the identity matrix. Since p is a ring homomorphism, it induces a ring homomorphism at the level of matricesand thus p ( L M ) = p ( L c ) = l c . Since p ( x n MQ ) = 1 , it follows that χ (Gr( M )) = p ( X TM ) = p ( L M ) = l s ( M ) and the proposition is proved. (cid:3) Example 6.9.
Let ( S, M ) be an unpunctured surface. Let Γ be a triangulation of ( S, M ) , let ( Q Γ , W Γ ) be the quiver with potential associated with this triangulationand let J Γ = J ( Q Γ ,W Γ ) be the corresponding Jacobian algebra. Then for any stringmodule M over J Γ , we have χ (Gr( M )) = l s ( M ) .7. More about the normalisation
We now describe some situations in which the normalisation can be omitted orcomputed combinatorially.7.1.
The hereditary case.
We first observe that the normalisation can be omittedin several cases.
Lemma 7.1.
Let ( Q , F, C , T ) be a realisable quadruple and let M be an unfrozenstring module. If the full subcategory of mod- B T formed by modules which aresupported on supp( M ) is hereditary, then n M = 0 .Proof. Let i ∈ supp( M ) , since M and S i are supported on supp( M ) , the truncatedEuler form h S i , M i only depends on the dimension vectors. But since g Q M is acyclic, P j ∈ Φ − ( i ) D S j , f M E also only depends on the dimension vectors, it thus follows that h S i , M i = P j ∈ Φ − ( i ) D S j , f M E and thus n M = 0 . (cid:3) Corollary 7.2.
Let ( Q , F, C , T ) be a realisable quadruple such that B T is heredi-tary. Then n M = 0 for every unfrozen string module M . (cid:3) The maximal factor method for modules over cluster-tilted algebras.
For any walk c in a locally finite quiver Q we set N c = (cid:2) , (cid:3) n Y i =0 A ( c i ) V c ( i + 1) ! (cid:20) (cid:21) ∈ L ( x Q ) so that (1.1) becomes L c = 1 Q ni =0 x t ( c i ) N c . Let η c ∈ N Q be such that N c = x η c Q P c ( x Q ) with P c ( x Q ) not divisible by any x i with i ∈ Q .Let C be a cluster category, T = T ⊕ · · · ⊕ T n be a tilting object in C , then B T = End C ( T ) is a cluster-tilted algebra and we denote by Q its ordinary quiver.Then ( Q, ∅ , C , T ) is a realisable quadruple and every B T -module is unfrozen.We now prove that for such modules, we can compute the normalisation combi-natorially : Proposition 7.3.
Consider the above notations and assume moreover that
End C ( T i ) ≃ k for any i ∈ { , . . . , n } . Then for every rigid string B T -module M we have n M = η s ( M ) .Proof. For any Laurent polynomial L ( x , . . . , x n ) ∈ Z [ x ± , . . . , x ± n ] , we denote by δ ( L ) ∈ Z n its denominator vector , that is, the unique vector ( d , . . . , d n ) such that L ( x , . . . , x n ) = f ( x , . . . , x n ) / Q ni =1 x d i i where f ( x , . . . , x n ) is a polynomial notdivisible by any x i .Since End C ( T i ) ≃ k for any i ∈ { , . . . , n } , it follows from [12] that for anyindecomposable rigid B T -module δ ( X TM ) = (dim Hom C ( T i , M )) ≤ i ≤ n where M is an indecomposable rigid object in C such that Hom C ( T, M ) = M .Since Hom C ( T, − ) induces an equivalence of categories C / (add T [1]) ≃ mod- B T ,it follows that δ ( X TM ) = (dim Hom B T ( P i , M )) ≤ i ≤ n where P i is the indecomposable projective B T -module at the point i . For any i ∈ { , . . . , n } , the dimension of Hom B T ( P i , M ) is the multiplicity of the simple B T -module S i as a composition factor of M . If M is a string module, this multiplicity isalso equal to the number of occurrences of i along the string s ( M ) . Thus, if s ( M ) = c · · · c n , the denominator of X TM in its irreducible form is equal to Q ni =0 x t ( c i ) .On the other hand the denominator of L M (in its irreducible form) is x − η s ( M ) Q ni =0 x t ( c i ) . Now it follows from Theorem 5.11 that X TM = x n M L M andthus the denominators coincide, that is to say n M = η s ( M ) . (cid:3) Examples
A first example in type A . We start with a very simple and detailedexample which is nevertheless instructive in order to understand the behaviour ofthe normalisation.
RIEZES, STRINGS AND CLUSTER VARIABLES 27
We consider the frozen quiver ( Q , F ) where γ (cid:0) (cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Q : 1 α / / β ^ ^ >>>>>>>> and F = { } .The corresponding cluster algebra is of geometric type with initial seed ( x , y , B ) where x = ( x , x ) , y = ( x ) and B = − − Its unfrozen part Q is of Dynkin type A . Note that Q is the ordinary quiverof a cluster-tilted algebra of type A whose relations are αβ = βγ = γα = 0 . Thus ( Q , F ) can be embedded in a realisable quadruple ( Q , F, C , T ) where C is a clustercategory of type A . There are three indecomposable modules supported on Q which are the simple S , the simple S and the projective k Q -module P and allthese modules are string modules.Since ( Q , F ) is not blown-up (there are two arrows entering or leaving the frozenpoint 3), we cannot simply apply Theorem 4.1. So we apply Theorem 5.11, whichmeans that we need to consider normalising factors.The string of S is e and thus, applying formula (1.1) to S , we get L S = x + x x .The quiver g Q S is γ ;1 −→ v −→ α ;1 . It is easily computed that n S = 0 so that X S = L S which is indeed the cluster variable in A ( Q , F ) corresponding to thesimple module S . We similarly prove that X S = L S = x + x x . The case of P is slightly more instructive. We apply the matrix formula and weget L P = x ( x + x + x ) x x . We can now use Proposition 7.3 in order to conclude thatthe normalising factor is x but we do the computation in order to see the blow-uptechnique working. The quiver supp( P ) is Q itself. Its blow-up along P is thus g Q P : 3 γ ;1 −→ v −→ v −→ β ;2 and f M is the representation e P : 0 −→ k k −→ k −→ .We compute that e L e P = x v x γ ;1 + x v x β ;2 + x γ ;1 x β ;2 x v x v so that the morphism π sending x v to x , x v to x and x γ ;1 , x β ;2 to x satisfies π ( e L f M ) = L M . And computing directly, or else applying Theorem 4.1, we get e X f M = e L e P .The normalisation of M is n M = (0 , , (see Example 5.8) so that we get X P = x L P = x + x + x x x which is indeed the cluster variable in A ( Q , F ) correspondingto this module.8.2. Regular cluster variables in type e A . Let Q be a euclidean quiver of type e A equipped with an acyclic orientation. In [5, Theorem 4], the authors provided acombinatorial formula for expressing all but finitely many cluster variables in thecoefficient-free cluster algebra associated with Q . The cluster variables they com-puted correspond in fact to the cluster characters associated with indecomposablepostprojective k Q -modules. Using their methods, it is possible to compute thecluster characters associated with indecomposable postprojective and preinjective k Q -modules but not the remaining cluster variables, namely those which correspondto indecomposable regular rigid k Q -modules. We now use Theorem 4.1 to completethe formula and compute cluster variables associated with regular modules.It is known that the Auslander-Reiten quiver of mod- k Q contains at most twoexceptional tubes (that is, tubes of rank ≥ ) and all the rigid indecomposableregular modules belong to these tubes. Every indecomposable regular module (rigidor not) in such a tube is a string module and its string may be completely described.For simplicity, we only compute the cluster variables associated with quasi-simpleregular modules in such tubes, this can easily be extended to any module in oneof the exceptional tubes. It is well-known (see for instance [4, §5.2]) that any suchmodule is of the form · · · k o o k / / k k / / · · · k / / k k / / k · · · o o so that, if we depict locally the quiver Q as · · · o o α / / α / / · · · α n − / / n − α n − / / n n + 1 · · · , o o the string is c = α · · · α n − and we compute L c = 1 Q n +1 i =1 x i (cid:2) , (cid:3) (cid:20) x
00 1 (cid:21) n − Y i =1 x i +1 n − X j =1 Q ni =1 x i x j x j +1 n − Y i =1 x i (cid:20) x n +1 (cid:21) (cid:20) (cid:21) . Now, if M c denotes the quasi-simple regular module associated with the string c ,then it follows from Lemma 7.1 that L c is the cluster character corresponding tothe module M c and thus, is a cluster variable in the cluster algebra A ( Q, ∅ ) .8.3. An example for the n -Kronecker quiver with principal coefficients. Let n ≥ and let K pp n be the quiver ′ (cid:15) (cid:15) ′ (cid:15) (cid:15) K pp n : 1 α / / / / α n / / with n arrows α , . . . , α n from 1 to 2. We set F = { ′ , ′ } so that K pp n is theprincipal extension of the n -Kronecker quiver K n . We fix two distinct arrows among α , . . . , α n which we denote by α and β .For any p ≥ , we define a string representation M p of K n by setting M p (1) = k p , M p (2) = k p +1 , M p ( α ) = 1 k p ⊕ and M p ( β ) = 0 ⊕ k p and we view M p as arepresentation of K pp n which is supported on the unfrozen part K n . For any p ≥ ,the string s ( M p ) is ( α − β ) p so that it has length p .We write y = x ′ and y = x ′ . Then, a direct computation shows that(8.7) L M p = 1 x p x p +12 (cid:2) , (cid:3) (cid:20) y x n − (cid:21) (cid:20) y + x n y y x n − y x y y x n (cid:21) p (cid:20) x (cid:21) . Applying Corollary 2.11, we see that there exists a realisable quadruple ( K pp n , ( K n ) ′ , C , T ) . Since K pp n is acyclic, it follows from [37] that B T is hereditaryand thus, Lemma 7.1 implies that the normalisation vanishes. Thus, X TM p = L M p is given by the formula (8.7) for every p ≥ . RIEZES, STRINGS AND CLUSTER VARIABLES 29
Cyclic cluster-tilted algebras of Dynkin type D . Let B be the quotientof the path algebra of the quiver Q : 1 / / / / / / · · · / / n w w by the ideal generated by all paths of length n − . Here we suppose that n ≥ . B is a cluster-tilted algebra of type D n if n ≥ and a cluster-tilted algebra of type A if n = 3 . Let M be the indecomposable module given by the Loewy series ... m ,where ≤ m ≤ n . Proposition 8.1.
Let Q denote the above quiver and let ( Q , F, C , T ) be a realisablequadruple with unfrozen part Q . For any i ∈ Q , let, as in Section 4.1, y i = Y α ∈ Q ( F,i ) x s ( α ) and z i = Y α ∈ Q ( i,F ) x t ( α ) . Let M be as above. Then (8.8) L M = m X ℓ =1 x x · · · x m +1 x ℓ x ℓ +1 ℓ Y i =2 y i ! m Y i = ℓ +1 z i ! Q mj =2 x j if m < n ; and if n = m then L M is given by (8.8) divided by x .Proof. We proceed by induction on m . If m = 2 , then M is simple and L M is givenby the exchange relation(8.9) L M = ( x y + x z ) 1 x . On the other hand, (8.8) gives (cid:18) x x x x x z + x x x x x y (cid:19) x , which is equal to the right-hand side of equation (8.9).Suppose now that m > . Let N M be the numerator of the Laurent polynomial L M . Our formula gives N M as the product of m − matrices :(8.10) (cid:2) x (cid:3) (cid:20) z y (cid:21) · · · (cid:20) x m x m − (cid:21) (cid:20) z m y m (cid:21) (cid:20) x m +1 (cid:21) . The product of the last three matrices in (8.10) is equal to (cid:20) x m x m +1 z m x m +1 z m + x m − y m (cid:21) = (cid:20) x m (cid:21) x m +1 z m + (cid:20) (cid:21) x m − y m . Let us denote the product of the first m − matrices by (cid:2) a b (cid:3) , so N M = (cid:2) a b (cid:3) (cid:18)(cid:20) x m (cid:21) x m +1 z m + (cid:20) (cid:21) x m − y m (cid:19) . Now let M ′ be the indecomposable module given by the Loewy series ... m − .Then, our formula gives N M ′ as a product of m − matrices, and the first m − matrices are just the same as the first m − matrices in (8.10), and therefore N M ′ = (cid:2) a b (cid:3) (cid:20) x m (cid:21) . Thus N M = N M ′ x m +1 z m + (cid:2) a b (cid:3) (cid:20) (cid:21) x m − y m , which, by induction, is equal to m − X ℓ =1 x x · · · x m x ℓ x ℓ +1 ℓ Y i =2 y i ! m − Y i = ℓ +1 z i ! x m +1 z m + x x · · · x m x m − x m m − Y i =2 y i ! x m − y m , which in turn is equal to m − X ℓ =1 x x · · · x m +1 x ℓ x ℓ +1 ℓ Y i =2 y i ! m Y i = ℓ +1 z i ! + x x · · · x m +1 x m x m +1 m Y i =2 y i ! , and this shows the formula (8.8). The statement for m = n follows from thenormalising factor. (cid:3) Acknowledgments
Work on this problem was started during the in Mar del Plata (Argentina); theauthors wish to thank the organisers for their kind invitation.The first author gratefully acknowledges partial support from the NSERC ofCanada, the FQRNT of Québec and the Université de Sherbrooke.This paper was written while the second author was at the Université de Sher-brooke as a CRM-ISM postdoctoral fellow under the supervision of the first author,Thomas Brüstle and Virginie Charette.The third author is supported by the NSF grants DMS-0908765 and DMS-1001637 and by the University of Connecticut.The fourth author gratefully acknowledges support from the NSERC of Canadaand Bishop’s University.
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