Hernandez-Leclerc modules and snake graphs
aa r X i v : . [ m a t h . QA ] O c t HERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS
BING DUAN, JIAN-RONG LI AND YAN-FENG LUO
Abstract.
In 2010, Hernandez and Leclerc studied connections between representationsof quantum affine algebras and cluster algebras. In 2019, Brito and Chari defined afamily of modules over quantum affine algebras, called Hernandez-Leclerc modules. Wecharacterize the highest ℓ -weight monomials of Hernandez-Leclerc modules. We give anon-recursive formula for q -characters of Hernandez-Leclerc modules using snake graphs,which involves an explicit formula for F -polynomials. We also give a new recursiveformula for q -characters of Hernandez-Leclerc modules. Contents
1. Introduction 21.1. Highest ℓ -weight monomials of Hernandez-Leclerc modules 21.2. Hernandez-Leclerc modules, snake graphs, and q -characters 31.3. A recursive formula for Hernandez-Leclerc modules 52. Preliminary 52.1. Cluster algebras 52.2. Category C ξ and Hernandez-Leclerc modules 72.3. Snake graphs 92.4. Continued fractions and F -polynomials 113. q -characters of Hernandez-Leclerc modules and snake graphs 123.1. A combinatorial description of Hernandez-Leclerc modules 123.2. Hernandez-Leclerc modules and snake graphs 143.3. q -characters of Hernandez-Leclerc modules in term of perfect matchings ofsnake graphs 173.4. Relation with Brito and Chari’s results 223.5. The highest and lowest ℓ -weight monomials of Hernandez-Leclerc modules 264. A new recursion for q -characters of Hernandez-Leclerc modules 33Acknowledgements 34References 35 Key words and phrases.
Cluster algebras; Quantum affine algebras; Hernandez-Leclerc modules; Snakegraphs; q -characters. Introduction
Quantum groups were introduced independently by Drinfeld [13, 14] and Jimbo [26].Let g be a simple Lie algebra over C and let U q ( b g ) be the corresponding untwisted quan-tum affine algebra with quantum parameter q ∈ C × not a root of unity. Denote by C the category of finite dimensional representations of U q ( b g ). It is well-known that C isnot semisimple but an abelian tensor category [23]. The isomorphism classes of finite-dimensional simple U q ( b g )-modules can be parametrized by Drinfeld polynomials in [3–5].Equivalently, each simple U q ( b g )-module can also be parametrized by the highest dominantmonomial of its q -character [22].Cluster algebras were introduced by Fomin and Zelevinsky in their seminal work [18]. Acluster algebra is a commutative ring with a set of distinguished generators called clustervariables which are defined through iterative processes known as mutations.A finite-dimensional U q ( b g )-module is said to be prime if it admits no nontrivial tensorfactorization [6]. A simple U q ( b g )-module is said to be real if its tensor square is also simple[28]. In [23], Hernandez and Leclerc introduced the notion of monoidal categorificationof cluster algebras. An abelian tensor category is said to be a monoidal categorificationof a cluster algebra if the Grothendieck ring of the category is isomorphic to the clusteralgebra and the classes of the real (respectively, real prime) simple modules correspondto cluster monomials (respectively, cluster variables).For each ℓ ∈ Z ≥ , Hernandez and Leclerc [23] introduced a full monoidal subcategory C ℓ of C whose objects are characterized by certain restrictions on the roots of the Drinfeldpolynomials of their composition factors.1.1. Highest ℓ -weight monomials of Hernandez-Leclerc modules. As a general-ization of C [23, 24], in [1] Brito and Chari introduced a subcategory C ξ and a quiver Q ξ depending on a choice of a height function ξ . These real prime simple U q ( b g )-modulesin C ξ are called Hernandez-Leclerc modules by Brito and Chari. Brito and Chari provedthat C ξ is a monoidal categorification of the cluster algebra A ( e x , Q ξ ) with coefficients oftype A [1, Section 1.3]. Hernandez-Leclerc modules are in bijection with cluster variablesand frozen variables, among the initial cluster variables and frozen variables correspondto some Kirillov-Reshetikhin modules. Let K ( ξ ) be the Grothendieck ring of C ξ . De-note by ι the algebraic isomorphism A ( e x , Q ξ ) → K ( ξ ) [1]. Moreover, Brito and Charigave a non-recursion for q -characters of Hernandez-Leclerc modules in term of the known q -characters of initial simple modules [1, Proposition in Section 2.5].In this paper, we use snake graphs to study q -characters of Hernandez-Leclec modules.Let I = { , , . . . , n } . We first give a combinatorial characterization of the highest ℓ -weight monomials of Hernandez-Leclerc modules. Theorem 1.1 (Theorem 3.1) . An Hernandez-Leclerc module corresponding to a clustervariable (excluding frozen variables) is a simple U q ( b g ) -module with the highest ℓ -weight ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 3 monomial Y i ,a Y i ,a . . . Y i k ,a k , where k ∈ Z ≥ , i j ∈ I , a j ∈ Z for j = 1 , , . . . , k , and (i) i < i < · · · < i k , (ii) ( a j − a j − )( a j +1 − a j ) < for ≤ j ≤ k − , (iii) | a j − a j − | = i j − i j − + 2 for ≤ j ≤ k . Hernandez-Leclerc modules, snake graphs, and q -characters. In [29, 30],Mukhin and Young gave a purely combinatorial q -character formula for snake modules oftypes A n and B n via path descriptions. A completely different approach to compute q -characters of Kirillov-Reshetikhin modules was developed by Hernandez and Leclerc [25].In fact, Hernandez and Leclerc proposed a geometric q -character formula conjecture forreal simple modules, which implies that the truncated q -character formula will play animportant role similar to the cluster character formula. Recently, Duan and Schiffler in[16] proved the geometric q -character formula for snake modules of types A n and B n , andcharacterized the associated general kernel. In [2], the authors gave an explicit q -characterformula of simple U q ( b sl n )-modules. The formula involves Kazhdan-Lusztig polynomials[27] which are hard to compute when the length of the highest ℓ -weight monomial of asimple module is large.Our aim is to give a non-recursive formula for q -characters of Hernandez-Leclec modulesusing snake graphs. In [17], Fomin, Shapiro, and Thurston introduced an important classof cluster algebras from surfaces with or without punctures. Each cluster variable (cluster)corresponds to a tagged arc (ideal triangulation) in the surface. In [31, 32], Musiker,Schiffler, and Williams constructed a combinatorial object associated to a non-initial arc,called a labeled snake graph, to compute the Laurent expansion of any non-initial clustervariable in a cluster algebra from a surface. Later Canakci and Schiffler studied abstractsnake graphs in a series of papers [7, 8, 10] and established interesting connections amongcluster variables, snake graphs, and continued fractions [9, 11]. In particular, identities inthe cluster algebra have been expressed in terms of snake graphs. Rabideau used continuedfractions to give a combinatorial formula for F -polynomials [33] in cluster algebras withprincipal coefficients from surfaces. Rabideau and Schiffler proved the constant numeratorconjecture for Markov numbers via snake graphs and continued fractions [34].Given a height function ξ , we construct a unique labeled snake graph for a non-initialHernandez-Leclerc module. Our idea is summarized by the diagram in Figure 1.Fix a simple root system { α i | i ∈ I } of type A n , let α i,j = α i + · · · + α j be apositive root for 1 ≤ i ≤ j ≤ n . Then the set of all cluster variables in A ( e x , Q ξ ) is { x [ − α i ] | i ∈ I } ∪ { x [ α i,j ] | ≤ i ≤ j ≤ n } . Denote by Match( G ) the set of all perfectmatchings of a snake graph G . For any P ∈ Match( G ), let x ( P ) (respectively, y ( P )) be BING DUAN, JIAN-RONG LI AND YAN-FENG LUO
Non-initial cluster variables o o Fomin, Shapiro, andThurston [17] / / Non-initial arcs (up to isotopic)
Musiker, Schiffler, andWilliams [31] (cid:15) (cid:15)
Non-initial Hernandez-Leclerc modules (cid:15) (cid:15)
Brito and Chari [1] O O Theorem 1.2 / / Labeled snake graphs
Figure 1.
Cluster variables, arcs, Hernandez-Leclerc modules, and labeledsnake graphs.the weight (respectively, height) monomial in the sense of Musiker, Schiffler, and Williams[31, 32].By applying the theory of cluster algebras, we give a q -character formula of an arbitraryHernandez-Leclerc module corresponding to a non-initial cluster variable (excluding frozenvariables) by perfect matchings of snake graphs. Theorem 1.2 (Theorems 3.8 and 3.17) . Let G be the labeled snake graph associated to x [ α i,j ] . Then x [ α i,j ] = 1 Q jℓ = i x ℓ P P ∈ Match( G ) x ( P ) y ( P ) L P ∈ Match( G ) y ( P ) ! , where the sign ⊕ appearing in the denominator refers to the addition of a tropical semifield.In particular, χ q ( ι ( x [ α i,j ])) = 1 Q jℓ = i χ q ( ι ( x ℓ )) χ q ι ( P P ∈ Match( G ) x ( P ) y ( P ) L P ∈ Match( G ) y ( P ) ) ! . There is a unique perfect matching P ∈ Match( G ) such that the highest or lowest ℓ -weightmonomial in χ q ( ι ( x [ α i,j ])) occurs in Q jℓ = i χ q ( ι ( x ℓ )) χ q ι ( x ( P ) y ( P ) L P ∈ Match( G ) y ( P )) ! . In [1, Proposition in Section 2.5], Brito and Chari gave a non-recursive formula of x [ α i,j ]using two sets Γ i,j and Γ ′ i,j . The set Γ i,j consists of (0 ,
1) sequences with ( j − i + 2) lengthsubject to four conditions, and Γ ′ i,j is determined by Γ i,j consisting of ( − , ,
1) sequenceswith the same length, see Section 3.4.On the other hand, we associate to x [ α i,j ] perfect matchings of the corresponding labeledsnake graph. Combining with Rabideau’s formula for F -polynomials [33, Theorem 3.4],Theorem 3.6 gives an explicit formula for F -polynomials. Moreover, we give the evaluationof F -polynomials at a tropical semifield, see Lemma 3.7. ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 5
A recursive formula for Hernandez-Leclerc modules.
We give a new recursiveformula for Hernandez-Leclerc modules by an induction on the length of the highest ℓ -weight monomials of Hernandez-Leclerc modules.By Brito and Chari’s results [1, Theorem 1, Corollary in Section 1.3], the class of eachHernandez-Leclerc module corresponds to a cluster variable x [ α i,j ], where j − Theorem 1.3 (Theorem 4.1) . Let j ∈ I be a source or sink vertex and j > i ∈ I . Then x [ α i,j ] x [ α j +1 ,j +1 ] = x [ α i,j +1 ]+ x [ α i, max { i − ,j • − } ] − δ i,j • x ′ min { , (1 − δ j • ,i • ) d j •− + δ j • ,i } max { i,j • } x [ − α j +2 ] d j +1 x ′ − d j +1 j +2 , (1.1) where δ i,j is the Kronecker symbol, d j = δ j,j ⋄ , and j • is the maximum source or sink vertexstrictly less than j in Q ξ . Comparing with a recursive formula given by Brito and Chari in [1, Proposition inSection 1.5], our recursive formula needs one more step of mutation for a height function ξ such that j is source or sink, see Remark 4.2 (2).The formula (1.1) provides a possibility of a combinatorial path formula in which weallow overlapping paths, generalizing Mukhin and Young’s path formula for snake modules[29, 30].The content of this paper is outlined as follows. In Section 2, we recall some back-ground on cluster algebras of geometric type, Hernandez-Leclerc modules, snake graphs,and F -polynomials. In Section 3, we characterize the highest ℓ -weight monomials ofHernandez-Leclerc modules and give a q -character formula in term of perfect matchingsof snake graphs. In Section 4, a new recursive formula for q -characters of Hernandez-Leclerc modules is introduced. 2. Preliminary
Cluster algebras.
We recall the definition of cluster algebras [18, 20].Let F be the field of rational functions in n independent variables over QP , where P is a semifield. For u , u , . . . , u r in P , we denote by F | P ( u , . . . , u r ) the evaluation of asubtraction-free rational expression F at u , . . . , u r .A tropical semifiled ( P , ⊕ , · ), where P = Trop( u , u , . . . , u r ), is an abelian multiplicativegroup freely generated by u , u , . . . , u r , the (auxiliary) addition ⊕ is defined by Y j u a j j ⊕ Y j u b j j = Y j u min( a j ,b j ) j . An important choice for coefficients or frozen variables in a cluster algebra is the tropicalsemifield, in this case, the corresponding cluster algebra is said to be of geometric type .In this paper, we pay attention to skew-symmetric cluster algebras of geometric type.
BING DUAN, JIAN-RONG LI AND YAN-FENG LUO
Let Q be a finite quiver without loops or 2-cycles. Assume without loss of generalitythat the vertex set of Q is { , . . . , m } , among vertices 1 , . . . , n ( n ≤ m ) are mutablevertices and vertices n + 1 , . . . , m are frozen vertices. Mutating Q at a mutable vertex k ,one obtains a new quiver µ k ( Q ) defined as follows:(i) add a new arrow i → j for each pair of arrows i → k → j , excluding that both i and j are frozen vertices;(ii) reverse the orientation of each arrow incident to k ; and(iii) remove all maximal pairwise disjoint 2-cycles. Definition 2.1. (Seeds) A seed in F is a pair ( e x , Q ), where(i) e x = { x , . . . , x n , x n +1 , . . . , x m } is a free generating set of F , called an extendedcluster . The subset x = { x , . . . , x n } is called a cluster and each element in x iscalled a cluster variable ; { x n +1 , . . . , x m } is a set of elements in P , called a coefficienttuple , among them each element is called a coefficient or frozen variable ; and(ii) Q is a quiver as above. Definition 2.2. (Seed mutations). Let ( e x , Q ) be a seed. The seed mutation µ k indirection k ∈ { , . . . , n } transforms ( e x , Q ) into the seed ( e x ′ , Q ′ ) defined as follows.(i) The extended cluster e x ′ = { x ′ , . . . , x ′ m } is defined by x ′ j = x j for j = k , whereas x ′ k is defined by the exchange relation x ′ k = Q i : i → k x i + Q j : k → j x j x k ;(ii) Q ′ = µ k ( Q ) defined as above.A cluster algebra is a ZP -subalgebra of F generated by all cluster variables obtainedfrom seed mutations.Cluster algebras of finite type are classified by the Dynkin diagrams [19], that is, there isan exchange matrix such that the Cartan counterpart of its principle part is one of Cartanmatrices of finite type. Under this correspondence, cluster variables in a cluster algebraof finite type are in bijection with almost positive roots in the root system associated tothe corresponding Cartan matrix. Theorem 2.3 ([19, Theorem 1.9]) . In a cluster algebra of finite type, the cluster variable x [ α ] for α = P i ∈ I a i α i is expressed in term of the initial cluster x as x [ α ] = P α ( x ) Q i ∈ I x a i i , where P α is a polynomial over ZP with nonzero constant term. In particular, x [ − α i ] = x i . In order to deal with coefficients or frozen variables, we need the following theoremfrom [20].
ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 7
Theorem 2.4 ([20, Theorem 3.7]) . Let A be a cluster algebra over an arbitrary semifield P , with a seed at an initial extended cluster { x , . . . , x n , y , . . . , y n } . Then the clustervariables in A can be expressed as follows: x ℓ = X ℓ | F ( x , . . . , x n , y , . . . , y n ) F ℓ | P ( y , . . . , y n ) , where X ℓ is the Laurent expression of x ℓ in the case of the principle coefficient, and F ℓ isthe specilization of X ℓ evaluating at x = · · · = x n = 1 , called F -polynomial. Category C ξ and Hernandez-Leclerc modules. Let I = { , , . . . , n } . Following[1], given a height function ξ : I → Z with | ξ ( i ) − ξ ( i + 1) | = 1 for 1 ≤ i ≤ n −
1, we oftenextend ξ to [0 , n + 1] by defining ξ (0) = ξ (2) and ξ ( n + 1) = ξ ( n − Q ξ with the vertex set I ∪ I ′ and arrows defined by i − i o o δ i,i ⋄ * * − δ i,i ⋄ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ i + 1 − δ i,i ⋄ h h i ′ O O ( i + 1) ′ if ξ ( i ) = ξ ( i + 1) + 1 and otherwise reverse all orientations, where i ⋄ is the minimuminteger ℓ ∈ [ i, n ] such that ξ ( ℓ ) = ξ ( ℓ + 2) if i < n and otherwise n ⋄ = n .A vertex i ∈ [1 , n ] is said to be a source or sink if i = n or ξ ( i ) = ξ ( i + 2), wherewe ignore all frozen vertices in Q ξ and require that 1 is a source or sink if and only if ξ (1) = ξ (3).Let j • = 0 for 1 ≤ j ≤ ⋄ and j • be the maximal source or sink of Q ξ satisfying j • < j for j > ⋄ . For k ≥ k = ( k + 1)(1 − δ k,k ⋄ ) + ( k • + 1) δ k,k ⋄ . Example 2.5. In A , let ξ (1 , , , , , , , ,
9) = ( − , − , − , − , − , − , − , − , − I ξ -4 -5 -6 -5 -4 -3 -4 -5 -6 i ⋄ i • i BING DUAN, JIAN-RONG LI AND YAN-FENG LUO
By definition, Q ξ is the following quiver1 (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) o o (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o / / (cid:15) (cid:15) ′ O O ′ O O ′ ′ _ _ ❄❄❄❄❄❄❄❄ ′ _ _ ❄❄❄❄❄❄❄❄ ′ O O ′ O O ′ O O ′ . Let P = Trop( x ′ j : j ∈ I ) be the tropical semifield generated by x ′ j , j ∈ I . Denote by A ( e x , Q ξ ) the cluster algebra with an initial seed ( e x , Q ξ ), where e x = { x , . . . , x n , x ′ , . . . , x ′ n } ,x , . . . , x n are cluster variables and x ′ , . . . , x ′ n are coefficients.In this paper, we fix a ∈ C × . For simplicity of notation, we write Y i,r for Y i,aq r for i ∈ I and r ∈ Z . Let P + be the free abelian monoid generated by variables Y i,ξ ( i ) ± for i ∈ I . Let C ξ be the full subcategory of C consisting of objects all of whose Jordan-H¨olderconstituents are indexed by elements of P + . Simple modules in C ξ are of the form L ( m ),where m ∈ P + and m is called the highest ℓ -weight of L ( m ). The elements in P + arecalled dominant monomials .Following [1], for 1 ≤ i < j ≤ n , let i < · · · < i k − be an ordered enumeration of thesubset { p : i < p < j | ξ ( p −
1) = ξ ( p + 1) } , and i = i , i k = j . Define an element ω ( i, j ) ∈ P + by ω ( i, j ) = Y i ,a Y i ,a · · · Y i k ,a k , where a = ξ ( i ) ± ξ ( i + 1) = ξ ( i ) ∓ a ℓ = ξ ( i ℓ ) ± ξ ( i ℓ ) = ξ ( i ℓ − ± ℓ ≥ Pr ξ = { Y i,ξ ( i ) ± | i ∈ I } ∪ { ω ( i, j ) | i, j ∈ I, i = j } and f = { f i = Y i,ξ ( i ) − Y i,ξ ( i )+1 | i ∈ I } . A simple module L ( m ) for m ∈ Pr ξ ∪ f is called an Hernandez-Leclerc module byBrito and Chari [1], which are precisely all the prime objects in this category. The simplemodules L ( f i ), i ∈ I , are Kirillov-Reshitikhin modules and L ( Y i,r ), i ∈ I , are fundamentalmodules. We are interested in L ( ω ( i, j )) for 1 ≤ i < j ≤ n .Following Fomin and Zelevinsky’s result [19], the set of cluster variables in A ( e x , Q ξ )is in bijection with the set Φ ≥− of almost positive roots in the root system of type A n .Let { α i | i ∈ I } be a set of simple roots of type A n and let α i,j = α i + · · · + α j for1 ≤ i ≤ j ≤ n . Denote all cluster variables and coefficients by { x i := x [ − α i ] , x [ α i,j ] , x ′ i | ≤ i ≤ j ≤ n } . Following [1], the cluster variable x [ α i,j ] is obtained by mutating the sequence i, i +1 , . . . , j at the initial cluster { x i | ≤ i ≤ n } .Let K ( ξ ) be the Grothendieck ring of C ξ . Denote by [ M ] ∈ K ( ξ ) the equivalence classof M ∈ C ξ . Brito and Chari proved that C ξ is a monoidal category of the cluster algebra A ( e x , Q ξ ). ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 9
Theorem 2.6. [1, Theorem 1, Corollary in Section 1.3]
Let ξ : I → Z be a height function.Then there is an isomorphism of rings ι : A ( e x , Q ξ ) → K ( ξ ) such that ι ( x [ − α i ]) = [ L ( Y i,ξ ( i +1) )] , ι ( x ′ i ) = [ L ( Y i,ξ ( i ) − Y i,ξ ( i )+1 )] ,ι ( x [ α i,i ⋄ ]) = [ L ( Y i,ξ ( i +1) ± )] , ξ ( i ) = ξ ( i + 1) ± ,ι ( x [ α i,k ]) = [ ω ( i, k )] , k = i ⋄ ,ι ( x ′ i x [ α ]) = [ x ′ i ω ] , i ∈ I, α ∈ Φ ≥− , ω = ι ( x [ α ]) . Moreover, ι sends a cluster variable (respectively, cluster monomial) to a real prime simpleobject (respectively, real simple object) of C ξ . In particular, C ξ is a monoidal categoricationof A ( e x , Q ξ ) . The proof given by Brito and Chari in [1] used the fact that the set of cluster monomialsforms a linear basis of cluster algebras of finite type [12, 15].2.3.
Snake graphs.
Fix an orthonormal basis of the plane R . A tile G is a square offixed side-length with four vertices and four edges in the plane whose sides are parallel ororthogonal to the chosen basis. Following [7], a snake graph is a connected graph consistingof finitely many tiles G , G , . . . , G d with d ≥
1, such that for each i = 1 , . . . , d − G i and G i +1 share exactly an edge e i and the edge is either the north edge of G i and the south edge of G i +1 or the east edge of G i and the west edge of G i +1 .(ii) G i and G j have no edge in common whenever | i − j | ≥ G i and G j are disjoint whenever | i − j | ≥ G = ( G , G , . . . , G d ) be a snake graph with tiles G , G , . . . , G d . The d − e , e , . . . , e d − are called interior edges of G and the rest of edges are called boundaryedges . Denote by S G (respectively, W G ) the south (respectively, west) edge of the first tileof G and denote by G N (respectively, G E ) the north (respectively, east) edge of the lasttile of G .A snake graph G is called straight if all its tiles lie in one column or one row, and asnake graph is called zigzag if no three consecutive tiles are straight [9, 10].A perfect matching P of a snake graph G is a subset of the set of edges of G such thateach vertex of G is exactly in one edge in P . Denote by Match( G ) the set of all perfectmatchings of G . A snake graph G has precisely two perfect matchings, called the minimalmatching P − and the maximal matching P + of G , which contain only boundary edges.A snake graph is called a labeled snake graph if each edge and each tile in the snakegraph carries a label or weight [8]. For our snake graphs, these labels are cluster variables.Without confusion, we still use G to denote a labeled snake graph. Assume that the edgesof a perfect matching P of G are labeled by v , v , . . . , v r . Following [31], one defines the weight monomial x ( P ) of P by x ( P ) = Q ri =1 x v i . Given a snake graph G , let e = S G and choose an edge e d ∈ {G N , G E } . In [9], Canakciand Schiffler defined a sign function f : { e , e , e , . . . , e d − , e d } → {±} , such that on every tile in G the north edge and the west edge have the same sign, the southedge and the east edge have the same sign, and the sign on the north edge is opposite tothe sign on the south edge.Let a i ∈ Z ≥ be the number of the sign in a maximal subsequence of constant signappearing in ( f ( e ) , f ( e ) , f ( e ) , . . . , f ( e d − ) , f ( e d )). As an illustrated example, the signfunction of the following snake graph is ( − , − , + , + , + , − , − , − , − , − ), and a = 2 , a =3 , a = 5. −−− −−− +++ +++ +++ −−− −−− −−− −−− −−− For a snake graph G with a sign function ( a , a , . . . , a n ), let ℓ i = P ij =1 a j for 1 ≤ i ≤ n and we agree that ℓ = 0. Following [9], one can define zigzag subsnake graphs H , . . . , H n of G as follows. Let H = ( G , G , . . . , G ℓ − ) ,... ... H i = ( G ℓ i − +1 , . . . , G ℓ i − ) ,... ... H n = ( G ℓ n − +1 , . . . , G d ) , where ( G j , . . . , G k ) is the zigzag subsnake graph with the tiles G j , G j +1 , . . . , G k if j ≤ k ,and ( G j +1 , . . . , G j ) is the single edge e j . The decomposition is in fact obtained by deletingthe sign-changed tiles.Following [33], once we choose the minimal perfect matching P − of a snake graph G ,then the minimal matching P − | H i of a subsnake graph H i is either the matching whichinherits from P − or the union of the matching which inherits from P − and a unique interioredge.For an arbitrary perfect matching P of G , the symmetric different P − ⊖ P is defined as P − ⊖ P = ( P − ∪ P ) \ ( P − ∩ P ) . ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 11
Lemma 2.7. [31, Lemma 4.8]
The set P − ⊖ P is the set of boundary edges of a (possiblydisconnected) subgraph G P of G , which is a union of cycles. These cycles enclose a set ofties ∪ j ∈ J G i j , where J is a finite index set. From now on, assume that the label of a tile G i is τ i in a labeled snake graph G .Following [31, Definition 4.9], with the notation above, the height monomial y ( P ) of aperfect matching P of G is defined by y ( P ) = Y j ∈ J y τ ij . Continued fractions and F -polynomials. Following [9, 33], a finite continuedfraction [ a , a , . . . , a n ] := a + 1 a + 1 . . . + 1 a n is said to be positive if each a i is a positive integer. Denote by N [ a , a , . . . , a n ] thenumerator of the continued fraction. Then N [ a , a , . . . , a n ] is computed by the recursion N [ a , a , . . . , a n ] = a n N [ a , a , . . . , a n − ] + N [ a , a , . . . , a n − ] , where N [ a ] = a and N [ a , a ] = a a + 1. We refer the reader to [34, Proposition 2.1]for more properties of continued fractions.For a positive continued fraction [ a , a , . . . , a n ], let ℓ i = P is =1 a s and we agree ℓ = 0.In [9], Canakci and Schiffler established a bijection between snake graphs and positivecontinued fractions via the sign function of a snake graph. Fix a positive continuedfraction [ a , a , . . . , a n ], the corresponding snake graph G [ a , a , . . . , a n ] consists of tiles G , G , . . . , G ℓ n − and has the sign function of the following form( − , . . . , − | {z } a , + , . . . , + | {z } a , − , . . . , − | {z } a , . . . , ± , . . . , ± | {z } a n ) . The snake graph G [ a , a , . . . , a n ] has N [ a , a , . . . , a n ] many perfect matchings [9, Theo-rem 3.4].For simplicity, let Q jℓ = i x ℓ = 1 for j < i . The following results from [33] will play animportant role in the subsequent proof. In [33] Rabideau required that S G ∈ P − for alabeled snake graph G . Lemma 2.8. [33, Lemma 3.3]
The F -polynomial associated to a zigzag snake graph ( G , G , . . . , G d ) (the label of G j is τ j ) is d X k =0 k Y j =1 y τ j or d +1 X k =1 d Y j = k y τ j , depending on whether the minimal perfect matching P − contains a pair of opposite bound-ary edges of G . In [33, Definition 3.1], Rabideau defined a continued fraction of Laurent polynomial[ L , L , . . . , L n ], where each L i = ϕ i C i and C i = ℓ i − Q j =1 y τ j if i is odd , ℓ i − Q j =1 y − τ j if i is even , ϕ i = ℓ i − P k = ℓ i − k Q j = ℓ i − +1 y τ j if i is odd , ℓ i P k = ℓ i − +1 ℓ i − Q j = k y τ j if i is even . Here ϕ i is in fact the F -polynomial associated to a certain zigzag snake graph. Note thatwe revise the subscripts of the letter y , because we agree that the label of G j is τ j . Theorem 2.9. [33, Theorem 3.4]
The F -polynomial associated to the snake graph of thepositive continued fraction [ a , a , . . . , a n ] , a > , is given by the following equation F ( G [ a , a , . . . , a n ]) = ( N [ L , L , . . . , L n ] if n is odd ,C − n N [ L , L , . . . , L n ] if n is even , where N [ L , L , . . . , L n ] is defined by the recursion N [ L , L , . . . , L n ] = L n N [ L , L , . . . , L n − ] + N [ L , L , . . . , L n − ] where N [ L ] = L and N [ L , L ] = L L + 1 . q -characters of Hernandez-Leclerc modules and snake graphs In this section, we give a formula for q -characters of Hernandez-Leclerc modules usingsnake graphs, which involves an explicit formula for F -polynomials.3.1. A combinatorial description of Hernandez-Leclerc modules.
We recall thedefinition of Hernandez-Leclerc modules in [1]. Let I = { , , . . . , n } . Recall that C ξ isthe full subcategory of C consisting of objects all of whose Jordan-H¨older constituents areindexed by elements of P + , see Section 2.2. Brito and Chari proved that C ξ is a monoidalcategorification of the cluster algebra A ( e x , Q ξ ). They call these modules in C ξ , whichcorrespond to cluster variables and frozen variables, Hernandez-Leclerc modules.From now on, we call an Hernandez-Leclerc module an HL-module for simplicity. ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 13
Theorem 3.1.
An HL-module corresponding to a cluster variable is a simple U q ( b g ) -module with the highest ℓ -weight monomial Y i ,a Y i ,a · · · Y i k ,a k , (3.1) where k ∈ Z ≥ , i j ∈ I , a j ∈ Z for j = 1 , , . . . , k , and (i) i < i < · · · < i k , (ii) ( a j − a j − )( a j +1 − a j ) < for ≤ j ≤ k − , (iii) | a j − a j − | = i j − i j − + 2 for ≤ j ≤ k .Proof. In [1], the definition of HL-modules depends on the choice of a map ξ : I → Z .We will prove that every simple U q ( b g )-module with the highest ℓ -weight monomial (3.1)determines a map ξ : I → Z and hence it is an HL-module in the sense of Brito andChari. Indeed, let m = Y i ,a Y i ,a · · · Y i k ,a k for i < i < · · · < i k . Let i = i , j = i k . Wesplit the proof into the following two cases. Case 1 . If a < a , we define ξ ( i ) = a + 1, ξ ( i + 1) = a + 2 , . . . , ξ ( i ) = a + i − i + 1.Condition (iii) implies that ξ ( i ) = a + i − i + 1 = a −
1. Using Condition (ii) and thesame argument as before, we define a map ξ : [ i, j ] → Z by ξ ( x ) = ( a ℓ − + x − i ℓ − + 1 , if x ∈ [ i ℓ − , i ℓ ]; a ℓ − x + i ℓ − , if x ∈ [ i ℓ , i ℓ +1 ].We extend ξ to the domain [1 , n ] subject to ( ξ ( x ) − ξ ( x + 1) = 1 if 1 ≤ x ≤ i − ,ξ ( x + 1) = ξ ( x −
1) if j ≤ x ≤ n − . So the vertices ( j − , j, . . . , n are sources or sinks. It follows from Theorem 2.6 that ι ( x [ α i,j ]) = ( [ L ( Y i,a )] , if j = i ⋄ ,[ ω ( i, j )] = [ L ( Y i ,a Y i ,a . . . Y i k ,a k )] if j = i ⋄ ,which is an HL-module in the sense of Brito and Chari. Case 2 . If a > a , we define ξ ( i ) = a − ξ ( i + 1) = a − , . . . , ξ ( i ) = a − i + i − ξ ( i ) = a − i + i − a + 1. Using Condition (ii) and thesame argument as before, we define a map ξ : [ i, j ] → Z by ξ ( x ) = ( a ℓ − − x + i ℓ − − , i ∈ [ i ℓ − , i ℓ ]; a ℓ + x − i ℓ + 1 , i ∈ [ i ℓ , i ℓ +1 ].We extend ξ to the domain [1 , n ] subject to ( ξ ( x + 1) − ξ ( x ) = 1 if 1 ≤ x ≤ i − ,ξ ( x + 1) = ξ ( x −
1) if j ≤ x ≤ n − . So the vertices ( j − , j, . . . , n are sources or sinks. It follows from Theorem 2.6 that ι ( x [ α i,j ]) = ( [ L ( Y i,a )] , if j = i ⋄ ,[ ω ( i, j )] = [ L ( Y i ,a Y i ,a . . . Y i k ,a k )] if j = i ⋄ ,which is an HL-module in the sense of Brito and Chari.Conversely, for ω i,ξ i ± ∈ Pr ξ , obviously, it can be written into the form (3.1). For any ω i,j = ω i ,a . . . ω i k ,a k ∈ Pr ξ , by the definition of ω i,j , we have i < i < . . . < i k , and the function ξ must be a strictly increasing height function or a strictly decreasingheight function on these intervals [ i , i ] , . . . , [ i k − , i k ], and the strictly increasing intervalsand the strictly decreasing intervals appear alternatively.Suppose that ξ is a strictly increasing (respectively, decreasing) function on the interval[ i ℓ , i ℓ +1 ] for a certain 1 ≤ ℓ ≤ k −
1. Then a ℓ = ξ ( i ℓ ) − , a ℓ +1 = ξ ( i ℓ +1 ) + 1(respectively, a ℓ = ξ ( i ℓ ) + 1 , a ℓ +1 = ξ ( i ℓ +1 ) − . Since ξ is a strictly decreasing (respectively, increasing) function on the interval [ i ℓ − , i ℓ ],we have ( a ℓ − a ℓ − )( a ℓ +1 − a ℓ ) = ( ξ ( i ℓ ) − ξ ( i ℓ − ) − ξ ( i ℓ +1 ) − ξ ( i ℓ ) + 2) < , and | a ℓ +1 − a ℓ | = | ξ ( i ℓ +1 ) − ξ ( i ℓ ) + 2 | = | ( i ℓ +1 − i ℓ ) + 2 | = i ℓ +1 − i ℓ + 2 , | a ℓ − a ℓ − | = | ξ ( i ℓ ) − ξ ( i ℓ − ) − | = | ξ ( i ℓ − ) + 2 − ξ ( i ℓ ) | = i ℓ − i ℓ − + 2 . Similarly, it holds for the case that ξ is a strictly decreasing (respectively, increasing)function on the interval [ i ℓ , i ℓ +1 ].Therefore, every HL-module has its highest ℓ -weight monomial of the form (3.1). (cid:3) Remark 3.2.
The same HL-module may correspond to different height functions ξ , evenin the same type.3.2. Hernandez-Leclerc modules and snake graphs.
In a cluster algebra of finitetype from a surface without punctures, arcs in an initial triangulation of the surface corre-spond to initial variables which are parameterized by negative simple roots [19, Theorem1.9]. The arc crossing initial arcs − α i , − α i +1 , . . . , − α j corresponds to the cluster variableparameterized by α i,j , where i ≤ j . Fix a height function ξ , we shall construct a uniquelabeled snake graph for a non-initial HL-module. In general we have the relationshipsshown in Figure 1. ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 15
From [17], also see [35, Chapter 3], it follows that up to rotation a triangulation of asurface determines a quiver, and the quiver completely reflects the configuration of arcsin the triangulation.Let ξ be a height function. Denote by Q a connected full subquiver of Q ξ , where weignore the frozen vertices. Assume without loss of generality that the set of vertices in Q is { i, i + 1 , . . . , j } from left to right in order. We define the snake graph G = ( G i , G i +1 , . . . , G j )associated to Q as follows. The first two tiles are placed in the same horizontal line, thatis, the east edge of G i and the west edge of G i +1 are the same. For each i + 2 ≤ ℓ ≤ j , thetile G ℓ is placed such that ( G ℓ − , G ℓ − , G ℓ ) is straight if the ( ℓ − G ℓ − , G ℓ − , G ℓ ) is zigzag. From now on, we use a digit “1” todenote the cluster variable x , and so on. Label each tile G ℓ by ℓ in the interior of thetile. For any two consecutive tiles ( G ℓ , G ℓ +1 ), each edge is labeled near the edge obeyingthe following rules.(i) If G ℓ and G ℓ +1 share exactly an edge e ℓ and the edge is the east edge of G ℓ andthe west edge of G ℓ +1 , then the north edge of G ℓ is labeled by ( ℓ + 1) and thesouth edge of G ℓ +1 is labeled by ℓ .(ii) If G ℓ and G ℓ +1 share exactly an edge e ℓ and the edge is the north edge of G ℓ andthe south edge of G ℓ +1 , then the east edge of G ℓ is labeled by ( ℓ + 1) and the westedge of G ℓ +1 is labeled by ℓ .We leave edges W G , S G , G N , and G E no label. In particular, every edge has no label fora snake graph consisting of a tile.The minimal (respectively, maximal) matching P − (respectively, P + ) of G is chosen asfollows. If there is an arrow ( i + 1) → i in Q , P − is defined as the unique matching whichcontains only boundary edges and also contains W G , P + is the other matching with onlyboundary edges. Otherwise P − is defined as the unique matching which contains onlyboundary edges and contains S G , P + is the other matching with only boundary edges. If Q consists of a single vertex, then we agree that S G ∈ P − , W G ∈ P + . Definition 3.3.
The labeled snake graph associated to the HL-module parameterized by α i,j is the labeled snake graph determined by the connected full subquiver with vertex set { i, i + 1 , . . . , j } in Q ξ .The following example explains a snake graph associated to an HL-module. Example 3.4.
Continue our previous Example 2.5 and consider the cluster variable x [ α , ]and the corresponding HL-module L ( Y , − Y , − Y , − Y , − ). The full subquiver of Q ξ withvertex set { , , , , , , } is the following quiver (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) o o (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o ′ O O ′ O O ′ ′ _ _ ❄❄❄❄❄❄❄❄ ′ _ _ ❄❄❄❄❄❄❄❄ ′ O O ′ O O ′ . Following Definition 3.3, the labeled snake graph associated to L ( Y , − Y , − Y , − Y , − )is shown as follows. The same HL-module may correspond to different labeled snake graphs, even in thesame type. Consider a height function ξ (1 , , , , , , , ,
9) = ( − , − , − , − , − , − , − , − , − A . The quiver Q ξ is the following quiver.1 (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) o o (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o / / (cid:15) (cid:15) o o ′ O O ′ O O ′ ′ _ _ ❄❄❄❄❄❄❄❄ ′ _ _ ❄❄❄❄❄❄❄❄ ′ O O ′ O O ′ ′ O O . By Theorem 2.6, we have ι ( x [ α , ]) = [ L ( Y , − Y , − Y , − Y , − )]. Using Definition 3.3, thelabeled snake graph associated to L ( Y , − Y , − Y , − Y , − ) is shown as follows. Lemma 3.5.
If a height function ξ is fixed, then an HL-module corresponding to anon-initial cluster variable (excluding frozen variables) determines a unique labeled snakegraph.Proof. Since a height function ξ is fixed, by Theorem 2.6, we have a cluster algebraisomorphism ι : A ( e x , Q ξ ) → K ( ξ ) and HL-modules corresponding to non-initial clustervariables are in bijection with almost positive roots. The result follows from Definition3.3. (cid:3) ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 17 q -characters of Hernandez-Leclerc modules in term of perfect matchingsof snake graphs. To our purpose, we first give the dual version of Theorem 2.9 bythe following theorem whose proof is completely similar to one given by Rabideau in[33, Theorem 3.4] except using the following formula from [7, 8]. F ( G [ a , a , . . . , a n ]) = y F ( G [ a , a , . . . , a n − ]) F ( H n ) + y F ( G [ a , a , . . . , a n − ]) , (3.2)where the variables y and y are defined as follows, y = 1. y = ( n is odd ,y τ ℓn − if n is even , y = (Q ℓ n − j = ℓ n − y τ j if n is odd , n is even . Theorem 3.6.
Suppose that in G [ a , a , . . . , a n ] ( a > the minimal perfect matchingcontains the west edge of the first tile. Then the F -polynomial associated to G [ a , a , . . . , a n ] is given by the following equation F ( G [ a , a , . . . , a n ]) = ( C − n N [ L , L , . . . , L n ] if n is odd , N [ L , L , . . . , L n ] if n is even , where each L i = ϕ i C i and C i = ℓ i − Q j = ℓ y − τ j if i is odd , ℓ i − Q j = ℓ y τ j if i is even , ϕ i = ℓ i P k = ℓ i − +1 ℓ i − Q j = k y τ j if i is odd , ℓ i − P k = ℓ i − k Q j = ℓ i − +1 y τ j if i is even . Proof.
We proceed by an induction on n . Consider the case that n = 1, using Lemma 2.8and the fact that the number of tiles in G [ a ] is a −
1, we have F ( G [ a ]) = a X k =1 a − Y j = k y τ j = ϕ = C − C ϕ = C − L . For n = 2, using (3.2) and the fact that ℓ = a and C = 1, we have F ( G [ a , a ]) = y F ( G [ a ]) F ( H ) + y = y τ a ( C − L ) ϕ + 1= L L + 1 = N [ L , L ] . Assume that n is odd and our formula holds for m < n . The minimal matching of H n is the matching which inherits from P − , shown in Figure 2, so F ( H n ) = ϕ n . Then by(3.2) F ( G [ a , a , . . . , a n ]) = F ( G [ a , a , . . . , a n − ]) F ( H n ) + Y ℓ n − j = ℓ n − y τ j F ( G [ a , a , . . . , a n − ])= N [ L , L , . . . , L n − ] ϕ n + Y ℓ n − j = ℓ n − y τ j C − n − N [ L , L , . . . , L n − ]= N [ L , L , . . . , L n − ] ϕ n + C − n N [ L , L , . . . , L n − ]= C − n ( L n N [ L , L , . . . , L n − ] + N [ L , L , . . . , L n − ])= C − n N [ L , L , . . . , L n ] . ......... ......... Figure 2.
The minimal perfect matching of H n . The dashed tile is asign-changed tile.Assume that n is even and our formula holds for m < n . The minimal matching of H n is a union of the matching which inherits from P − and e ℓ n − , shown in Figure 3, so F ( H n ) = ϕ n . Then by (3.2) F ( G [ a , a , . . . , a n ]) = y τ ℓn − F ( G [ a , a , . . . , a n − ]) F ( H n ) + F ( G [ a , a , . . . , a n − ])= y τ ℓn − C − n − N [ L , L , . . . , L n − ] ϕ n + N [ L , L , . . . , L n − ]= N [ L , L , . . . , L n − ] C n ϕ n + N [ L , L , . . . , L n − ]= L n N [ L , L , . . . , L n − ] + N [ L , L , . . . , L n − ]= N [ L , L , . . . , L n ] . ......... ......... Figure 3.
The minimal perfect matching of H n . The dashed tile is asign-changed tile. (cid:3) ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 19
In practice, we always assume that a > a , a , . . . , a n ].Otherwise, we take the rotation of a snake graph by 180 degrees or the flips at thelines y = − x , it follows from [11, Proposition 3.1 (b) and (c)] that G [ a , a , . . . , a n ] ∼ = G [ a n , . . . , a , a ]. In addition, [ a , a , . . . , a n − ,
1] = [ a , a , . . . , a n − + 1]. Lemma 3.7.
Let G be the snake graph associated to the HL-module parameterized by α i,j and F = P P ∈ Match( G ) y ( P ) be its F-polynomial. Then F | P = y ( P + ) | P = ( y i y i +1 · · · y j ) | P . Proof.
Following [31], all perfect matchings in Match( G ) form a poset with the minimalperfect matching P − and the maximal perfect matching P + . Recall that in [33] a tile G can be turned if two of its edges are in a perfect matching P . Let P ′ be the perfectmatching obtained by replacing the two edges of G in P with the other two edges of G . We say that P ′ is obtained from P by turning the tile G . Every perfect matching isobtained from P − or P + by turning a sequence of tiles.The height monomial y ( P ′ ) of P ′ is defined recursively by y ( P − ) = 1 and if P ′ is above P and obtained by turning a tile G ℓ then y ( P ′ ) = y ℓ y ( P ). In the following, we will provethat for each step of turning a tile, y ( P ) | P ≤ y ( P ′ ) | P , meaning that y ( P ′ ) | P contains allpossible factors with multiplicities appearing in y ( P ) | P . As a conclusion, F | P = y ( P + ) | P .In our setting, if P is above P − and obtained by turning a tile G ℓ , then y ( P ) = y ℓ andeither ℓ
6∈ { i, j } is a source or ℓ ∈ { i, j } such that ℓ → l for a certain l ∈ I . In general, if P ′ is above P and obtained by turning a tile G ℓ then y ( P ′ ) = y ℓ y ( P ), and at least one ofthe following two cases occur:(i) either ℓ
6∈ { i, j } is a source or ℓ ∈ { i, j } such that ℓ → l for a certain l ∈ I , where y l is not a factor in y ( P ),(ii) there exists a certain l ∈ I (not unique) such that l → ℓ in Q ξ , where y l is a factorin y ( P ).For the case (i), y ℓ = x ′ ℓ or all possible factors in the denominator of y ℓ cannot becancelled by y ( P ), and hence y ( P ) | P ≤ y ( P ′ ) | P . For the case (ii), all possible arrows nearthe vertex ℓ are shown as follows. l / / (cid:15) (cid:15) ℓ / / (cid:15) (cid:15) k (cid:15) (cid:15) l ′ ℓ ^ ^ ❂❂❂❂❂❂❂❂ k ′ ^ ^ ❃❃❃❃❃❃❃ , l / / (cid:15) (cid:15) ℓ (cid:15) (cid:15) k o o l ′ ℓ ′ ^ ^ ❃❃❃❃❃❃❃❃ k ′ O O , l / / ℓ / / (cid:15) (cid:15) k (cid:15) (cid:15) l ′ O O ℓ ′ k ′ _ _ ❄❄❄❄❄❄❄❄ , l / / ℓ (cid:15) (cid:15) k o o l ′ O O ℓ ′ k ′ O O ,k (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ ℓ o o (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ l o o k ′ O O ℓ ′ O O l ′ O O , k (cid:15) (cid:15) ℓ o o (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ l o o k ′ ℓ ′ O O l ′ O O , k (cid:15) (cid:15) / / ℓ (cid:15) (cid:15) l o o k ′ ℓ ′ _ _ ❄❄❄❄❄❄❄❄ l ′ O O , k / / ℓ (cid:15) (cid:15) l o o k ′ O O ℓ ′ l ′ O O . So checking case by case from up to down and from left to right y ( P ′ ) | P = ( x ′ k x ′ ℓ y ( P )) | P = y ( P ) | P , because x ′ ℓ is cancelled by the numerator in y l ,y ( P ′ ) | P = ( 1 x ′ ℓ y ( P )) | P = y ( P ) | P , because x ′ ℓ is cancelled by the numerator in y l ,y ( P ′ ) | P = ( x ′ k x ′ ℓ y ( P )) | P = x ′− ℓ y ( P ) | P ,y ( P ′ ) | P = ( 1 x ′ ℓ y ( P )) | P = x ′− ℓ y ( P ) | P ,y ( P ′ ) | P = ( x ′ ℓ x ′ l y ( P )) | P = y ( P ) | P , because x ′ l is cancelled by the numerator in y l ,y ( P ′ ) | P = ( x ′ ℓ x ′ l y ( P )) | P = y ( P ) | P , because x ′ l is cancelled by the numerator in y l ,y ( P ′ ) | P = ( 1 x ′ ℓ y ( P )) | P = y ( P ) | P , because x ′ ℓ is cancelled by the numerator in y k ,y ( P ′ ) | P = ( 1 x ′ ℓ y ( P )) | P = x ′− ℓ y ( P ) | P , the seventh equation is because both G l and G k have been turned from P − before turningthe tile G ℓ . (cid:3) Theorem 3.8.
Let C ξ be the full subcategory introduced by Brito and Chari [1] and G bethe labeled snake graph associated to x [ α i,j ] . Then x [ α i,j ] = 1 Q jℓ = i x ℓ P P ∈ Match( G ) x ( P ) y ( P ) L P ∈ Match( G ) y ( P ) ! , where the sign ⊕ appearing in the denominator refers to the (auxiliary) addition in atropical semifield P = Trop( u , u , . . . , u r ) ( r ∈ Z ≥ ) and y ℓ = Q j → ℓ u j Q ℓ → j u − j is aLaurent monomial in r variables u , u , . . . , u r for i ≤ ℓ ≤ j . In particular, χ q ( ι ( x [ α i,j ])) = 1 Q jℓ = i χ q ( ι ( x ℓ )) χ q ι ( P P ∈ Match( G ) x ( P ) y ( P ) L P ∈ Match( G ) y ( P ) ) ! . Proof.
The proof of the first part follows [31, Theorem 4.10] and [20, Theorem 3.7] (seeTheorem 2.4). The second one follows from [1, Theorem 1, Corollary in Section 1.3] (seeTheorem 2.6). (cid:3)
We explain Theorem 3.8 by the following example.
ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 21 x x x x x x x x x y x x x x x y y x x x y x x y y x x x y y y x x x x y y x x x y y x x x y y y x x x y y y x x y y y x x y y y y x x x x y y y y x x x y y y y x x x y y y y y x x x x y y y y x x x y y y y y x x x y y y y y y x x x x y y y x x x y y y y x x x y y y y y x x x x x y y y y y y x x x x x y y y y y y y Table 1.
Perfect matchings of the labeled snake graph associated to x [ α , ]. Example 3.9.
Continue our previous Examples 3.4. Since x , x are not mutated, our co-efficients are in Trop( x ′ , x ′ , x ′ , x ′ , x ′ , x ′ , x ′ , x ′ , x ′ , x , x ). By Theorem 3.8, we substitutevariables by y = x ′ x ′ , y = x ′ , y = x ′ x ′ , y = x ′ x ′ , y = 1 x ′ , y = x ′ x ′ , y = x ′ x x ′ . All possible perfect matchings of the labeled snake graph associated to the HL-module L ( Y , − Y , − Y , − Y , − ) are shown in Table 1. In Table 1 we list the monomial x ( P ) y ( P )in the below of P ∈ Match( G ). By Theorem 3.6, the associated F -polynomial is given by F ( y , y , y , y , y , y , y ) = F ( G [2 , , C − N [ L , L , L ] , where L = 1 + y , L = y (1 + y + y y ) , L = y − y − y − y − y − y − (1 + y + y y ) . By computation, we have F ( G [2 , , y ) y (1+ y + y y )(1+ y + y y )+ y y y y y y (1+ y )+(1+ y + y y ) . Using Lemma 3.7, we have F | P ( y , y , y , y , y , y , y ) = ( y y y y y y y ) | P = 1 x ′ x ′ . Therefore by Theorem 3.8, x [ α , ] = 1 x x x x x x x X P ∈ Match( G ) x ( P ) y ( P )( ⊕ P ∈ Match( G ) y ( P ))= x x ′ x ′ x x + x x x ′ x ′ x x x + x x ′ x ′ x x + x x ′ x ′ x ′ x x x x + x x x ′ x ′ x ′ x x x x x + x x ′ x ′ x ′ x x x x + x x ′ x ′ x ′ x x x + x x ′ x ′ x ′ x x x x + x x x ′ x ′ x ′ x x x x + x x x ′ x ′ x ′ x x x x + x x x ′ x ′ x ′ x x x x x + x x x x ′ x ′ x ′ x x x x x + x x ′ x ′ x ′ x x x + x x ′ x ′ x ′ x x x x + x x x ′ x ′ x ′ x x x x + x x ′ x ′ x ′ x x x + x x x ′ x ′ x ′ x x x x + x x x ′ x ′ x ′ x x x x + x ′ x ′ x ′ x x x + x x ′ x ′ x ′ x x x x + x x ′ x ′ x ′ x x x x + x x ′ x ′ x x + x x x ′ x ′ x x . Replacing x ℓ , x ′ ℓ by q -characters of the corresponding initial simple modules, we obtainthe q -character of L ( Y , − Y , − Y , − Y , − ).3.4. Relation with Brito and Chari’s results.
In [1, Proposition in Section 2.5], Britoand Chari gave a non-recursive formula of x [ α i,j ] using two sets Γ i,j and Γ ′ i,j . Recall thatin [1, Section 2.3], the set Γ i,j for i, j ∈ I is defined as follows. Γ i,j = { } if j < i andif i ≤ j then Γ i,j consists of (0 ,
1) sequences ε := ( ε i , . . . , ε j +1 ) with ( j − i + 2) lengthsubject to the following four conditions: ε i + · · · + ε i ⋄ ≤ ≤ ε i + · · · + ε i ⋄ +1 i ⋄ ≤ j, (3.3) ε m +1 + · · · + ε ( m +1) ⋄ ≤ ≤ ε m +1 + · · · + ε ( m +1) ⋄ +1 i ⋄ ≤ m = m ⋄ < j • , (3.4) ε max { i,j • +1 } + · · · + ε j ≤ , (3.5) ε j +1 = ( j = j ⋄ , − ( ε max { i,j • +1 } + · · · + ε j ) otherwise . (3.6) ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 23
The set Γ ′ i,j is determined by Γ i,j consisting of ( − , ,
1) sequences ε ′ := ( ε ′ i , . . . , ε ′ j +1 )[1, Section 2.4]:(1) if i • = m • or ε max { i, ( m • ) • +1 } + · · · + ε m • = 1, then ε ′ m = ( ( δ m,m ⋄ − ε m +1 − δ m,m ⋄ if ε max { i,m • +1 } + · · · + ε m = 0 ,δ m,m ⋄ − ( ε m + ε m +1 ) if ε max { i,m • +1 } + · · · + ε m = 1 , (2) if m • ≥ i and ε max { i, ( m • ) • +1 } + · · · + ε m • = 0, then ε ′ m = δ m,m ⋄ (1 − ε m +1 ), and(3) if j = j ⋄ , then ε ′ j +1 = 1 − ( ε max { i,j • +1 } + · · · + ε j ), otherwise ε ′ j +1 = ε max { i,j • +1 } + · · · + ε j .Following [1, Section 2.5], for any ε ∈ Γ i,j and the corresponding ε ′ ∈ Γ ′ i,j , one defines m εi,j = x (1 − ε i ) i − x ε ′ i i · · · x ε ′ j j x ε ′ j +1 j +1 , f εi,j = ( x ′ i ) ε i · · · ( x ′ j ) ε j ( x ′ j +1 ) (1 − δ j,j ⋄ ) ε j +1 . Example 3.10.
For x [ α , ] in Example 3.9, the sets Γ , and Γ ′ , , monomials m ε , and f ε , are listed in Table 2. The ( i + 1)-th row (1 ≤ i ≤
23) in Table 2 corresponds to the i -th perfect matching in Table 1 from up to down and from left to right.In general, we have the following theorem. Theorem 3.11.
Let G := G [ a , a , . . . , a n ] be the snake graph associated to x [ α i,j ] ( i ≤ j ) .Then | Γ i,j | = | Match( G ) | = N [ a , a , . . . , a n ] .Proof. By [1, Lemma in Section 2.3], we have | Γ i,j | = | Γ i,j − | + | Γ i,j • − | for j > i and | Γ i,j | = 1 for j < i .Recall that G has N [ a , a , . . . , a n ] many perfect matchings [9, Theorem 3.4] and N [ a , a , . . . , a n ] = N [ a , a , . . . , a n − ,
1] = N [ a , a , . . . , a n −
1] + N [ a , a , . . . , a n − ] . In the following, we prove that | Γ i,j | = N [ a , a , . . . , a n ] by an induction on j − i . If j = i , then N [2] = 2 and by the proof of [1, Proposition in Section 2.5], we haveΓ i,i = ( { (0 , , (1 , } if i = i ⋄ , { (0 , , (1 , } if i = i ⋄ , Γ ′ i,i = ( { ( − , , ( − , } if i = i ⋄ , { ( − , , ( − , } if i = i ⋄ . Assume that our formula holds for j − i ≤ k and | Γ i,i + k | = N [ a , a , . . . , a n ], where G [ a , a , . . . , a n ] is the snake graph associated to x [ α i,i + k ]. If i + k = ( i + k ) ⋄ , then | Γ i,i + k +1 | = | Γ i,i + k | + | Γ i, ( i + k +1) ⋄ − | = N [ a , a , . . . , a n ] + N [ a , a , . . . , a n − ]= N [ a , a , . . . , a n + 1] . In this case, G [ a , a , . . . , a n + 1] is the snake graph associated to x [ α i,i + k +1 ]. Γ , Γ ′ , m ε , f ε , (0 , , , , , , ,
1) (0 , − , , , , , − , x − x x − x ′ x ′ (0 , , , , , , ,
0) (0 , − , , , , − , − , x − x x − x − x x ′ x ′ (0 , , , , , , ,
0) (0 , − , , , , − , , x − x − x x ′ x ′ (0 , , , , , , ,
1) ( − , − , − , , , , − , x − x − x − x x − x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , − , − , , , − , − , x − x − x − x x − x − x x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , − , − , , , − , , x − x − x − x − x x ′ x ′ x ′ (1 , , , , , , ,
1) ( − , , − , , , , − , x − x − x x − x ′ x ′ x ′ (0 , , , , , , ,
1) ( − , , − , − , , , − , x − x − x − x x − x ′ x ′ x ′ (1 , , , , , , ,
1) ( − , , − , − , , , − , x − x x − x − x x − x ′ x ′ x ′ (1 , , , , , , ,
0) ( − , , − , , , − , − , x − x − x x − x − x x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , , − , − , , − , − , x − x − x − x x − x − x x ′ x ′ x ′ (1 , , , , , , ,
0) ( − , , − , − , , − , − , x − x x − x − x x − x − x x ′ x ′ x ′ (1 , , , , , , ,
0) ( − , , − , , , − , , x − x − x − x x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , , − , − , , − , , x − x − x − x − x x ′ x ′ x ′ (1 , , , , , , ,
0) ( − , , − , − , , − , , x − x x − x − x − x x ′ x ′ x ′ (1 , , , , , , ,
1) ( − , , , − , , , − , x − x x − x − x ′ x ′ x ′ (1 , , , , , , ,
0) ( − , , , − , , − , − , x − x x − x − x − x x ′ x ′ x ′ (1 , , , , , , ,
0) ( − , , , − , − , − , , x − x x − x − x − x x ′ x ′ x ′ (0 , , , , , , ,
1) ( − , , , − , , , − , x − x − x − x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , , , − , , − , − , x − x − x − x − x x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , , , − , − , − , , x − x − x − x − x x ′ x ′ x ′ (0 , , , , , , ,
0) ( − , , , , − , , , x − x − x x ′ x ′ (1 , , , , , , ,
0) ( − , , , , − , , , x − x x − x x ′ x ′ Table 2.
The sets Γ , and Γ ′ , , monomials m ε , and f ε , .If i + k = ( i + k ) ⋄ , then | Γ i,i + k +1 | = | Γ i,i + k | + | Γ i, ( i + k +1) ⋄ − | = N [ a , a , . . . , a n ] + | Γ i,i + k − | = N [ a , a , . . . , a n − ,
1] + N [ a , a , . . . , a n − N [ a , a , . . . , a n − , . In this case, G [ a , a , . . . , a n − ,
2] is the snake graph associated to x [ α i,i + k +1 ]. (cid:3) We refine Theorem 3.11 as follows.
Theorem 3.12.
Let G = ( G , . . . , G j − i +1 ) be the labeled snake graph associated to x [ α i,j ] .Then ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 25 |{ ( ε i , . . . , ε i ⋄ +1 ) | ( ε i , . . . , ε i ⋄ +1 , . . . ) ∈ Γ i,j }| = | Match( G , . . . , G i ⋄ − i +2 ) | , (3.7) |{ ( ε m +1 , . . . , ε ( m +1) ⋄ +1 ) | ( . . . , ε m +1 , . . . , ε ( m +1) ⋄ +1 , . . . ) ∈ Γ i,j }| = | Match( G m − i +2 , . . . , G ( m +1) ⋄ − i +2 ) | , (3.8) |{ ( ε max { i,j • +1 } , . . . , ε j +1 ) | ( . . . , ε max { i,j • +1 } , . . . , ε j +1 ) ∈ Γ i,j }| = | Match( G max { i,j • +1 }− i +1 , . . . , G j − i +1 ) | . (3.9) Proof.
Indeed, by (3.3), the set in the left hand side of (3.7) is equivalent to { ( ε i , . . . , ε i ⋄ +1 ) ∈ Z ( i ⋄ − i +2) ≥ | ( ε i + · · · + ε i ⋄ , ε i ⋄ +1 ) ∈ { (0 , , (1 , , (1 , }} , the cardinality of the set is 2( i ⋄ − i ) + 3; by (3.4), the set in the left hand side of (3.8) isequivalent to n ( ε m +1 , . . . , ε ( m +1) ⋄ +1 ) ∈ Z (( m +1) ⋄ − m +1) ≥ | ( ε m +1 + · · · + ε ( m +1) ⋄ , ε ( m +1) ⋄ +1 ) ∈ { (0 , , (1 , , (1 , }} , the cardinality of the set is 2(( m + 1) ⋄ − m ) + 1; by (3.5)–(3.6), the set in the left handside of (3.9) is equivalent to n ( ε max { i,j • +1 } , . . . , ε j +1 ) ∈ Z ( j − max { i,j • +1 } +2) ≥ | ( ε max { i,j • +1 } , . . . , ε j , ε j +1 ) ∈ { (0 , , (1 , }} , if j = j ⋄ , n ( ε max { i,j • +1 } , . . . , ε j +1 ) ∈ Z ( j − max { i,j • +1 } +2) ≥ | ( ε max { i,j • +1 } , . . . , ε j , ε j +1 ) ∈ { (0 , , (1 , } , if j = j ⋄ , the cardinality of the set is ( j − max { i, j • + 1 } + 2).In the other hand, the snake graph in (3.7) or (3.8) is a union of a zigzag snake graphand a tile, whereas the snake graph in (3.9) is a zigzag snake graph. A zigzag snake graphwith d tiles has ( d + 1) many perfect matchings. Finally, we apply the following formula N [ a,
2] = 2 a + 1 , where a is one less than the number of tiles in a snake graph. (cid:3) Remark 3.13.
Let G be the labeled snake graph associated to x [ α i,j ] and Γ i,j the setdefined in the beginning of this section. There is a partial order on the set Match( G ) [31].We expect that the following are true and they are equivalent.(1) There exsits a partial order on Γ i,j such that there is a bijection from Γ i,j toMatch( G ) preserving the order. (2) For any ε ∈ Γ i,j , there exists a unique perfect matching P ε ∈ Match( G ) such that m εi,j f εi,j = x ( P ε ) y ( P ε )( Q jℓ = i x ℓ ) y ( P + ) | P . The highest and lowest ℓ -weight monomials of Hernandez-Leclerc modules. In this subsection, we determine the highest and lowest ℓ -weight monomials in the q -character of an HL-module using perfect matchings of snake graphs. Lemma 3.14.
Let Q ξ be a quiver associated to a height function ξ . Assume that there isno source or sink vertex in the interval ( i, j ) . Then there exists a unique perfect matching P such that x ( P ) y ( P )( x i x i +1 . . . x j ) F | P ( y i , y i +1 , . . . , y j ) = x ′ i x (1 − δ j,j ⋄ ) j +1 x i or x ′ i x ′ i +1 x (1 − δ j,j ⋄ ) j +1 x i x i +1 . Proof. If ξ ( i ) = ξ ( i + 1) + 1, then there is a full subquiver of Q ξ , which is one of thefollowing quivers( i − i o o " " ❊❊❊❊❊❊❊❊❊❊ ( i + 1) o o ❋❋❋❋❋❋❋❋❋❋❋ . . . o o ●●●●●●●●●● ( j − o o " " ❋❋❋❋❋❋❋❋❋ j o o δ j,j ⋄ + + − δ j,j ⋄ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ( j + 1) − δ j,j ⋄ i i i ′ O O ( i + 1) ′ O O . . . O O ( j − ′ O O j ′ O O ( j + 1) ′ , (3.10)( i − i o o / / ( i + 1) / / (cid:15) (cid:15) . . . / / (cid:15) (cid:15) ( j − / / (cid:15) (cid:15) j (cid:15) (cid:15) − δ j,j ⋄ ( j + 1) δ j,j ⋄ u u i ′ O O ( i + 1) ′ . . . c c ❋❋❋❋❋❋❋❋❋❋❋ ( j − ′ c c ●●●●●●●●●● j ′ b b ❋❋❋❋❋❋❋❋❋ ( j + 1) ′ − δ j,j ⋄ g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ . (3.11)If ξ ( i ) = ξ ( i + 1) −
1, then there is a full subquiver of Q ξ , which is one of the followingquivers ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 27 ( i − / / i / / (cid:15) (cid:15) ( i + 1) / / (cid:15) (cid:15) . . . / / (cid:15) (cid:15) ( j − / / (cid:15) (cid:15) j (cid:15) (cid:15) − δ j,j ⋄ ( j + 1) δ j,j ⋄ u u i ′ ( i + 1) ′ b b ❊❊❊❊❊❊❊❊❊❊ . . . c c ❋❋❋❋❋❋❋❋❋❋❋ ( j − ′ c c ●●●●●●●●●● j ′ b b ❋❋❋❋❋❋❋❋❋ ( j + 1) ′ − δ j,j ⋄ g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ , (3.12)( i − / / i (cid:15) (cid:15) ( i + 1) o o ❋❋❋❋❋❋❋❋❋❋❋ . . . o o ●●●●●●●●●● ( j − o o " " ❋❋❋❋❋❋❋❋❋ j o o δ j,j ⋄ + + − δ j,j ⋄ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ( j + 1) − δ j,j ⋄ i i i ′ ( i + 1) ′ O O . . . O O ( j − ′ O O j ′ O O ( j + 1) ′ . (3.13)By our Definition 3.3, the labeled snake graph associated to ι ( x [ α i,j ]) is a zigzag labeledsnake graph G = ( G i , G i +1 , . . . , G j ). Let P − (respectively, P + ) be the minimal (respec-tively, maximal) perfect matching of G . Define another labeled perfect matching P of G as follows: P = P + for (3.10) , W G ∪ e ∪ P − | ( G\ G i ) for (3.11) ,P − for (3.12) , W G ∪ e ∪ P + | ( G\ G i ) for (3.13) . Then by Lemma 3.7, we have F | P ( y i , y i +1 , . . . , y j ) = x i − x δ j,j ⋄ j +1 x ′ (1 − δ j,j ⋄ ) j +1 for (3.10) , x i − x ′ i +1 x (1 − δ j,j ⋄ ) j +1 for (3.11) , x ′ i x (1 − δ j,j ⋄ ) j +1 for (3.12) , x ′ i x δ j,j ⋄ j +1 x ′ (1 − δ j,j ⋄ ) j +1 for (3.13) . By computation, we have x ( P ) x i x i +1 . . . x j = x i for (3.10) and (3.12) , x i x i +1 for (3.11) and (3.13) ,y ( P ) F | P ( y i , y i +1 , . . . , y j ) = ( x ′ i x (1 − δ j,j ⋄ ) j +1 for (3.10) and (3.12) ,x ′ i x ′ i +1 x (1 − δ j,j ⋄ ) j +1 for (3.11) and (3.13) . Therefore x ( P ) y ( P )( x i x i +1 . . . x j ) F | P ( y i , y i +1 , . . . , y j ) = x ′ i x (1 − δ j,j ⋄ ) j +1 x i for (3.10), (3.12) ,x ′ i x ′ i +1 x (1 − δ j,j ⋄ ) j +1 x i x i +1 for (3.11), (3.13) . Finally, the uniqueness of P follows from the uniqueness of the perfect matching withthe height monomial y ( P ), because the set of all perfect matchings in a zigzag snakegraph forms a total order, and a weight monomial does not affect the associated heightmonomial. (cid:3) Denote by G = ( G i , G i +1 , . . . , G j ) the snake graph with sign function ( a , a , . . . , a l )associated to the HL-module parameterized by α i,j . We have the following observation:The vertices ( i + ℓ − , ( i + ℓ − , . . . , ( i + ℓ l − −
1) are sources or sinks in Q ξ . Henceby Definition 3.3, these tiles G i + ℓ − , G i + ℓ − , . . . , G i + ℓ l − − are sign-changed tiles.Let H t = ( G i + ℓ t − , G i + ℓ t − +1 , . . . , G i + ℓ t − ) , ≤ t ≤ l, be all the (labeled) subsnake graphs in the decomposition of G . Let P − (respectively, P + ) be the minimal (respectively, maximal) perfect matching of G . Denote by P − | H t (respectively, P + | H t ) the minimal (respectively, maximal) perfect matching obtained byrestricting P − (respectively, P + ) to H t , where 1 ≤ t ≤ l . Define another (labeled) perfectmatching P of G as follows. If i is a source or sink in Q ξ , we define P | H = ( W G ∪ e ∪ P − | ( H \ G i ) if i is a source , W G ∪ e ∪ P + | ( H \ G i ) if i is a sink , and otherwise let P | H = ( P − | H if i → i ′ in Q ξ ,P + | H if i ′ → i in Q ξ . ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 29
For 1 < t ≤ l , let P | H t = ( P − | H t if ( i + ℓ t − ) → ( i + ℓ t − ) ′ in Q ξ ,P + | H t if ( i + ℓ t − ) ′ → ( i + ℓ t − ) in Q ξ . Define P as the gluing of P | H t for 1 ≤ t ≤ l and it is a perfect matching of G . Definition 3.15. If i is a source or sink in Q ξ , then we define the revised height monomial e y ( P | H ) = y i if i is a source ,y ( P | H ) if i is a sink and l = 1 ,y i + ℓ − y ( P | H ) if i is a sink and l > . Otherwise, for 1 ≤ t ≤ l we define the revised height monomial e y ( P | H t ) = i + ℓ t − ) → ( i + ℓ t − ) ′ in Q ξ ,y ( P | H t ) if ( i + ℓ t − ) ′ → ( i + ℓ t − ) in Q ξ and t = l,y i + ℓ t − y ( P | H t ) if ( i + ℓ t − ) ′ → ( i + ℓ t − ) in Q ξ and t = l. We immediately have the following lemma.
Lemma 3.16.
With the notation above, there exists a unique perfect matching P ∈ Match( G ) such that x ( P ) y ( P )( j Q s = i x s ) F | P ( y i , y i +1 , . . . , y j ) = x ′ i x ′ i +1 x ′ i + ℓ x ′ i + ℓ . . . x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 x i x i +1 x i + ℓ x i + ℓ . . . x i + ℓ l − if i is a source or sink ,x ′ i x ′ i + ℓ x ′ i + ℓ . . . x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 x i x i + ℓ x i + ℓ . . . x i + ℓ l − otherwise . Proof.
By the definition of P , if i is a source or sink in Q ξ , then we have x ( P | H ) = ℓ − Q s =2 x i + s if l = 1 , ℓ − Q s =2 x i + s if l > , e y ( P | H ) = y i if i is a source , ℓ − Q s =1 y i + s if i is a sink and l = 1 , ℓ − Q s =1 y i + s if i is a sink and l > , and otherwise for 1 ≤ t ≤ lx ( P | H t ) = ℓ t − Q s = ℓ t − +1 x i + s if t = l, ℓ t − Q s = ℓ t − +1 x i + s if t < l, e y ( P | H t ) = i + ℓ t − ) → ( i + ℓ t − ) ′ in Q ξ , ℓ t − Q s = ℓ t − y i + s if ( i + ℓ t − ) ′ → ( i + ℓ t − ) in Q ξ and t = l, ℓ t − Q s = ℓ t − y i + s if ( i + ℓ t − ) ′ → ( i + ℓ t − ) in Q ξ and t = l. By Lemma 3.7, we have F | P ( y i , y i +1 , . . . , y j ) = y ( P + ) | P = l − Y t =1 ( ℓ t − Y s = ℓ t − y i + s ) | P ( ℓ l − Y s = ℓ l − y i + s ) | P . We consider zigzag snake graphs( G i + ℓ t − , G i + ℓ t − +1 , . . . , G i + ℓ t − ) = H t ∪ G i + ℓ t − for 1 ≤ t ≤ l − G i + ℓ l − , G i + ℓ l − +1 , . . . , G i + ℓ l − ) = H l . By checking case by case, for 1 ≤ t ≤ l − x ( P | H t ) is just the weight monomial on H t ∪ G i + ℓ t − with the height monomial e y ( P | H j ), and x ( P | H l ) is the weight monomial on H l with the height monomial e y ( P | H l ).Since these vertices ( i + ℓ − , ( i + ℓ − , . . . , ( i + ℓ l − −
1) are sources or sinks in Q ξ , by Lemma 3.14, we have x ( P ) y ( P )( j Q s =1 x s ) F | P ( y i , y i +1 , . . . , y j ) = l − Y t =1 x ( P | H t ) e y ( P | H t )( ℓ t − Q s = ℓ t − x i + s )( ℓ t − Q s = ℓ t − y i + s ) | P x ( P | H l ) e y ( P | H l )( ℓ l − Q s = ℓ l − x i + s )( ℓ l − Q s = ℓ l − y i + s ) | P = x ′ i x ′ i +1 x i x i +1 l − Q t =2 x ′ i + ℓ t − x i + ℓ t − ! x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 x i + ℓ l − if i is a source or sink , l − Q j =1 x ′ i + ℓ j − x i + ℓ j − ! x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 x i + ℓ l − otherwise . ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 31
Finally, the uniqueness of P follows from the uniqueness of the perfect matching of each H t for 1 ≤ t ≤ l . (cid:3) By Theorem 2.6 and Theorem 3.1, assume without loss of generality that ι ( x [ α i,j ]) = [ L ( Y i ,a Y i ,a . . . Y i k ,a k )]for i = i < i < · · · < i k . Denote by G = ( G i , G i +1 , . . . , G j ) the snake graph withsign function ( a , a , . . . , a l ) associated to the HL-module L ( Y i ,a Y i ,a . . . Y i k ,a k ). Thenby Lemma 3.16, either l = k or | l − k | = 1, and l = k if and only if i is not a source orsink and j is a source or sink. Theorem 3.17.
With the notation above, the highest or lowest ℓ -weight monomial in the q -character of an arbitrary HL-module L ( Y i ,a Y i ,a . . . Y i k ,a k ) occurs in χ q ( ι ( x ′ i x ′ i +1 x ′ i + ℓ x ′ i + ℓ . . . x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 )) χ q ( ι ( x i x i +1 x i + ℓ x i + ℓ . . . x i + ℓ l − )) if i is a source or sink ,χ q ( ι ( x ′ i x ′ i + ℓ x ′ i + ℓ . . . x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 )) χ q ( ι ( x i x i + ℓ x i + ℓ . . . x i + ℓ l − )) otherwise . Proof.
By Theorem 3.8, we have χ q ( j Y ℓ = i ι ( x ℓ )) χ q ( ι ( x [ α i,j ])) = χ q ι ( P P ∈ Match( G ) x ( P ) y ( P ) L P ∈ Match( G ) y ( P ) ) ! . (3.14)By [21, Corollary 6.9], for an arbitrary simple U q ( b g )-module L ( m ), the lowest ℓ -weightmonomial in χ q ( L ( m )) is the product of the lowest ℓ -weight monomials of fundamentalmodules whose highest ℓ -weight monomials are factors of m , and the highest or lowest ℓ -weight monomial is unique. Hence the lowest ℓ -weight monomial in both sides of Equation(3.14) should be same. By Lemma 3.16, there exists a unique perfect matching P of thesnake graph associated to the HL-module L ( Y i ,a Y i ,a . . . Y i k ,a k ) such that x ( P ) y ( P ) F | P ( y i , y i +1 , . . . , y j ) = x ′ i x ′ i +1 x ′ i + ℓ x ′ i + ℓ . . . x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 x i x i +1 x i + ℓ x i + ℓ . . . x i + ℓ l − ( j Q s = i x s ) if i is a source or sink ,x ′ i x ′ i + ℓ x ′ i + ℓ . . . x ′ i + ℓ l − x (1 − δ j,j ⋄ ) j +1 x i x i + ℓ x i + ℓ . . . x i + ℓ l − ( j Q s = i x s ) otherwise . By Theorem 2.6, for any 1 ≤ ℓ ≤ k , ι ( x i ℓ ) = [ L ( Y i ℓ ,ξ ( i ℓ +1) )] , ι ( x ′ i ℓ ) = [ L ( Y i ℓ ,ξ ( i ℓ ) − Y i ℓ ,ξ ( i ℓ )+1 )] . Comparing the highest ℓ -weight monomial and the lowest ℓ -weight monomial in both sidesof Equation (3.14), and by their uniqueness property, our result holds. (cid:3) Example 3.18.
Continue our previous Example 3.9, the minimal perfect matching P − of G and each P − | H t ( t = 1 , ,
3) are shown in the left and right hand side of the followingfigure respectively.
By definition, the required perfect matching P of G is By the definition of x ( P | H t ) for t = 1 , , x ( P | H ) = x , x ( P | H ) = x x , x ( P | H ) = x , e y ( P | H ) = y y , e y ( P | H ) = 1 , e y ( P | H ) = y y . Using Lemma 3.16, we have x ( P ) y ( P )( Q s =1 x s ) F | P ( y , y , . . . , y ) = ( x y y x x ( y y ) | P )( x x x x x ( y y y ) | P )( x y y x x ( y y ) | P )= x ′ x ′ x ′ x x x x . Therefore by Theorem 3.17, the highest ℓ -weight monomial and the lowest ℓ -weightmonomial in the q -character of L ( Y , − Y , − Y , − Y , − ) occur in χ q ( ι ( x ′ x ′ x ′ x )) χ q ( ι ( x x x )) = χ q ( L ( Y , − Y , − ) L ( Y , − Y , − ) L ( Y , − Y , − ) L ( Y , − )) χ q ( L ( Y , − ) L ( Y , − ) L ( Y , − )) , and they are( Y , − Y , − )( Y , − Y , − )( Y , − Y , − )( Y , − )( Y , − )( Y , − )( Y , − ) = Y , − Y , − Y , − Y , − , ( Y − , Y − , )( Y − , Y − , )( Y − , Y − , )( Y − , ) Y − , Y − , Y − , = Y − , Y − , Y − , Y − , , respectively. ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 33 A new recursion for q -characters of Hernandez-Leclerc modules In this section, we give a recursive formula for HL-modules by an induction on thelength of the highest ℓ -weight monomials of HL-modules.Let d j = δ j,j ⋄ for j ∈ I . We have the following theorem. Theorem 4.1.
Let j ∈ I be a source or sink and j > i ∈ I . Then x [ α i,j ] x [ α j +1 ,j +1 ] = x [ α i,j +1 ]+ x [ α i, max { i − ,j • − } ] − δ i,j • x ′ min { , (1 − δ j • ,i • ) d j •− + δ j • ,i } max { i,j • } x [ − α j +2 ] d j +1 x ′ − d j +1 j +2 . (4.1) Proof.
By [1, Lemma in Section 2.2], if there is an arrow ( j − → j in Q ξ , then aftermutating the sequence i, ( i + 1) , . . . , ( j −
1) we obtain the following arrows at vertices j and ( j + 1):max { i − , j • − } a j ( ( ( j − j o o d j − (cid:15) (cid:15) ( j + 1) o o d j +1 + + − d j +1 ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ( j + 2) − d j +1 k k max { i, j • } ′ b j : : ✉✉✉✉✉✉✉✉✉✉✉ j ′ ( j + 1) ′ O O ( j + 2) ′ , where a j = 1 − δ i,j • and b j = min { , (1 − δ j • ,i • ) d j • − + δ j • ,i } .We mutate vertices j and ( j + 1) in order and obtain the following arrows connectingto the vertex j :max { i − , j • − } j a j u u b j z z ✉✉✉✉✉✉✉✉✉✉✉ d j +1 ) ) − d j +1 ( j + 1) o o ( j + 2)max { i, j • } ′ ( j + 2) ′ . Nextly, we mutate the vertex j and compare the denominators in the exchange relationusing Theorem 2.3, then x [ α i,j ] x [ α j +1 ,j +1 ] = x [ α i,j +1 ] + x [ α i, max { i − ,j • − } ] a j x ′ b j max { i,j • } x [ − α j +2 ] d j +1 x ′ − d j +1 j +2 . If there is an arrow j → ( j −
1) in Q ξ , then we reverse all the orientations and obtainthe same equation. (cid:3) Remark 4.2. (1) The q -character of ι ( x [ α i,j +1 ]) is the product of q -characters of ι ( x [ α i,j ]) and ι ( x [ α j +1 ,j +1 ]) except those terms appearing in the product of q -characters of ι ( x [ α i, max { i − ,j • − } ] a j ), ι ( x ′ b j max { i,j • } ), ι ( x [ − α j +2 ] d j +1 ), and ι ( x ′ − d j +1 j +2 ). (2) Our recursive formula (4.1) is different from a recursive formula given by Brito andChari [1, Proposition in Section 1.5], which are exchange relations correspondingto the mutation sequence ( i, i + 1 , . . . , j + 1), whereas our recursive formula needsto mutate ( i, i + 1 , . . . , j, j + 1 , j ) for a height function ξ such that j is source orsink.(3) It is possible by Equation (4.1) for us to give a combinatorial path formula inwhich we allow overlapping paths, generalizing Mukhin and Young’s path formulafor snake modules [29, 30].In practice, given an HL-module, we construct a height function ξ such that ι ( x [ α i,j +1 ])is an HL-module (up to isomorphic) for some i < j , and j, ( j + 1) being sources or sinks,see our construction in Theorem 3.1. Hence Equation (4.1) reduces to x [ α i,j +1 ] = x [ α i,j ] x [ α j +1 ,j +1 ] − x [ α i, max { i − ,j • − } ] a j x ′ b j max { i,j • } x [ − α j +2 ] d j +1 giving a recursive formula for the q -character of an HL-module by an induction on thelength of its highest ℓ -weight monomial.We end this section with an example illustrating Theorem 4.1. Example 4.3.
Let L ( Y , − Y , − Y , − ) be an HL-module in type A . By Theorem 3.1, wetake ξ (1) = − , ξ (2) = − , ξ (3) = − , ξ (4) = − , and we have a convention that ξ (0) = − ξ (5) = −
6. The quiver Q ξ is as follows.1 (cid:15) (cid:15) o o / / (cid:15) (cid:15) o o ′ ′ O O ′ ′ O O . Take j = 2 being a source in Q ξ and by Theorem 4.1, we have L ( Y , − Y , − Y , − ) = L ( Y , − Y , − ) L ( Y , − ) − L ( Y , − Y , − ) L ( Y , − ) . The q -character of L ( Y , − Y , − Y , − ) is the product of q -characters of L ( Y , − Y , − ) and L ( Y , − ), except those terms which are in the product of q -characters of L ( Y , − Y , − )and L ( Y , − ). In the product of q -characters of L ( Y , − Y , − ) and L ( Y , − ), there are 400monomials, among them 75 monomials are the same as the monomials in the product of q -characters of L ( Y , − Y , − ) and L ( Y , − ). Acknowledgements
The authors are supported by the National Natural Science Foundation of China (no.11771191, 12001254). Jian-Rong Li is supported by the Austrian Science Fund (FWF): M2633-N32 Meitner Program. Bing Duan is grateful to Professor Ralf Schiffler for helpfuldiscussions about snake graphs and cluster expansion formulas.
ERNANDEZ-LECLERC MODULES AND SNAKE GRAPHS 35
References [1] M. Brito, V. Chari,
Tensor products and q -characters of HL-modules and monoidal categorifica-tions , J. ´Ec. polytech. Math. 6 (2019), 581–619.[2] W. Chang, B. Duan, C. Fraser, and J.-R. Li, Quantum affine algebras and Grassmannians , Math.Z. 296 (2020), no. 3–4, 1539–1583.[3] V. Chari, A. Pressley,
Quantum affine algebras , Comm. Math. Phys. 142 (1991), no. 2, 261–283.[4] V. Chari, A. Pressley,
A guide to quantum groups , Cambridge University Press, Cambridge, 1994.[5] V. Chari, A. Pressley,
Quantum affine algebras and their representations . Representations of groups(Banff, AB, 1994), 59–78, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.[6] V. Chari, A. Pressley,
Factorization of representations of quantum affine algebras , Modular inter-faces (Riverside, CA, 1995), 33–40, AMS/IP Stud. Adv. Math., 4, Amer. Math. Soc., Providence,RI, 1997.[7] I. Canakci, R. Schiffler,
Snake graph calculus and cluster algebras from surfaces , J. Algebra 382(2013), 240–281.[8] I. Canakci, R. Schiffler,
Snake graph calculus and cluster algebras from surfaces II: self-crossingsnake graphs , Math. Z. 281 (2015), no. 1–2, 55–102.[9] I. Canakci, R. Schiffler,
Cluster algebras and continued fractions , Compos. Math. 154 (2018), no.3, 565–593.[10] I. Canakci, R. Schiffler,
Snake graph calculus and cluster algebras from surfaces III: Band graphsand snake rings , Int. Math. Res. Not. IMRN 2019, no. 4, 1145–1226.[11] I. Canakci, R. Schiffler,
Snake graphs and continued fractions , European J. Combin. 86 (2020),103081, 19 pp.[12] G. Cerulli Irelli, B. Keller, D. Labardini-Fragoso, P.-G. Plamondon,
Linear independence of clustermonomials for skew-symmetric cluster algebras , Compos. Math. 149 (2013), no. 10, 1753–1764.[13] V. G. Drinfeld,
Hopf algebras and the quantum Yang-Baxter equation , (Russian) Dokl. Akad. NaukSSSR 283 (1985), no. 5, 1060–1064.[14] V. G. Drinfeld,
A new realization of Yangians and of quantum affine algebras , (Russian) Dokl.Akad. Nauk SSSR 296 (1987), no. 1, 13–17; translation in Soviet Math. Dokl. 36 (1988), no. 2,212–216.[15] L. Demonet,
Categorification of skew-symmetrizable cluster algebras , Algebr. Represent. Theory14 (2011), no. 6, 1087–1162.[16] B. Duan, R. Schiffler,
A geometric q -character formula for snake modules , J. London Math. Soc.(2) 102 (2020) 846–878.[17] S. Fomin, M. Shapiro, D. Thurston, Cluster algebras and triangulated surfaces. I. Cluster com-plexes , Acta Math. 201 (2008), no. 1, 83–146.[18] S. Fomin, A. Zelevinsky,
Cluster algebras I: Foundations , J. Amer. Math. Soc. 15 (2002), no. 2,497–529.[19] S. Fomin, A. Zelevinsky,
Cluster algebras. II. Finite type classification , Invent. Math. 154 (2003),no. 1, 63–121.[20] S. Fomin, A. Zelevinsky,
Cluster algebras. IV. Coefficients , Compos. Math. 143 (2007), no. 1,112–164.[21] E. Frenkel and E. Mukhin,
Combinatorics of q -characters of finite-dimensional representations ofquantum affine algebras , Comm. Math. Phys. 216 (2001), no. 1, 23–57.[22] E. Frenkel and N. Reshetikhin, The q -characters of representations of quantum affine algebras anddeformations of W-algebras , Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp.163–205.[23] D. Hernandez and B. Leclerc, Cluster algebras and quantum affine algebras , Duke Math. J. 154(2010), no. 2, 265–341.[24] D. Hernandez and B. Leclerc,
Monoidal categorifications of cluster algebras of type A and D , Sym-metries, Integrable Systems and Representations, Springer Proceedings in Mathematics Statistics40 (2013), 175–193.[25] D. Hernandez and B. Leclerc,
A cluster algebra approach to q -characters of Kirillov-Reshetikhinmodule , J. Eur. Math. Soc. 18 (2016), no. 5, 1113–1159.[26] M. Jimbo, A q -difference analogue of U ( g ) and the Yang-Baxter equation , Lett. Math. Phys. 10(1985), no. 1, 63–69.[27] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. 53(1979), no. 2, 165–184.[28] B. Leclerc,
Imaginary vectors in the dual canonical basis of U q ( n ), Transform. Groups 8 (1) (2003)95–104.[29] E. Mukhin and C. A. S. Young, Path description of type B q -characters , Adv. Math. 231(2012),no. 2, 1119–1150.[30] E. Mukhin and C. A. S. Young, Extended T-systems , Selecta Math. (N.S.) 18 (2012), no. 3, 591–631.[31] G. Musiker, R. Schiffler, L. Williams,
Positivity for cluster algebras from surfaces , Adv. Math. 227(2011), no. 6, 2241–2308.[32] G. Musiker, R. Schiffler, L. Williams,
Bases for cluster algebras from surfaces , Compos. Math. 149(2013), no. 2, 217–263.[33] M. Rabideau,
F-polynomial formula from continued fractions , J. Algebra 509 (2018), 467–475.[34] M. Rabideau, R. Schiffler,
Continued fractions and orderings on the Markov numbers , Adv. Math.370 (2020), 107231.[35] R. Schiffler,
Quiver representations , CMS Books in Mathematics/Ouvrages de Math´ematiques dela SMC. Springer, Cham, 2014.
Bing Duan, School of mathematics and statistics, Lanzhou University, 730000, P. R.China; School of physical science and technology, Lanzhou University, 730000, P. R.China.
Email address : [email protected] Jian-Rong Li, Institute of Mathematics and Scientific Computing, University of Graz,Graz 8010, Austria.
Email address : [email protected] Yan-Feng Luo, School of mathematics and statistics, Lanzhou University, 730000, P.R. China.
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