Difference equations for graded characters from quantum cluster algebra
aa r X i v : . [ m a t h . R T ] J un DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROMQUANTUM CLUSTER ALGEBRA
PHILIPPE DI FRANCESCO AND RINAT KEDEM
Abstract.
We introduce a new set of q -difference operators acting as raising operatorson a family of symmetric polynomials which are characters of graded tensor products ofcurrent algebra g [ u ] KR-modules [FL99] for g = A r . These operators are generalizationsof the Kirillov-Noumi [KN99] Macdonald raising operators, in the dual q -Whittaker limit t → ∞ . They form a representation of the quantum Q -system of type A [DFK14]. Thissystem is a subalgebra of a quantum cluster algebra, and is also a discrete integrablesystem whose conserved quantities, analogous to the Casimirs of U q ( s l r +1 ), act as differenceoperators on the above family of symmetric polynomials. The characters in the specialcase of products of fundamental modules are class I q -Whittaker functions, or charactersof level-1 Demazure modules or Weyl modules. The action of the conserved quantitieson these characters gives the difference quantum Toda equations [Eti99]. We obtain ageneralization of the latter for arbitrary tensor products of KR-modules. Introduction
We consider the set of all symmetric polynomials in r + 1 variables with coefficients in Z [ q ] which arise as graded characters of tensor products of g [ u ]-modules, as defined by[FL99]. Here, g = sl r +1 , g [ u ] are polynomials in u with coefficients in g , and we restrict ourattention to tensor products of Kirillov-Reshetikhin (KR) modules [CM06]. In this casethese are simply finite-dimensional, irreducible g -modules with highest weights which aremultiples of a fundamental weight, with an induced g [ u ]-action.The definition of the grading in [FL99] was motivated by the action of the affine algebra b g on conformal blocks in WZW conformal field theory. It is described entirely in termsof the action of g [ u ] on tensor products of finite-dimensional modules. In special cases,the graded tensor product is a Demazure module of g [ u ] [FL07] or a Weyl module [CL06].In special, stabilized limits, their characters coincide with the characters of affine algebramodules.It was conjectured in [FL99], and subsequently proved [AK07, DFK08], that the gradingis equivalent to that of the quantum algebra action on the analogous tensor product inthe crystal limit, related to the (finite) quantum spin chain whose Hilbert space is thetensor product of the associated quantum group modules [OSS01]. In the case of sl n , the Date : June 7, 2016. coefficients of the Schur functions in the expansion of the graded characters, the gradedmultiplicities, are generalized Kostka polynomials [SW99].In our previous work [DFK14], we gave a simple formulation of these characters in termsof generators of a non-commutative algebra, the quantum Q -system, which is a subalgebraof a quantum cluster algebra [BZ05]. The graded tensor product mutiplicities are computedas linear functionals of the corresponding monomial in generators of the quantum clusteralgebra (see Theorem 2.9 below).In this paper, we turn to the consequences of this formulation, and consider the actionof two natural operators on this product: Multiplication by generators of the algebra,and the action of the conserved quantities of the quantum Q -system considered as a non-commutative integrable, discrete evolution. We find that the action by the conservedquantities is a generalization, in the case when the tensor product includes non-fundamentalmodules, of the q -deformed quantum Toda Hamiltonians [Eti99]. The action by generatorsof the algebra is given by q -difference operators, which in the case of fundamental modulesis the dual q -Whittaker limit of the Macdonald raising operators of Kirillov and Noumi[KN99].Let us summarize our results briefly. Let n = { n ( α ) ℓ , ≤ ℓ ≤ k, α ∈ [1 , r ] } be a set ofnon-negative integers, where k is some positive integer which we call the level . The tensorproduct of g -modules M n = ⊗ ≤ α ≤ r ⊗ ≤ ℓ ≤ k V ( ℓω α ) ⊗ n ( α ) ℓ , where ω α are the fundamental weights and V ( λ ) is the irreducible g -module with highestweight λ , can be endowed with the structure of a g [ t ]-module and a g -equivariant grading[FL99]. The graded components M n [ j ] are g -modules, and the graded character of M n aredefined as χ n ( q, z ) = X j ≥ q j ch z M n [ j ]where ch z is the classical character, expanded in terms of Schur functions of z = ( z , ..., z r +1 ).The first operator which acts on the characters is a generalized q -deformed Toda operator: Theorem 3.11.
Let k ≥ and let n ( α ) k and n ( α ) k − be greater than or equal to − δ k, for all α . The graded characters χ n def = χ n ( q − , z ) satisfy the following difference equation: r +1 X α =1 χ n + ǫ α − ,k − − ǫ α,k − + ǫ α,k − ǫ α − ,k − r X α =1 q k − − P ki =1 in ( α ) i χ n + ǫ α − ,k − − ǫ α,k − + ǫ α +1 ,k − ǫ α,k = e ( z ) χ n where the vector ǫ α,i is defined so that ( ǫ α,i ) ( β ) j = δ β,α δ j,i , and e ( z ) = z + z + · · · + z r +1 . IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA 3
In the special case when k = 1, we show in Section 4 that the graded characters are special q -Whittaker functions, and the difference equations in Theorem 3.11 are the U q ( sl r +1 )difference Toda equation [Eti99].The second operator which acts on the graded characters adds a factor to the tensorproduct. This too can be expressed as the action of a q -difference operator on χ n ( q − ; z ),which we call a raising operator. As a result, there is an expression for the solutions of thedifference equations of Theorem 3.11 as the product of raising operators on the constantfunction 1. When k = 1, the difference operators are a dual q -Whittaker limit degenerationof the difference Macdonald raising operators of Kirillov and Noumi [KN99].Let q = v − r − and let D i ( z , ..., z r +1 ) = ( vz , ..., qvz i , ..., vz r +1 ). Given a subset I ⊂ [1 , r + 1], let I be its complement, and denote(1.1) z I = Y i ∈ I z i , D I = Y i ∈ I D i , and a I ( z ) = Y i ∈ Ij ∈ ¯ I z i z i − z j . Define the difference operators D α,n acting on the space of Laurent polynomials in z =( z , ..., z r +1 ) with coefficients in Z [ v, v − ] as follows:(1.2) D α,n = v − Λ α,α n − P rβ =1 Λ α,β X I ⊂ [1 ,r +1] | I | = α ( z I ) n a I ( z ) D I α ∈ [0 , r + 1] , n ∈ Z . The matrix Λ is given in Equation (2.3).The crucial property satisfied by the operators D α,k is that they obey the dual quan-tum Q -system relations (see Theorem 5.3). The main result of Section 5, relying on thisobservation, is the following expression for the graded characters: Theorem 5.7.
The graded characters for s l r +1 at level k are given by: χ n ( q − , z ) = v P i,j,α,β n ( α ) i Min( i,j )Λ α,β n ( β ) j + P i,α,β n ( α ) i Λ α,β + P α Λ α,α + P α<β Λ α,β × r Y α =1 ( D α,k ) n ( α ) k r Y α =1 ( D α,k − ) n ( α ) k − · · · r Y α =1 ( D α, ) n ( α )1 Q -system. The particular case of characters for fundamental KR-modules isaddressed in Section 4, where they are identified with q -Whittaker functions, by identifyingthe difference equation they obey with the q -deformed quantum Toda equation. In Section5, we present the general construction of the characters by iterated action of q -differenceoperators on the constant 1. This is proved by showing that the latter satisfy the dualquantum Q -system relations and realize the action of the quantum Q -system generators PHILIPPE DI FRANCESCO AND RINAT KEDEM on characters by adding one extra factor in the tensor product. Details of the proofs aregiven in Appendices A and B.
Acknowledgments.
We thank O.Babelon, M.Bergvelt, A.Borodin, I. Cherednik, I.Corwin,V. Pasquier, and S.Shakirov for discussions at various stages of this work. R.K.’s researchis supported by NSF grant DMS-1404988. P.D.F. is supported by the NSF grant DMS-1301636 and the Morris and Gertrude Fine endowment. R.K. would like to thank theInstitut de Physique Th´eorique (IPhT) of Saclay, France, for hospitality during variousstages of this work. 2.
Notations and definitions
The starting point for the results of this paper is the algebra called the quantum Q-system. This is a subalgebra of the quantum cluster algebra [BZ05] associated with the A r Q-system [Ked08, DFK11, DFK14]. The main results of the paper are representationsof this algebra and its conserved quantities acting as difference operators on the space ofsymmetric Laurent polynomials in r + 1 variables.2.1. Quantum Q -system. The A r quantum Q -system is a non-commuative algebra A generated by invertible elements { Q α,k : α ∈ [1 , r ] , k ∈ Z } subject to the quantum Q -system relations:(2.1) v Λ α,α Q α,k +1 Q α,k − = Q α,k − Q α +1 ,k Q α − ,k , Q ,k = Q r +1 ,k = 1 , as well as the commutation relations(2.2) Q α,k Q β,k ′ = v Λ α,β ( k ′ − k ) Q β,k ′ Q α,k ( | k − k ′ | ≤ | α − β | + 1) . The matrix Λ is proportional to the inverse of the Cartan matrix of A r :(2.3) Λ α,β = Min( α, β ) ( r + 1 − Max( α, β )) , ( α, β ∈ [1 , r ]) . Here, v is an invertible central element of the algebra. Note that all the variables Q α,k fordifferent α and fixed k commute with each other.The algebra A , which contains the inverses of all its generators, is finitely generated byany set of 2 r generators and their inverses which belong to the same “cluster” [DFK11]. Forexample, the set S = { Q α, , Q α, : α ∈ [1 , r ] } . Since A is a subalgebra of a quantum clusteralgebra, all other generators Q α,k with k ∈ Z are Laurent polynomials in the generating set.In Section 3.1, we provide a brief review of the discrete integrable structure of the quan-tum Q -system. Explicit solutions were worked out in detail in Ref. [DF11]. IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA 5
The constant term of elements in A . Starting with the quantum Q-system, weshowed in [DFK14] that there is a linear functional from monomials in positive powers ofthe generators of A to characters of graded tensor products of KR-modules. In [DFK14]showed this for all simply-laced algebras, but in the current context, we concentrate ontype A exclusively.Quantum cluster algebras have a Laurent property, generalizing the one for commutativecluster algebras [FZ02]. As a consequence, denoting by Z v := Z [ v, v − ]: Lemma 2.1.
Given a set of initial data S = { Q α, , Q α, : α ∈ [1 , r ] } , any solution Q α,k of the quantum Q -system can be expressed as a Laurent polynomial of the elements of S ,with coefficients in Z v . Thus, any polynomial in the generators of A can be expressed as a Laurent polynomialin the elements of S . Since these elements q -commute according to Equation (2.2), we candefine a normal ordering for any monomial in the elements of S as follows. Definition 2.2.
The normal ordered expression of a monomial in the generators in S is the expression obtained, using the commutation relations (2.2) , when all the Q α, arewritten to the left of all the Q β, , for all α, β . The normal ordering extends to any Laurent polynomial or series in the elements of S .Normal ordering is necessary in order to give a unique meaning to the evaluation of apolynomial in S at some central value of the subset { Q α, } α , because these generatorsdo not commute with the generators { Q α, } α . The evaluation occurs only after normalordering. Definition 2.3.
The linear map ev : Z v [ { Q ± α, , Q ± α, } α ] → Z v [ { Q ± α, } α ] is given by (1) normalordering the Laurent polynomial of the variables in S , and then (2) setting Q α, = 1 for all α in the normal-ordered expression. A closely related map is the following:
Definition 2.4.
The map ev : Z v [ Q ± α, , Q ± α, ] → Z v [ Q ± α, ] is given by (1) normal orderingthe Laurent polynomial of the variables in S , and then (2) setting Q α, = v − P β Λ α,β for all α in the normal-ordered expression. The two evaluation maps are related in the following manner:
Lemma 2.5.
For any Laurent polynomial f ∈ Z v [ Q ± α, , Q ± α, ] , we have: ev r Y β =1 Q β, f ! = r Y β =1 Q β, ! ev ( f ) Proof.
The commutation relations (2.2) imply: (cid:16)Q rβ =1 Q β, (cid:17) Q α, = v − P β Λ α,β Q α, Q rβ =1 Q β, . (cid:3) PHILIPPE DI FRANCESCO AND RINAT KEDEM
We also extend the notion of the evaluation map to any Laurent polynomials or seriesin the generators of S .There is a stronger version of Lemma 2.1 in the case of the quantum Q-system, which isa polynomiality property due to the specific form of the quantum Q -system (see Corollary5.13 of [DFK14]): Lemma 2.6.
Let f be a polynomial of the variables { Q α,k , α ∈ [1 , r ] , k ≥ } , obeying thequantum Q -system. Then ev ( f ) ∈ Z v [ { Q β, , β ∈ [1 , r ] } ] , namely it is a polynomial of thevariables { Q β, } β ∈ [1 ,r ] , with coefficients which are Laurent polynomials in v . As a consequence of Lemma 2.5, we can restate the polynomiality property as: ev ( Q rβ =1 Q β, f ) ∈ (cid:16)Q rβ =1 Q β, (cid:17) Z v [ { Q β, } β ∈ [1 ,r ] ] is a polynomial of the variables { Q β, } β ∈ [1 ,r ] , which is a multi-ple of Q rβ =1 Q β, .The following two definitions concern the evaluation of Laurent series, which we use onlywhere these evaluations converge. Definition 2.7.
Given a Laurent series f in { Q − α, , α ∈ [1 , r ] } with coefficients in Z v , themap CT(f ) is the constant term in Q α, for all α . Definition 2.8.
Given a Laurent series f in { Q − α, , α ∈ [1 , r ] } with coefficients in Z v [ Q ± α, ] ,we define the linear map φ = CT ◦ ev sends such a series to an element in Z v by firstevaluating the normal ordered expression of f at all Q α, = 1 , and then extracting theconstant term in all Q α, . Constant term identity for graded tensor product multiplicities.
The maintool in the paper [DFK14] was an expression of any graded tensor product multiplicity asthe constant term of a corresponding monomial in the generators of A . Fix a dominant g -weight λ = P α ℓ α ω α ∈ P + and a set of non-negative integers n = { n ( α ) i } as before. Thegraded tensor product multiplicities M n ,λ ( q ) := X j q j dim(Hom g ( M n [ j ] , V ( λ ))can be expressed as follows: Theorem 2.9. [DFK14]
The Graded multiplicities of the irreducible components in thelevel k M -sum formula of [KR87] can be expressed as M n ,λ ( q − ) = v P α,β,i n ( α ) i Λ α,β + ( P α ℓ α Λ α,α + P i,j,α,β n ( α ) i Min( i,j )Λ α,β n ( β ) j ) φ r Y α =1 Q α, Q − α, ! k Y i =1 r Y α =1 ( Q α,i ) n ( α ) i ! r Y α =1 lim k →∞ ( Q α,k Q − α,k +1 ) ℓ α +1 ! , (2.4) where Q α,n are the generators of A , and (2.5) q = v − r − IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA 7 is central.
We will use both variables, v and q throughout the paper. One can show that thepolynomial M n ,λ is a function of q only.Note that in Equation (2.4), the monomial involving the Q α,i becomes a polynomial of theinitial data variables Q α, after the evaluation step of Definition 2.3, as a consequence of thepolynomiality Lemma 2.6, and the obvious property that ev ( f g ) = ev ( ev ( f ) g ). Howeverthe “tails” lim k →∞ Q α,k Q − α,k +1 are Laurent series in Q − α, , for α ∈ [1 , r ]. Therefore, theconstant terms pick only finitely many contributions.Finally, (2.4) may be translated into an analogous expression for the graded character,by using:(2.6) χ n ( q − ; z ) = X λ ∈ P + M n ,λ ( q − ) s λ ( z )where s λ ( z ) = ch z V ( λ ) is a Schur function, P + the set of positive weights of g .3. From conserved quantities to difference equations
In this section, we derive difference equations for the graded characters χ n ( q ; z ), by usingthe explicit conserved quantities of the quantum Q -system (2.1), viewed as a discrete,integrable evolution equation in A [DFK11].3.1. Conserved quantities of the quantum Q -system. Discrete integrability of thesystem (2.1) means that there are r discrete, algebraically independent conserved quantities, C m , m = 1 , , ..., r which commute with each other. Moreover, these are coefficients of alinear recursion relation satisfied by the generators { Q ,k } .Let us recall the explicit formulas for the conserved quantities [DFK11]. For each n ∈ Z ,define the weights y i ( n ) ∈ A as the following ordered monomials: y α − ( n ) = Q α,n +1 Q − α − ,n +1 Q − α,n Q α − ,n , ( α = 1 , , ..., r + 1);(3.1) y α ( n ) = − Q α +1 ,n +1 Q − α,n +1 Q − α,n Q α − ,n , ( α = 1 , , ..., r ) . (3.2) Theorem 3.1. [DFK11]
Modulo the quantum Q-system, the following elements of A areindependent of n : (3.3) C m := X ≤ i
Lemma 3.2.
The conserved quantity C is (3.4) C = r +1 X α =1 Q − α,n Q α − ,n (cid:0) v r Q α,n +1 Q − α − ,n +1 − v − Q α +1 ,n +1 Q − α,n +1 (cid:1) .. Proof.
Fix n and use the expression for the weights (3.1) as well as the commutationrelations (2.2): C = r +1 X α =1 y α − ( n ) + r X α =1 y α ( n )= r +1 X α =1 v Λ α,α − α,α − +Λ α − ,α − Q − α,n Q α − ,n Q α,n +1 Q − α − ,n +1 − r X α =1 v Λ α,α +1 − Λ α − ,α +1 − Λ α,α +Λ α − ,α Q − α,n Q α − ,n Q α +1 ,n +1 Q − α,n +1 . (3.5)The Lemma follows from the identities:Λ α,α − α,α − + Λ α − ,α − = r, Λ α,α +1 − Λ α − ,α +1 − Λ α,α + Λ α − ,α = − . (cid:3) Define the elements of A (3.6) θ α,k = Q α,k Q − α,k +1 , ξ α,k = v Λ α,α θ α,k . In [DFK14], we showed that ξ α,k is expressible as a formal power series of the variables Q − α, with no constant term, with coefficients which are Laurent polynomials of the Q β, ’s, andthat the limit k → ∞ exists. We denote it by(3.7) ξ α = lim k →∞ ξ α,k . As the conserved quantities C m are independent of n , they may be evaluated in the limit n → ∞ . We have: Lemma 3.3. (3.8) y α := lim n →∞ y α ( n ) = 0 and y α − := lim n →∞ y α − ( n ) = v r ξ α − ξ − α Moreover, the odd variables y , y , ..., y r +1 commute among themselves, and C m is their m -th elementary symmetric function: (3.9) C m = e m ( y , y , ..., y r +1 ) = v mr/ e m ( ξ ξ − , ξ ξ − , ..., ξ r ξ − r +1 ) IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA 9
Proof.
Using the commutation relations (2.2), we note that y α − ( n ) = v r ξ − α,n ξ α − ,n y α ( n ) = v r ξ − α +1 ,n ξ α,n ( ξ α,n ξ − α,n − − Q -system relation to rewrite − Q α +1 ,n Q α − ,n Q − α,n = 1 − v Λ α,α Q α,n +1 Q α − ,n − Q − α,n = 1 − ξ α,n ξ − α,n − The limits (3.8) follow from (3.7). Finally the commutations follow from (2.2), and (3.3)clearly reduces to the m -th elementary symmetric function of the odd y ’s. (cid:3) Generating series.
We define generating series for the characters (2.6). First, define τ ( z ) = τ ( z , ..., z r +1 ) as τ ( z ) := v P α Λ α,α + P α<β Λ α,β X λ ∈ P + r Y α =1 ( ξ α ) ℓ α +1 s λ ( z )= v r +14 ( r +23 ) X λ ∈ P + r Y α =1 ( ξ α ) ℓ α +1 s λ ( z ) , (3.10)with ξ α as in (3.7). Here, P + is the set of dominant integral weights of sl r +1 , whereas s λ ( z )is the Schur function parameterized by partitions λ of length r + 1 or less, with the usualcorrespondence between sl r +1 weights and the set of such partitions, that is, ℓ α = λ α − λ α +1 .Fix k ≥ u = { u α,i : α ∈ [1 , r ] , i ∈ [1 , k ] } :(3.11) G ( k ) ( u ) = φ r Y α =1 Q α, ! k Y i =1 r Y α =1 − u α,i Q α,i !! τ ( z ) ! . Here, each rational function is defined to be a series in the variables u α,i , and the productover i is ordered from left to right.The coefficient of Q α,i u n ( α ) i α,i in the formal series expansion of G ( k ) ( u ) is defined to be G ( k ) n , with n = { n α,i } α ∈ [1 ,n ]; i ∈ [1 ,k ] :(3.12) G ( k ) n = φ r Y α =1 Q α, ! k Y i =1 r Y α =1 Q n ( α ) i α,i !! τ ( z ) ! with the function φ defined in 2.8. The normalization of τ ( z ) in (3.10) is chosen so that G (1)0 = G (1) (0) = 1.Comparing with Equation (2.4), using the commutation relations between Q α, and Q β, ,we see that these coefficients are the renormalized characters of Equation (2.6):(3.13) χ n ( q − , z ) = v P α,β,i n ( α ) i Λ α,β + n · (Λ ⊗ A ) n G ( k ) n ( z ) . The action of the conserved quantities at infinity.
We have the following the-orem for the action of the conserved quantities C m on τ ( z ) . Let e m ( z ) denote the m thelementary symmetric function in the r + 1 variables z , ..., z r +1 . Then Theorem 3.4.
The conserved quantities (3.3) of the A r quantum Q -system act on thefunction τ ( z ) as: (3.14) C m τ ( z ) = v mr e m ( z ) τ ( z ) + R m ( z ) where R m ( z ) is a sum of power series of the ξ α , with each of the summands independent ofat least one of the { ξ α } α ∈ [1 ,r ] .Proof. Recall the expression (3.10) for τ ( z ). Using (3.9), we write explicitly: v − mr C m v − r +14 ( r +23 ) τ ( z ) = X ℓ ,..,ℓ r ≥ r Y α =1 ξ ℓ α +1 α X ≤ i
Corollary 3.5.
When evaluated inside the generating function (3.11) , each conserved quan-tity C m acts on τ ( z ) as the scalar v mr e m ( z ) , namely: (3.15) φ r Y α =1 Q α, ! k Y i =1 r Y α =1 − u α,i Q α,i ! C m τ ( z ) ! = v mr e m ( z ) G ( k ) ( u ) Proof.
Using (3.14), note that as each summand of R m ( z ) has at least one missing ξ α ,the corresponding constant term in Q α, must vanish, as the rest of the power series onlygenerates positive powers of Q α, , once left evaluated at Q α, = 1. (cid:3) Difference equations from conserved quantities.
We now use a standard ar-gument to reformulate the conserved quantities of the quantum Q -system into differenceequations for the quantities G ( k ) n of (3.12). IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA11
Theorem 3.6.
For k ≥ , the coefficients G ( k ) n , n = { n ( α ) i } i ∈ [1 ,k ]; α ∈ [1 ,r ] , n ( α ) k , n ( α ) k − ≥ , obeythe following difference equation: v r e ( z ) G ( k ) n = r +1 X α =1 v r + P β (Λ α,β − Λ α − ,β ) n ( β ) k G ( k ) n + ǫ α − ,k − − ǫ α,k − + ǫ α,k − ǫ α − ,k − r X α =1 v − P β (Λ α,β − Λ α − ,β ) n ( β ) k G ( k ) n + ǫ α − ,k − − ǫ α,k − + ǫ α +1 ,k − ǫ α,k (3.16) where we use the notation ǫ β,m for the vector with entries ( ǫ β,m ) ( α ) i = δ α,β δ i,m . For k = 1 ,the coefficients G (1) n , n = { n ( α ) } α ∈ [1 ,r ] , n ( α ) ≥ , obey the difference equation: v r e ( z ) G (1) n = v r + P β Λ ,β ( n ( β ) +1) G (1) n + ǫ + r +1 X α =2 (cid:16) v r + P β (Λ α,β − Λ α − ,β )( n ( β ) +1) − v − P β (Λ α +1 ,β − Λ α,β )( n ( β ) +1) (cid:17) G (1) n + ǫ α − ǫ α − (3.17) with the notation ǫ β for the vector with entries ( ǫ β ) ( α ) = δ α,β .Proof. We compute in two ways the quantity B = φ r Y α =1 Q α, ! k Y i =1 r Y α =1 ( Q α,i ) n ( α ) i ! C τ ( z ) ! First, we find B = v r e ( z ) G ( k ) n by direct application of (3.15). Second, we use the expres-sion (3.4) with n = k − C . Using the notation: h M i n = φ r Y α =1 Q α, ! k Y i =1 r Y α =1 ( Q α,i ) n ( α ) i ! M τ ( z ) ! for any Laurent monomial M of the Q ’s, we have: h Q − α,k − Q α − ,k − Q α,k Q − α − ,k i n = v P β (Λ α,β − Λ α − ,β ) n ( β ) k h i n + ǫ α − ,k − − ǫ α,k − + ǫ α,k − ǫ α − ,k h Q − α,k − Q α − ,k − Q α +1 ,k Q − α,k i n = v P β (Λ α,β − Λ α − ,β ) n ( β ) k h i n + ǫ α − ,k − − ǫ α,k − + ǫ α +1 ,k − ǫ α,k The case k = 1 must be treated separately, as the insertion of Q α, amounts to a factor v − P β Λ α,β , coming from commutation of Q − α,k − Q α − ,k − through Q β ( Q β,k ) n ( β ) k . The Theoremfollows. (cid:3) Example 3.7.
When r = 1 (case of s l ), we have for n i ≡ n (1) i , and Λ , = 1 : v n k +1 G ( k ) n ,...,n k − − ,n k +1 + v − n k G ( k ) n ,...,n k − +1 ,n k − − v n k − G ( k ) n ,...,n k − − ,n k − = v ( z + z − ) G ( k ) n ,...,n k − ,n k with z = z = z − , whereas for k = 1 , n ≡ n : (3.18) v n +2 G (1) n +1 + ( v − n − v n ) G (1) n − = v ( z + z − ) G (1) n More generally, repeating this with the other conserved quantities C m , m ≥ D ( m ) G ( k ) n = v mr e m G ( k ) n , where the differenceoperators D ( m ) form a commuting family for m = 1 , , ..., r , and D (1) acts on the function G ( k ) n of n via the l.h.s. of eq.(3.16). Example 3.8.
When r = 2 and k = 2 (case of s l , level ), we have the following recursionrelations in the variables n ( α )1 = n α and n ( α )2 = p α , α = 1 , , obtained respectively byinserting the conserved quantities C and C of Example ?? : G (2) n − ,p ; n +1 ,p + v − n G (2) n +1 ,p − n − ,p +1 + v − n − p G (2) n ,p +1; n ,p − − v − G (2) n − ,p ; n − ,p +1 − v − − n G (2) n +1 ,p − n ,p − = v − − n − p e ( z ) G (2) n ,p ; n ,p G (2) n ,p − n ,p +1 + v − p G (2) n − ,p +1; n +1 ,p − + v − n − p G (2) n +1 ,p ; n − ,p − v − G (2) n ,p − n +1 ,p − − v − − p G (2) n − ,p +1; n − ,p = v − − n − p e ( z ) G (2) n ,p ; n ,p with e ( z ) = z + z + z and e ( z ) = z z + z z + z z , z z z = 1 . For later use, let us focus on the level 1 higher difference equations obtained by inserting C m , m ∈ [1 , r ] into the bracket h· · · i n defined above. As apparent from eq.(3.17) of Theorem3.6, the difference equation for G (1) allows to express G (1) n + ǫ as a linear combination of theshifted functions G (1) n + ǫ α +1 − ǫ α , α = 1 , , ..., r , as well as G (1) n . Similarly, due to the form of theconserved quantities as functions of the Q α,n ’s, the level 1 C m difference equation allows toexpress G (1) n + ǫ m as a linear combination of shifted functions of the form: G (1) n + P ≤ i ≤ m ǫ αi +1 − ǫ αi with 1 ≤ α < · · · < α m ≤ r , as well as G (1) n . Combining all the equations for m = 1 , , ..., r provides therefore a recursive method for computing all G (1) n . Indeed, defining σ ( n ) = P α n ( α ) , we see that each equation is a three term recursion in the variable σ ( n ), as theterm n + ǫ m has a value of σ G (1) n for σ ( n ) ≤ N , we therefore deduce G (1) n for all values σ ( n ) = N + 1. We have thefollowing: Theorem 3.9.
The difference equations obtained by inserting C m , m = 1 , , ..., r at level1 determine the functions G (1) n uniquely.Proof. We must examine the initial conditions for G (1) n . We note that for any n with some n ( α ) = −
1, the function G (1) n must vanish. Indeed, by definition it is the constant term in Q α, of an expression with no non-negative power of Q α, (as the insertion of Q − α, cancelsthe prefactor Q α, , and the contributions from τ ( z ) only provide strictly negative powers of IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA13 Q α, ). We conclude that all values of G (1) n = 0 for σ ( n ) = − , G (1)0 , ,..., = 1 by thenormalization of φ . With these initial data, the r difference equations determine a uniquesolution G (1) n for all n = ( n ( α ) ) α ∈ [1 ,r ] and n ( α ) ≥ α . (cid:3) Example 3.10.
When r = 2 and k = 1 (case of s l , level ), we have Λ , = Λ , = 2 and Λ , = Λ , = 1 . Denoting by n = n (1)1 and p = n (2)1 , we have the following recursionrelation for G n,p ≡ G (1) n,p : (3.19) v G n +1 ,p + ( v − n − G n − ,p +1 + v − − n ( v − p − G n,p − = v − n − p − e ( z ) G n,p This equation does not determine G n,p entirely. We also have to consider the “conjugateequation”, obtained by insertion of the second conserved quantity C : (3.20) v G n,p +1 + ( v − p − G n +1 ,p − + v − − p ( v − n − G n − ,p = v − n − p − e ( z ) G n,p These two equations are readily seen to be three-term linear recursion relations in the vari-able j = σ ( n, p ) = n + p , namely allow to express a single function with σ = j + 1 interms of functions with σ = j, j − . Together with the initial data G − ,p = G n, − = 0 for all n, p ≥ and G , = 1 which determine all functions with σ = − , , the two aboveequations therefore determine G n,p completely. For instance, using the equations for allvalues of σ = n + p indicated, we get: σ = 0 : G , = v − e G , = v − e σ = 1 : G , = v − ( v − e + (1 − v − ) e ) G , = v − ( v − e e + 1 − v − ) G , = v − ( v − e + (1 − v − ) e ) with the shorthand e = z + z + z and e = z z + z z + z z . Note that the two equationsdetermining G , are compatible, as a consequence of the commutation of C and C whichimplies e G , = e G , . Theorem 3.6 may be immediately translated in terms of graded characters χ n ( q − , z ) byuse of the formula (3.13), which results straightforwardly into the following: Theorem 3.11.
The graded characters χ n ≡ χ n ( q − , z ) , n = ( n ( α ) i ) α ∈ [1 ,r ]; i ∈ [1 ,k ] , n ( α ) k , n ( α ) k − ≥ − δ k, , satisfy the following difference equation for k ≥ : r +1 X α =1 χ n + ǫ α − ,k − − ǫ α,k − + ǫ α,k − ǫ α − ,k − r X α =1 q k − − P ki =1 in ( α ) i χ n + ǫ α − ,k − − ǫ α,k − + ǫ α +1 ,k − ǫ α,k = e ( z ) χ n (3.21) with the convention that ( ǫ α,i ) ( β ) j = δ β,α δ j,i , for β ∈ [1 , r ] and j ∈ [1 , k ] , ǫ ,i = ǫ r +1 ,i = 0 forall i , and e ( z ) = z + z + ... + z r +1 . The higher conserved quantities give rise to higher difference equations for χ n . Example 3.12.
In the case r = 1 ( s l ), we have: χ n ,...,n k − − ,n k +1 + χ n ,...,n k − +1 ,n k − − q k − − P ki =1 in i χ n ,...,n k − − ,n k − = ( z + z − ) χ n ,...,n k − ,n k For k = 1 , this reduces to: (3.22) χ n +1 + (1 − q − n ) χ n − = ( z + z − ) χ n Difference equations for characters of level-1 (Weyl) modules
Level-1 difference equations and the q -deformed open Toda chain. Whenspecialized to level k = 1, the difference equation of Theorem 3.11 takes a particularlysimple form. In this section, we show that this difference equation is the eigenvalue equa-tion for the q-deformed Toda operator for U q ( sl r +1 ) of [Eti99], after applying a suitableautomorphism, and performing a number of specializations.We start with a few definitions. Definition 4.1.
We introduce the following difference operators acting on functions of thevariable x = ( x , ..., x r ) : S α ( f )( x ) = f ( x , ..., x α − , x α − , x α +1 , ..., x r ) ( α = 1 , , ..., r )(4.1) S ( f )( x ) = f ( x ) , S r +1 ( f )( x ) = f ( x )(4.2) T α = S α +1 S − α ( α = 0 , , ..., r )(4.3) Definition 4.2.
The ˜ q -deformed difference (open) Toda Hamiltonian [Eti99] for U ˜ q ( sl r +1 ) is the following operator acting on functions of x , for fixed parameters ˜ q, ν α ∈ C ∗ : (4.4) H ˜ q = r X α =0 T α + (˜ q − ˜ q − ) r X α =1 ν α ˜ q − x α T α − T α In [Eti99], this Hamiltonian is related to the so-called relativistic Toda operator by useof an automorphism.
Definition 4.3.
We introduce the following automorphism τ of the algebra T generated by T α , α = 0 , , ..., r and U α = ˜ q − x α , α = 1 , , ..., r : (4.5) τ ( T α ) = T α τ ( U α ) = U α T α T − α − In particular, the automorphism τ respects the commutation relations T α U β = q δ β,α +1 − δ β,α U β T α . The image of H q under τ is the following Hamiltonian:(4.6) H ′ ˜ q = τ ( H ˜ q ) = T + r X i =1 (cid:0) q − ˜ q − ) ν α ˜ q − x α (cid:1) T α IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA15
On the other hand, it is easy to rewrite the level-1 difference equation (3.21) of Theorem3.11 for χ n ≡ χ n ( q − , z ), n = { n ( α ) } α ∈ [1 ,r ] as:(4.7) r X α =0 T − α − r X α =1 q − n ( α ) T − α ! χ n = e ( z ) χ n where the operator T α acts on functions of n by ( T α f )( n ) = f ( n + ǫ α − ǫ α +1 ).We see that if we pick(4.8) x = − n , ˜ q = q − , (˜ q − ˜ q − ) ν α = − q -Toda Hamilton-ian (4.6), with eigenvalue e ( z ). The level-1 graded character is therefore a ˜ q -Whittakerfunction.4.2. Whittaker functions and fusion products.
In [DFKT14], we have obtained theso-called fundamental q-Whittaker functions W λ ( x ) for U q ( sl r +1 ) by explicitly construct-ing Whittaker vectors in a Verma module V λ with generic highest weight λ , using a pathmodel. These form a basis of the eigenspace of the q-Toda Hamiltonian (4.4) for eigenvalue E λ = P ri =0 q λ + ρ | ω i +1 − ω i ) where ω i are the fundamental weights of A r . The dimension of thiseigenspace is the order of the Weyl group, here ( r + 1)!, as we may generate other indepen-dent solutions W s ( λ + ρ ) − ρ ( x ) by Weyl group reflections s , while preserving E s ( λ + ρ ) − ρ = E λ .Identifying z i = ˜ q λ + ρ | ω i − ω i − ) for i = 1 , , ..., r + 1, we deduce that the graded level-1character χ n is a linear combination of the image of the fundamental ˜ q -Whittaker functionsunder the automorphism τ , with the additional specialization (4.8).Let us illustrate this in the case of U q ( sl ). The ˜ q -Toda eigenvector equation is: W λ ( x −
1) + (1 + (˜ q − ˜ q − ) ν ˜ q − x ) W λ ( x + 1) = ( p + p − ) W ( x ) , p = ˜ q − λ +12 Applying the automorphism τ and using the specialization (4.8), we obtain a transformedfundamental ˜ q -Whittaker function W ′ λ ( n ), with the following series expansion (valid for | q | > W ′ λ ( n ) = τ ( W λ )( − n ) = p n − X a ∈ Z + q − a ( n +1) Q ai =1 (1 − q − i )(1 − p q − i )Analogously, we have the Weyl-reflected fundamental q-Whittaker function: W ′− λ − ( n ) = p − n X a ∈ Z + q − a ( n +1) Q ai =1 (1 − q − i )(1 − p − q − i )The functions W ′ λ ( n ) , W ′− λ − ( n ) form a basis of the eigenspace of the transformed ˜ q -TodaHamiltonian with same eigenvalue, namely(4.9) τ ( H ˜ q ) W ′ ( n ) = W ′ ( n + 1) + (1 − q − n ) W ′ ( n −
1) = ( p + p − ) W ′ ( n ) This coincides with the level-1 difference equation (3.22) with p = z . Looking for a linearcombination χ n = c λ ( p, q ) W ′ λ ( n ) + c − λ − ( p, q ) W ′− λ − ( n ) for say n = 0 ,
1, we find thecoefficients: c λ ( p, q ) = p (1 − p − ) Q ∞ i =1 (1 − p − q − i ) , c − λ − ( p, q ) = c λ ( p − , q ) = p − (1 − p ) Q ∞ i =1 (1 − p q − i )Remarkably, we have realized the graded character, which is polynomial in q − , p, p − asa linear combination of two infinite series of q − (the fundamental q-Whittaker functions).The cancellations occurring are the q-deformed version of the so-called class 1 regular-ity condition on Whittaker functions. So we may view the graded character as a classone specialized q-Whittaker function. We expect this to generalize to U q ( sl r +1 ) (see also[GLO10, GLO11] for analogous considerations).5. The solution for sl r +1 In this section, we introduce a generalization of the specialized Macdonald difference op-erators (corresponding to their “dual Whittaker limit” t → ∞ ), and use them to constructa solution of the difference equations for the sl r +1 graded characters, by iterated actionon the constant function 1. We shall proceed in several steps. After introducing the newdifference operators, we show that they satisfy the dual quantum Q -system. This allows toconsider them as raising operators for graded characters, Theorems 5.6 and 5.7, which areproved in two separate steps, first only for level k = 1 and then for general level k ≥ A realization of the dual quantum Q -system via generalized Macdonaldoperators.Definition 5.1. Recall the notations (1.1) z I , D I , a I ( z ) . We have the following sequenceof operators D α,n , α = 0 , , ..., r + 1 and n ∈ Z : (5.1) D α,n = v − Λ α,α n − P rβ =1 Λ α,β X I ⊂ [1 ,r +1] | I | = α ( z I ) n a I ( z ) D I In particular we have D ,n = 1 and D r +1 ,n = ( z z · · · z r +1 ) n D D · · · D r +1 = ( z z · · · z r +1 ) n = 1Recall the standard definition of the difference Macdonald operators for s l r +1 [Mac95]:(5.2) M q,tα = X I ⊂ [1 ,r +1] | I | = α Y i ∈ Ij I tz i − z j z i − z j Γ I IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA17 where Γ I = Q i ∈ I Γ i , and Γ i ( z ) = ( z , ..., z i − , q z i , z i +1 , ..., z r +1 ). This expression allows toidentify our operators D α,n for n = 0 as: D α, = v − P rβ =1 Λ α,β M α ∆ α M α := lim t →∞ t − α ( r +1 − α ) M q,tα = X I ⊂ [1 ,r +1] | I | = α a I ( z ) Γ I (5.3)where ∆( z ) = ( vz , ..., vz r +1 ). The operators M q,tα as well as their limits M α , α = 0 , , ..., r +1 are known to form a commuting family. Remark 5.2.
In the theory of Macdonald polynomials and difference operators, the limit t → ∞ may be thought of as a “dual Whittaker limit”. Indeed, as pointed out below, theduality of Macdonald polynomials P q − ,t − λ = P q,tλ , allows to relate our limit t → ∞ to theso-called q-Whittaker limit t → . Let A ∗ be the algebra generated by { Q ∗ α,k : α ∈ [1 , r ] , k ∈ Z } over Z v modulo the idealgenerated by the relations Q ∗ α,n Q ∗ β,p = v − Λ α,β ( p − n ) Q ∗ β,p Q ∗ α,n ( | p − n | ≤ | β − α | + 1)(5.4) v Λ α,α Q ∗ α,n − Q ∗ α,n +1 = ( Q ∗ α,n ) − Q ∗ α +1 ,n Q ∗ α − ,n Equivalently, the second relation may be rewritten, using (5.4) as:(5.5) v − Λ α,α Q ∗ α,n +1 Q ∗ α,n − = ( Q ∗ α,n ) − v − r − Q ∗ α +1 ,n Q ∗ α − ,n We refer to this as the dual quantum Q -system. The algebra A ∗ is isomorphic to thealgebra A op , with the opposite multiplication to A .We have the following main result. Theorem 5.3.
We have a polynomial representation π of A ∗ , with π ( Q ∗ α,n ) = D α,n of (5.1) . That is, acting by left multiplication on the space C [ z ] , the operators D α,n obey thedual quantum Q -system relations for A r : D α,n D β,p = v − Λ α,β ( p − n ) D β,p D α,n ( | p − n | ≤ | β − α | + 1)(5.6) v − Λ α,α D α,n +1 D α,n − = ( D α,n ) − v − r − D α +1 ,n D α − ,n (5.7)Note that when n = p = 0 the relation (5.6) boils down to the commutation of thespecialized Macdonald operators at t → ∞ , as M α ∆ = ∆ M α .The remainder of this section is devoted to the proof of this theorem.Let us define for any disjoint sets I , J of indices the quantities:(5.8) a I,J ( z ) = Y i ∈ Ij ∈ J z i z i − z j , b I,J ( z ) = Y i ∈ Ij ∈ J z i z i − qz j , c I,J ( z ) = Y i ∈ Ij ∈ J qz i qz i − z j Note that in this notation a I ( z ) of eq.(1.1) is simply a I, ¯ I ( z ).We have the following two lemmas. Lemma 5.4.
Fix integers ≤ a ≤ b and z = ( z , z , ..., z a + b ) . Then we have: (5.9) X I ∪ J =[1 ,a + b ] , I ∩ J = ∅| I | = a, | J | = b ( z J ) p ( a I,J ( z ) b J,I ( z ) − q pa a J,I ( z ) b I,J ( z )) = 0 ( | p | ≤ b − a + 1) Lemma 5.5.
Fix an integer a ≥ and z = ( z , z , ..., z a ) . Then we have: (5.10) X I ∪ J =[1 , a ] , I ∩ J = ∅| I | = | J | = a a I,J ( z ) b J,I ( z ) (cid:18) − q a z I z J (cid:19) = X I ∪ J =[1 , a ] , I ∩ J = ∅| I | = a +1 , | J | = a − a I,J ( z ) b J,I ( z )The above Lemmas 5.4 and 5.5 are proved in Appendix A below. Let us now turn tothe proof of Theorem 5.3. Let us first compute the quantity D α,n D β,p . Substituting thedefinition (5.1), we get: D α,n D β,p = X I,J ⊂ [1 ,r +1] | I | = α, | J | = β ( z I ) n a I ( z ) D I ( z J ) p a J ( z ) D J = X K ⊂ L ⊂ [1 ,r +1] | L |≤ α + β, | K |≤ α,β X I ∪ J L \ K,I ∩ J ∅ ( I = K ∪ I , J = K ∪ J ( z I z K ) n ( z J z K ) p a K ∪ I , ¯ L ∪ J v αβp D K D I a K ∪ J , ¯ L ∪ I D J = v αβp X K ⊂ L ⊂ [1 ,r +1] | L |≤ α + β, | K |≤ α,β ( z L z K ) n X I ∪ J L \ K,I ∩ J ∅| I | = α −| K | , | J | = β −| K | ( z J ) p − n a K, ¯ L a I , ¯ L a K,J a I ,J c K, ¯ L a K,I a J , ¯ L b J ,I D I D J = v αβp X K ⊂ L ⊂ [1 ,r +1] | L |≤ α + β, | K |≤ α,β ( z L z K ) n a K, ¯ L c K, ¯ L a K,L \ K a L \ K, ¯ L X I ∪ J L \ K,I ∩ J ∅| I | = α −| K | , | J | = β −| K | ( z J ) p − n a I ,J b J ,I D K D L where we have replaced the sum over I, J by one over K = I ∩ J and L = I ∪ J first, andthen written the disjoint unions I = K ∪ I , J = K ∪ J , ¯ I = ¯ L ∪ J , and ¯ J = ¯ L ∪ I .Note that we have isolated a factor u K,L ( n ) := ( z L z K ) n a K, ¯ L c K, ¯ L a K,L \ K a L \ K, ¯ L which doesnot depend on I , J . We may now write: v n Λ α,α + p Λ β,β + P γ Λ α,γ +Λ β,γ (cid:8) D α,n D β,p − v − Λ α,β ( p − n ) D β,p D α,n (cid:9) = X K ⊂ L ⊂ [1 ,r +1] | L |≤ α + β, | K |≤ α,β u K,L ( n ) X I ∪ J L \ K,I ∩ J ∅| I | = α −| K | , | J | = β −| K | ( z J ) p − n (cid:16) v αβp a I ,J b J ,I − v αβn − Λ α,β ( p − n ) a J ,I b I ,J (cid:17) D K D L = v αβp X K ⊂ L ⊂ [1 ,r +1] | L |≤ α + β, | K |≤ α,β u K,L ( n ) X I ∪ J L \ K,I ∩ J ∅| I | = α −| K | , | J | = β −| K | ( z J ) p − n (cid:16) a I ,J b J ,I − q α ( p − n ) a J ,I b I ,J (cid:17) D K D L = 0where we have first used Λ α,β + αβ = α ( r + 1) for α ≤ β , q = v − ( r +1) , and then appliedLemma 5.4 for every fixed pair K, L to the second summation, with a = α − | K | ≤ b = β − | K | and | p − n | ≤ b − a + 1 = β − α + 1. The relation (5.6) follows. IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA19
Analogously, we compute: v n Λ α,α +2 P β Λ α,β (cid:8) ( D α,n ) − v − Λ α,α D α,n +1 D α,n − (cid:9) = X K ⊂ L ⊂ [1 ,r +1] | L |≤ α, | K |≤ α u K,L ( n ) X I ∪ J L \ K,I ∩ J ∅| I | = | J | = α −| K | a I ,J b J ,I (cid:18) v nα − v ( n − α − Λ α,α z I z J (cid:19) D K D L = v nα X K ⊂ L ⊂ [1 ,r +1] | L |≤ α, | K |≤ α u K,L ( n ) X I ∪ J L \ K,I ∩ J ∅| I | = | J | = α −| K | a I ,J b J ,I (cid:18) − q α z I z J (cid:19) D K D L (5.11)and finally v n Λ α,α +2 P β Λ α,β − r − D α +1 ,n D α − ,n = v n v (Λ α +1 ,α +1 +Λ α − ,α − ) n + P β Λ α +1 ,β +Λ α − ,β D α +1 ,n D α − ,n = v nα X K ⊂ L ⊂ [1 ,r +1] | L |≤ α, | K |≤ α u K,L ( n ) X I ∪ J L \ K,I ∩ J ∅| I | = α +1 −| K | , | J | = α − −| K | a I ,J b J ,I D K D L (5.12)where we have used the relations2 + Λ α +1 ,α +1 + Λ α − ,α − − α,α = 0Λ α +1 ,β + Λ α − ,β − α,β = − ( r + 1) δ α,β The relation (5.7) follows by identifying equations (5.11) and (5.12) by applying Lemma5.5 for a = α − | K | to the second summation for K, L fixed. This completes the proof ofTheorem 5.3.5.2.
Graded characters and difference raising operators.
The main results.
In this section, we show that, in a way analogous to how theKirillov-Noumi difference operators are raising operators for Macdonald polynomials [KN99],our generalized degenerate Macdonald operators are raising operators for the graded char-acters.
Theorem 5.6.
For n = { n ( α ) i } α ∈ [1 ,r ]; i ∈ Z > , the coefficients G ( k ) n (3.12) for A r at level k aregiven by the iterated action of the generalized Macdonald operators (5.1) on the constantfunction : (5.13) G ( k ) n = r Y α =1 ( D α,k ) n ( α ) k r Y α =1 ( D α,k − ) n ( α ) k − · · · r Y α =1 ( D α, ) n ( α )1 Theorem 5.7.
The graded characters for s l r +1 at level k are given by: χ n ( q − , z ) = v P i,j,α,β n ( α ) i Min( i,j )Λ α,β n ( β ) j + P i,α,β n ( α ) i Λ α,β + P α Λ α,α + P α<β Λ α,β × r Y α =1 ( D α,k ) n ( α ) k r Y α =1 ( D α,k − ) n ( α ) k − · · · r Y α =1 ( D α, ) n ( α )1 z z · · · z r +1 = 1, we may restate this result in terms of the familyof difference operators M α,n defined as:(5.15) M α,n = X I ⊂ [1 ,r +1] | I | = α ( z I ) n a I ( z ) Γ I = v Λ α,α n + P β Λ α,β D α,n ∆ − α These satisfy a renormalized version of the dual quantum Q -system: M α,n M β,p = q Min( α,β )( p − n ) M β,p M α,n ( | p − n | ≤ | β − α | + 1)(5.16) q α M α,n +1 M α,n − = ( M α,n ) − M α +1 ,n M α − ,n ( α ∈ [1 , r ]; n ∈ Z )(5.17)with M ,n = 1 and M r +1 ,n = ( z z · · · z r +1 ) n ∆ − r − . Note also that M α, is equal to thedegenerate Macdonald operator M α of eq.(5.3). We have: Corollary 5.8.
The graded characters for s l r +1 at level k are given by: χ n ( q − , z ) = q − P i,j,α,β n ( α ) i Min( i,j )Min( α,β ) n ( β ) j + P i,α iαn ( α ) i × r Y α =1 ( M α,k ) n ( α ) k r Y α =1 ( M α,k − ) n ( α ) k − · · · r Y α =1 ( M α, ) n ( α )1 Proof.
We use the relation (5.15) to rewrite the result of Theorem 5.7. We make use of thecommutation relation ∆ M α,n = v nα M α,n ∆, and of Λ α,β + αβ = ( r + 1)Min( α, β ). (cid:3) Remark 5.9.
The iterated action of the raising operators M α,n on the function resultsclearly in a symmetric polynomial of the z ’s with coefficients that are polynomial in q . Onthe other hand, the prefactor is a negative integer power of q , as X i,j,α,β n ( α ) i Min( i, j )Min( α, β ) n ( β ) j − X i,α iαn ( α ) i = X i,α iα n ( α ) i ( n ( α ) i − X i Proof in the case of level . Let us now turn to the proof of Theorem 5.6. We willproceed in two steps. First, we will show the theorem in the case k = 1 only. The idea isto show that the expression (5.13) satisfies all the difference equations that determine G (1) n .To this end we use the conserved quantities of the dual quantum Q -system, easily obtainedby applying the anti-homorphism ∗ : A → A ∗ , Q α,k Q ∗ α,k such that ( AB ) ∗ = B ∗ A ∗ for all A, B ∈ A and v ∗ = v , and then evaluating in the polynomialrepresentation where π ( Q ∗ α,k ) = D α,k . The quantities Y α ( n ) = π ( y α ( n ) ∗ ) of (3.1-3.2) areexpressed in terms of D α,n , D α,n +1 as: Y α − ( n ) = D α − ,n D − α,n D − α − ,n +1 D α,n +1 Y α ( n ) = − D α − ,n D − α,n D − α,n +1 D α +1 ,n +1 Here, we use the formal (left and right) inverse D − α,k of the difference operator D α,k definedas follows. If | v | > 1, setting I α = { , , ..., α } , we write the convergent series: D − α,k = z kI α a I α ( z ) D I α X I ⊂ [1 ,r +1] , | I | = α D − I α z kI a I ( z ) z kI α a I α ( z ) D I − = X n ≥ X I ⊂ [1 ,r +1] , | I | = α D − I α z kI a I ( z ) z kI α a I α ( z ) D I n D − I α z kI α a I α ( z )If | v | < 1, we must use ¯ I r − α +1 = { r + 1 , r, ..., r − α + 2 } instead of I α .Noting that ∗ is an anti-homorphism which inverts the order of weights, we get thefollowing: Lemma 5.10. The conserved quantities C m = π ( C ∗ m ) , m = 0 , , ..., r +1 of the dual quantum Q -system are expressed in terms of the operators D α,n , D α,n +1 as: (5.19) C m = X Hard Particle configurations i
For all m = 0 , , ..., r + 1 , the conserved quantity C m (5.19) of the dualquantum Q -system acts on functions of z by multiplication by v mr times the m -th elemen-tary symmetric function e m ( z ) , namely C m = v mr X ≤ i
We will compute the action of C m expressed as (5.19) in the limit when n → ∞ . Wemust estimate the operator D α,n when n becomes large. To this end, and without loss ofgenerality, let us assume the modules of the z i ’s are strictly ordered, say | z | > | z | > · · · > | z r +1 | > 0. Then for large n the expression for D α,n is dominated by the contribution ofthe subset I α = { , , ..., α } , and we have D α,n ∼ v − Λ α,α n − P β Λ α,β ( z I α ) n a I α ( z ) D I α hence lim n →∞ D − α,n +1 D α,n = v − Λ α,α z − I α This gives lim n →∞ Y α − ( n ) = v r z α ( α = 1 , , ..., r + 1)lim n →∞ Y α ( n ) = 0 ( α = 1 , , ..., r )as the latter is proportional to ( z α +1 /z α ) n → n → ∞ . As before, the hard particlemodel reduces to that on the odd vertices of G r which are not connected by edges, hence thepartition functions are simply the elementary symmetric functions of the variables v r z α , α = 1 , ..., r + 1 and the theorem follows. (cid:3) We are now ready to prove Theorem 5.6 in the case of level k = 1. We will show thatthe function (5.13) for k = 1 satisfies the same difference equation (3.17) as in Theorem3.6 and its higher m versions.First, we may identify the action of the conserved quantity C m on the function τ ( z )within the constant term evaluation of Corollary 3.5 with that of the conserved quantity C m on functions of z of Theorem 5.11 above: in both cases, the action is by multiplicationby v mr e m ( z ). This involves writing the conserved quantity at n → ∞ in both cases.Second, if we use the expression of the conserved quantity C m (resp. C m ) as a functionof Q α, , Q α, (resp. D α, , D α, ), we obtain the exact same combinations of shift operators.This shows that the difference equations obeyed by (3.12) and (5.13) at level k = 1 areidentical. To complete the analysis, we should in principle examine the initial conditions.We have seen that G (1) n = 0 as soon as any of the n ( α ) are equal to − 1. Let us now showthat these conditions are not necessary to fix the solution, as each such term comes with avanishing prefactor, and therefore drops out of the difference equation.This fact relies on an important result of Ref. [DFK14], which was instrumental inproving the polynomiality property for the associated quantum cluster algebra. It relieson the Laurent polynomiality property which asserts that any cluster variable may be IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA23 expressed as a Laurent polynomial of any seed variables. The following Lemma was derivedby combining the Laurent property of the quantum cluster algebra for initial data S = { Q α, , Q α, } as well as for initial data S − = { Q α, − , Q α, } . Lemma 5.12. ( [DFK14] , Lemma 5.9 and its proof.) For any polynomial p of the { Q α,i } with coefficients in Z v , there exists a unique expression of the form: (5.21) p = X A ∪ B =[1 ,r ]; A ∩ B = ∅ ; m α ( A,B ) ≥ Y α ∈ A Q m α ( A,B ) α, − ! c A,B m ( { Q γ, } ) Y β ∈ B Q m β ( A,B ) β, ! where the coefficients c A,B m are Laurent polynomials of the variables { Q γ, } . In other words, any occurrence of Q − α, in the Laurent polynomial expression of p maybe replaced by a term Q α, − , for which coefficients remain Laurent polynomials of thevariables { Q γ, } . This powerful property can be applied to the conserved quantities aswell. Indeed, each quantity C m of (3.3) is a Laurent polynomial of the initial data S aswell as of S − depending on whether it is expressed at n = 0 or n = − 1. Repeating theargument leading to Lemma 5.12, we also find that each C m may be expressed in a uniqueway in the form (5.21). Let us examine the expression of C m as a Laurent polynomial ofthe initial data S more closely. From the hard particle condition and the explicit form of y i (0) (3.1-3.2), we see that the terms containing negative powers of Q α, in C m must be ofthe form c A,B ( { Q γ, } ) (cid:16) Q α ∈ A Q − α, (cid:17)(cid:16) Q β ∈ B Q β, (cid:17) , for some disjoint subsets A, B ⊂ [1 , r ], aseach particle is exclusive of its neighbors on the graph. Such terms may be rewritten as (cid:16) Q α ∈ A Q α, − (cid:17) c ′ A,B ( { Q γ, } ) (cid:16) Q β ∈ B Q β, (cid:17) according to the above. Now consider the level 1quantity G (1) n = φ (cid:16)(cid:16) Q β Q β, (cid:17)(cid:16) Q α Q n ( α ) α, (cid:17) τ ( z ) (cid:17) and insert C m as before. We get: v mr e m ( z ) G (1) n = φ r Y β =1 Q β, ! r Y α =1 Q n ( α ) α, ! C m τ ( z ) ! Suppose some n ( α ) = 0. The insertion of C m , expressed in terms of S variables, willintroduce terms of the form G (1) n with n ( α ) = − 1, whenever Q − α, occurs in C m . These areprecisely the unwanted terms, for which we showed that G (1) n = 0. However, we need notimpose this condition. Indeed, by the above argument we may replace the terms with Q − α, in C m with Q α, − , up to a change of coefficient c A,B → c ′ A,B . This gives a contribution of the form: φ (cid:16) Y β Q β, (cid:17)(cid:16) Y γ = α Q n ( γ ) γ, (cid:17) Q α, − p τ ( z ) ! = v − P γ = α Λ α,γ n ( γ ) φ (cid:16) Y β Q β, (cid:17) Q α, − (cid:16) Y γ = α Q n ( γ ) γ, (cid:17) p τ ( z ) ! = 0by first using the commutation relation (2.2), and then noting that Q α, Q α, − = v − Λ α,α ( Q α, − Q α +1 , Q α − , ) causes the evaluation to vanish. Hence the terms which would have created G (1) n with n ( α ) = − s l (3.18) (where the coefficient of the unwanted termvanishes for n = 0), and for s l (3.19-3.20) (where the coefficients of the unwanted termsvanish when n = 0 or p = 0).The same holds for the difference equations satisfied by (5.13) at k = 1. To prove it, werepeat the above argument, and note that unwanted terms from C m take the form π ( p ∗ ) Y γ = α D n ( γ ) γ, ! D α, − p ∗ of the { Q ∗ α,i } . This is due to the fact that D α, − Lemma 5.13. For any α ∈ [1 , r ] , we have the following identity: (5.22) X I ⊂ [1 ,r +1] | I | = α ( z I ) p a I ( z ) = (cid:26) p = 00 for p = − , − , ..., α − r − . This implies immediately the following: Corollary 5.14. We have D α, − p for all p = 1 , , ..., r + 1 − α , and D α, v − P β Λ α,β . The only initial data needed to feed the level 1 difference equations is therefore G (1)0 = 1,and the solution is uniquely determined by the equations. The corresponding function(5.13) for n = 0 is also trivially equal to 1, and Theorem 5.6 follows in the level 1 case.5.2.3. Proof for general level k ≥ . Let V = Z v [ z ] S r +1 , the space of symmetric polynomialsin z with coefficients in Z v . Using the map φ of Definition 2.8, we construct the map Ψfrom A + , the space of polynomials in Q α,k with coefficients in Z v , to V as follows. IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA25 Definition 5.15. For all p ∈ Z v [ { Q α,k | α ∈ [1 , r ] , k ≥ } ] , we define: (5.23) Ψ( p ) := φ r Y β =1 Q β, p τ ( z ) ! In particular, this allows to rewrite (3.12) as:(5.24) G n ( q − , z ) = Ψ (cid:18)Y Q n ( α ) i α,i (cid:19) and we have the normalization condition Ψ(1) = 1.Let V denote the image of A + under Ψ. V is a right module over A op+ where thesuperscript op denotes the opposite multiplication, under the action: Q α,k ◦ Ψ( p ) = Ψ( p Q α,k ) Theorem 5.16. The operators D α,k act on V by left-multiplication, and form a represen-tation of the action of A op+ on V , such that: Q α,k ◦ Ψ( p ) = D α,k Ψ( p ) .Proof. We use the anti-homomorphism ∗ that maps Q α,k Q ∗ α,k and reverses the orderof multiplication, while preserving v , and compose it with the representation π . To anypolynomial p of the { Q α,i } with coefficients in Z v we associate the polynomial p ∗ of the Q ∗ α,i by p ∗ ( { Q ∗ α,i } ) = p ( { Q α,i } ) ∗ , and finally π ( p ∗ ) by the substitution Q ∗ α,i → D α,i , namely π ( p ∗ ) = p ∗ ( { D α,i } ). We wish to prove that Ψ( p ) = π ( p ∗ ) 1. By Lemma 5.12 we may write: p = X A ∪ B =[1 ,r ]; A ∩ B = ∅ ; m α ( A,B ) ≥ Y α ∈ A Q m α ( A,B ) α, − ! c A,B m ( { Q γ, } ) Y β ∈ B Q m β ( A,B ) β, ! where the coefficients c A,B m are Laurent polynomials of the { Q γ, } , for any polynomial p ofthe { Q α,i } obeying the quantum Q -system relations. As moreover Ψ( Q α, − f ) = 0 for anypolynomial f , we see that Ψ( p ) = Ψ( ϕ ( p ))where ϕ ( p ) = X m α ( ∅ , [1 ,r ]) ≥ c ∅ , [1 ,r ] m ( { Q γ, } ) r Y α =1 Q m α ( ∅ , [1 ,r ]) α, The map ϕ is simply the truncation to the polynomial part of p in the variables Q α, . Let ϕ ∗ the corresponding truncation of any Laurent polynomial of { Q ∗ α, , Q ∗ α, } to it polynomialpart in { Q ∗ α, } . We have π ( p ∗ ) 1 = ϕ ∗ ( π ( p ∗ )) 1where we have used ϕ ( p ) ∗ = ϕ ∗ ( p ∗ ), and π ( f ∗ ) D α, − f ∗ of the Q ∗ α,i . By definition of Ψ and φ and the evaluation ev , we maynow evaluate ϕ ( p ) at Q α, = v − P β Λ α,β without altering Ψ( p ) = Ψ( ev ( ϕ ( p ))). Note that ev ( f ) ∗ = ev ∗ ( f ∗ ) where ev ∗ is the right evaluation at Q ∗ α, = v − P β Λ α,β (after the dualnormal ordering that puts all Q ∗ α, to the right). Finally, from Corollary 5.14 we have: π ( p ∗ ) 1 = ev ∗ ( ϕ ( π ( p ∗ ))) 1. The two polynomials ev ( ϕ ( p )) and ev ∗ ( ϕ ( π ( p ∗ ))) are the same polynomial of respectively { Q α, } and { D α, } with coefficients in Z v . Therefore, the state-ment Ψ( p ) = π ( p ∗ ) 1 needs only be proved for a polynomial p ∈ Z v [ { Q α, , α ∈ [1 , r ] } ], andin fact for any monomial of the form Q α Q m α α, with m α ≥ 0. This is exactly the level 1 caseof Theorem 5.6, which was proved in Sect. 5.2.2 above. The Theorem follows by using theanti-homomorphism property π (( p Q α,k ) ∗ ) = D α,k π ( p ∗ ). (cid:3) Finally, noting that Ψ(1) = 1, and applying Theorem 5.16 iteratively, leads straightfor-wardly to Theorem 5.6 for arbitrary level k .5.3. Level one case and degenerate Macdonald polynomials. When restricted tolevel 1, the formula of Corollary 5.8 for graded characters reduces to the following, for n = { n ( α ) } α ∈ [1 ,r ] :(5.25) χ n ( q − , z ) = q − P α,β n ( α ) Min( α,β ) n ( β ) + P α αn ( α ) r Y α =1 ( M α, ) n ( α ) M α, as in (5.15). We have the following: Theorem 5.17. The level one s l r +1 graded characters (5.25) are eigenfunctions of thedegenerate Macdonald difference operators M α, = M α of (5.3) , namely: M α, χ n ( q − , z ) = E α, n χ n ( q − , z ) , E α, n = q P β Min( α,β ) n ( β ) Proof. Starting from formula (5.25), we compute: M α, χ n ( q − , z ) = q − P α,β n ( α ) Min( α,β ) n ( β ) + P α αn ( α ) M α, r Y β =1 ( M β, ) n ( β ) q P β Min( α,β ) n ( β ) χ n ( q − , z )by use of the commutation relations (5.16), and the fact that M α, (cid:3) Recall that the symmetric A r ( q, t )-Macdonald polynomials P q,tλ ( z ) of the variables z =( z , ..., z r +1 ), indexed by partitions λ = ( λ ≥ λ ≥ · · · ≥ λ r +1 ≥ M q,tα , α = 1 , , ..., r ,and whose leading term is the symmetric monomial m λ = Q i z λ i i + permutations. TheMacdonald polynomials P q,tλ ( z ) satisfy the following duality property [Mac95]:(5.26) P q,tλ ( z ) = P q − ,t − λ ( z )Comparing this with the result of Theorem 5.17, we conclude: IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA27 Corollary 5.18. The level one A r graded characters χ n ( q − , z ) are the following degeneratelimits of the Macdonald polynomials: (5.27) χ n ( q − , z ) = lim t →∞ P q,tλ ( z ) = P q − , λ ( z ) where the correspondence between n and λ is via: λ = n (1) + n (2) + · · · + n ( r ) , λ = n (2) + · · · + n ( r ) , . . . λ r = n ( r ) , λ r +1 = 0Note that we have picked λ r +1 = 0, as the variables z satisfy z z · · · z r +1 = 1, so that P q,tλ ,..,λ r ,λ r +1 ( z ) = ( z z · · · z r +1 ) λ r +1 P q,tλ ,..,λ r , ( z ) is independent of λ r +1 . Remark 5.19. From eq. (5.27) , we may identify the graded level one character χ n ( q, z ) with the Whittaker limit t → of the Macdonald polynomial lim t → P q,tλ ( z ) . This shows inparticular that χ n ( q, z ) is a polynomial of q . Remark 5.20. The raising operators M α, coincide with the raising operators K + α forMacdonald polynomials introduced by Kirillov and Noumi [KN99] , in the limit t → ∞ , aswell as with the dual raising operators K − α in the Whittaker limit t → . Conclusion In this paper we have used the constant term identity [DFK14] for graded tensor productmultiplicities involving solutions of the A r quantum Q -system to: (1) derive differenceequations for the corresponding graded characters, and (2) write expressions for the gradedcharacters in terms of generalized degenerate Macdonald q -difference operators, which forma representation of the dual quantum Q -system. This latter construction establishes in thecase of sl r +1 an intriguing bridge between two standard mathematical theories: on one handthat of A r Macdonald operators and polynomials, and on the other hand that of quantumcluster algebras, specifically that of the A r quantum Q -system.Our q -difference operators coincide in the initial cluster { M α, , M α, } to respectively theMacdonald operators and the Kirillov-Noumi raising operators for Macdonald polynomials[KN99], both in the dual Whittaker limit t → ∞ . We may view the finite t case as adeformation of our initial cluster. The algebraic framework of Macdonald theory is theDouble Affine Hecke Algebra (DAHA) [Che05]. In a forthcoming paper [DFK16] we definenatural t -deformations of our q -difference operators in the context of DAHA, that reduceto M α,n (5.15) in the limit t → ∞ . The DAHA relations are in a sense the natural t -deformation of the quantum Q -system. This should extend to other types than A r as well,for which both quantum Q -systems and DAHA structures are known.Finally, the explicit representation of graded characters as iterated action of q -differenceoperators on the constant 1 may be useful to explore the so-called conformal limit, in whichsome n ( α ) i are taken to infinity (infinite tensor products, see e.g. [FF03] for the case of sl ). Appendix A. Proof of Lemmas 5.4 and 5.5 The proof of Lemmas 5.4 and 5.5 goes as follows. First we rewrite the statement of theLemmas as a vanishing condition for the antisymmetrized version of some rational fractionof the z ’s. Then we show that all residues at the poles of this antisymmetrized expressionvanish. Finally we conclude that the result is proportional to the antisymmetrization of apolynomial of the z ’s with a too small degree, which must therefore vanish.A.1. Antisymmetrization: general properties. For any function f ( z ) of the variables z = ( z , ..., z N ) we define the symmetrization ( S ) and antisymmetrization ( AS ) operatorsas: S ( f )( z ) = 1 N ! X σ ∈ S N f ( z σ (1) , ..., z σ ( N ) )(A.1) AS ( f )( z ) = 1 N ! X σ ∈ S N sgn( σ ) f ( z σ (1) , ..., z σ ( N ) )(A.2)We have the following immediate result: Lemma A.1. For any function f I,J ( z ) of z indexed by two subsets I, J of [1 , N ] we have: X I,J ⊂ [1 ,N ] ,I ∩ J = ∅| I | = a, | J | = N − a f I,J ( z ) = (cid:18) Na (cid:19) S ( f I ,J ( z )) where I = [1 , a ] and J = [ a + 1 , N ] . Lemma A.1 allows to rephrase the statements of Lemmas 5.4 and 5.5 as identities onsymmetrized expressions.For z = ( z , ..., z N ), we define the Vandermonde determinant ∆( z ) = Q ≤ i The non-zero anti-symmetric polynomial P of z of smallest total degree,namely such that AS ( P ) = P , is proportional to the Vandermonde determinant of the z ’s,up to a constant independent of the z ’s. This implies the following: Corollary A.3. For any polynomial P ( z ) of total degree strictly less than N ( N − / , wehave AS ( P ) = 0 . IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA29 A.2. Proof of Lemma 5.4. For any integers b ≥ a ≥ 0, and p ≥ m ≥ I = [1 , a ], J = [ a + 1 , a + b ], let us define ϕ m,pa,b ( z ) = S z mI z pJ Y i ∈ I j ∈ J z i z i − z j z j z j − qz i We note that as m ≤ p , then: ϕ m,pa,b ( z ) = ( z · · · z a + b ) m ϕ ,p − ma,b ( z )We also define:(A.3) ψ m,pa,b ( z ) := ϕ m,pa,b ( z ) − q a ( p − m ) ϕ p,mb,a ( z )Using Lemma A.1, it is straightforward to show that the statement of Lemma 5.4 is equiv-alent to:(A.4) ψ m,pa,b ( z ) = 0In the following, we use the notation ∆ I = Q ≤ k ≤ ℓ ≤ n ( z i k − z i ℓ ) for any ordered set I = { i , i , ..., i n } . With I , J as above, we now express:∆( z ) ϕ m,pa,b ( z ) = AS ∆( z ) z mI z pJ Y i ∈ I j ∈ J z i z i − z j z j z j − qz i = AS ∆ I ∆ J z m + bI z pJ Y i ∈ I j ∈ J z j z j − qz i The only possible poles of ∆( z ) ϕ p,qa,b ( z ) are for z i → qz j for i = j . Let us computethe residue at the pole z → qz in z . Pick two ordered sets I ′ , J ′ with I ′ ∩ J ′ = ∅ , I ′ ∪ J ′ = [1 , a + b ], | I ′ | = a , | J ′ | = b , and such that 1 is the first element of I ′ = { } ∪ I and 2 the last element of J ′ = J ∪ { } . For any subset L ⊂ [1 , N ], we denote by AS L theantisymmetrization over the set { z i } i ∈ L . We computeRes z → qz ∆( z ) ϕ m,pa,b ( z ) = z m + b ( qz ) p +1 AS [3 ,a + b ] ∆ I Y i ∈ I ( z − z i )∆ J Y j ∈ J ( z j − qz ) z m + bI z pJ Y i ∈ I j ∈ J z j z j − qz i Y i ∈ I qz qz − qz i Y j ∈ J z j z j − qz = q p +1 z m + p + b + a ∆( z ′ ) ϕ m +1 ,p +1 a − ,b − ( z ′ ) where z ′ = ( z , ..., z a + b ). Using (A.3), we deduce that:Res z → qz ∆( z ) ψ m,pa,b ( z ) = ∆( z ′ ) n q p +1 z m + p + b + a ϕ m +1 ,p +1 a − ,b − ( z ′ ) − q a ( p − m ) q m +1 z m + p + b + a ϕ p +1 ,m +1 b − ,a − ( z ′ ) o = q p +1 z m + p + b + a ∆( z ′ ) ψ m +1 ,p +1 a − ,b − ( z ′ )We now proceed by induction on a . For a = 0, we have: ϕ m,p ,b ( z ) = S (( z · · · z b ) p ) = ( z · · · z b ) p = ϕ p,mb, ( z )hence ψ m,p ,b ( z ) = 0. Assuming that ψ m +1 ,p +1 a − ,b − ( z ′ ) = 0, we see that the residue at z → qz of ψ m,pa,b ( z ) vanishes, hence the is no pole of the form 1 / ( z − qz ) in the antisymmetrizedexpression. By symmetry, this holds for any pole z i → qz j . We conclude that ψ m,pa,b ( z ) is apolynomial. Using the antisymmetrization formula, we easily get:∆( z ) ϕ m,pa,b ( z ) = AS ∆ I ∆ J z m + bI z p + aJ Y i ∈ I j ∈ J z j − qz i = ( z · · · z a + b ) p + a AS ∆ I ∆ J z b − a − ( p − m ) I Y i ∈ I j ∈ J z j − qz i Similarly: ∆( z ) ϕ p,mb,a ( z ) = ( z · · · z a + b ) p + a AS ∆ I ∆ J z b − a − ( p − m ) I Y i ∈ J j ∈ I z j − qz i Finally, we have:∆( z ) ψ m,pa,b ( z )( z · · · z a + b ) p + a − = AS ∆ I ∆ J z b − a +1 − ( p − m ) I z J Y i ∈ I j ∈ J z j − qz i − q a ( p − m ) Y i ∈ J j ∈ I z j − qz i where the r.h.s. is a polynomial, as b − a + 1 − ( p − m ) ≥ z i = qz j .Writing N = a + b , its total degree is: N ( N − ma + pb − N ( p + a − 1) = N ( N − − ( p − m ) a − N ( a − < N ( N − a ≥ 1. The degree of the polynomial is therefore too small, and it must vanish byCorollary A.3. The Lemma 5.4 follows. IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA31 A.3. Proof of Lemma 5.5. We proceed analogously. For I = [1 , a ] and J = [ a + 1 , a ],we define θ a ( z ) = S Y i ∈ I j ∈ J z i z i − z j z j z j − qz i (cid:18) − q a z I z J (cid:19) We also have: ∆( z ) θ a ( z ) = AS ∆ I ∆ J z aI Y i ∈ I j ∈ J z j z j − qz i (cid:18) − q a z I z J (cid:19) Let us compute the residue of the pole of this expression at z → qz . As before, we picktwo ordered sets I ′ and J ′ of cardinality a such that I ′ ∩ J ′ = ∅ , I ′ ∪ J ′ = [1 , a ] and 1 isthe first element of I ′ = { } ∪ I and 2 the last element of J ′ = J ∪ { } . We compute:Res z → qz ∆( z ) θ a ( z ) = qz a +11 AS ∆ I Y i ∈ I ( z − z i )∆ J Y j ∈ J ( z j − qz ) z aI Y i ∈ I j ∈ J z j z j − qz i Y i ∈ I qz qz − qz i Y j ∈ J z j z j − qz (cid:18) − q a − z I z J (cid:19) = qz a ( z · · · z a ) AS ∆ I ∆ J z a − I Y i ∈ I j ∈ J z j z j − qz i (cid:18) − q a − z I z J (cid:19) = qz a ( z · · · z a )∆( z ′ ) θ a − ( z ′ )where we denote by z ′ = ( z , z , ..., z a ).Likewise, we define for I = [1 , a + 1] and J = [ a + 2 , a ]: ϕ a ( z ) = S Y i ∈ I j ∈ J z i z i − z j z j z j − qz i We also have: ∆( z ) ϕ a ( z ) = AS ∆ I ∆ J z a − I Y i ∈ I j ∈ J z j z j − qz i Let us compute the residue of the pole of this expressions at z → qz . We pick two orderedsets I ′ and J ′ such that I ′ ∩ J ′ = ∅ , I ′ ∪ J ′ = [1 , a ], | I ′ | = a + 1, | J ′ | = a − 1, and 1 is the first element of I ′ = { } ∪ I and 2 the last element of J ′ = J ∪ { } . We compute:Res z → qz ∆( z ) ϕ a ( z ) = qz a AS ∆ I Y i ∈ I ( z − z i )∆ J Y j ∈ J ( z j − qz ) z a − I Y i ∈ I j ∈ J z j z j − qz i Y i ∈ I qz qz − qz i Y j ∈ J z j z j − qz = qz a AS ∆ I ∆ J z a − I z J Y i ∈ I j ∈ J z j z j − qz i = qz a ( z z · · · z a )∆( z ′ ) ϕ a − ( z ′ )We conclude thatRes z → qz ∆( z ) { θ a ( z ) − ϕ a ( z ) } = qz a ( z z · · · z a )∆( z ′ ) { θ a − ( z ′ ) − ϕ a − ( z ′ ) } We proceed by induction on a . For a = 1 we have( z − z ) θ ( z , z ) = AS (cid:18) z z z − qz (1 − q z z ) (cid:19) = z − z Analogously, we find ϕ ( z , z ) = S (1) = 1hence θ ( z , z ) − ϕ ( z , z ) = 0. Assuming that θ a − ( z ′ ) − ϕ a − ( z ′ ) = 0, we deduce that∆( z )( θ a ( z ) − ϕ a ( z )) has no pole at z = qz . By symmetry, it has no pole at any z i = qz j ,hence it is a polynomial. Finally we write:∆( z )( θ a ( z ) − ϕ a ( z ))( z z · · · z a ) a − = AS ∆ I ∆ J z I Y i ∈ I j ∈ J z j − qz i ( z J − q a z I ) − AS ∆ I ∆ J z J Y i ∈ I j ∈ J z j − qz i where the r.h.s. is a polynomial of total degree N ( N − / − a ( a − < N ( N − / a ≥ N = 2 a . By Corollary A.3, the result must vanish, and the Lemma 5.5 follows. IFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA33 Appendix B. Proof of Lemma 5.13 Notations are as in Sect. A.1. Let us consider for α ∈ [1 , N ] and p ∈ Z the quantity A α,p ( z ) = X I ⊂ [1 ,N ] | I | = α ( z I ) p a I ( z )Picking the particular subset I α = { , , ..., α } , we may also write A α,p ( z ) = (cid:18) Nα (cid:19) S (( z I α ) p a I α ( z ))We now wish to eliminate the denominators in this (symmetric) expression. We use that AS (∆( z ) f ( z )) = ∆( z ) S ( f ( z )) for any f to rewrite:∆( z ) A α,p ( z ) = (cid:18) Nα (cid:19) AS (cid:0) ( z I α ) p + N − α ∆( z , ..., z α )∆( z α +1 , ..., z N ) (cid:1) The function to be antisymmetrized is a polynomial if p ≥ α − N , and then it has totaldegree α ( p + N − α ) + α ( α − N − α )( N − α − α p + N ( N − p = − , − , ..., α − N the antisymmetrized expressionmust vanish.When p = 0, the degree is exactly N ( N − / z ) A α,p ( z ) is proportionalto ∆( z ). The proportionality constant is fixed by evaluating A α, in the successive limits z → ∞ , z → ∞ , ..., z α → ∞ , and we finally get A α, = 1.This completes the proof of Lemma 5.13. References [AK07] Eddy Ardonne and Rinat Kedem. Fusion products of Kirillov-Reshetikhin modules andfermionic multiplicity formulas. J. Algebra , 308(1):270–294, 2007.[BZ05] Arkady Berenstein and Andrei Zelevinsky. Quantum cluster algebras. Adv. Math. , 195(2):405–455, 2005.[Che05] Ivan Cherednik. Double affine Hecke algebras , volume 319 of London Mathematical SocietyLecture Note Series . Cambridge University Press, Cambridge, 2005.[CL06] Vyjayanthi Chari and Sergei Loktev. Weyl, Demazure and fusion modules for the current algebraof sl r +1 . Adv. Math. , 207(2):928–960, 2006.[CM06] Vyjayanthi Chari and Adriano Moura. The restricted Kirillov-Reshetikhin modules for thecurrent and twisted current algebras. Comm. Math. Phys. , 266(2):431–454, 2006.[DF11] Philippe Di Francesco. Quantum A r Q -system solutions as q-multinomial series. Electron. J.Combin. , 18(1):Paper 176, 17, 2011.[DFK08] Philippe Di Francesco and Rinat Kedem. Proof of the combinatorial Kirillov-Reshetikhin con-jecture. Int. Math. Res. Not. IMRN , (7):Art. ID rnn006, 57, 2008. [DFK11] Philippe Di Francesco and Rinat Kedem. Non-commutative integrability, paths and quasi-determinants. Adv. Math. , 228(1):97–152, 2011.[DFK14] Philippe Di Francesco and Rinat Kedem. Quantum cluster algebras and fusion products. Int.Math. Res. Not. IMRN , (10):2593–2642, 2014.[DFK16] Philippe Di Francesco and Rinat Kedem. Quantum Q -systems, DAHA and quantum toroidalalgebras. work in progress , 2016.[DFKT14] Philippe Di Francesco, Rinat Kedem, and Bolor Turmunkh. A path model for whittaker vectors. preprint arXiv:1407.8423 [math.RT] , 2014.[Eti99] Pavel Etingof. Whittaker functions on quantum groups and q -deformed Toda operators. In Differential topology, infinite-dimensional Lie algebras, and applications , volume 194 of Amer.Math. Soc. Transl. Ser. 2 , pages 9–25. Amer. Math. Soc., Providence, RI, 1999.[FF03] B. Fe˘ıgin and E. Fe˘ıgin. Integrable c sl -modules as infinite tensor products. In Fundamentalmathematics today (Russian) , pages 304–334. Nezavis. Mosk. Univ., Moscow, 2003.[FL99] Boris Feigin and Sergey Loktev. On generalized Kostka polynomials and the quantum Verlinderule. In Differential topology, infinite-dimensional Lie algebras, and applications , volume 194 of Amer. Math. Soc. Transl. Ser. 2 , pages 61–79. Amer. Math. Soc., Providence, RI, 1999.[FL07] G. Fourier and P. Littelmann. Weyl modules, Demazure modules, KR-modules, crystals, fusionproducts and limit constructions. Adv. Math. , 211(2):566–593, 2007.[FZ02] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc. ,15(2):497–529 (electronic), 2002.[GLO10] Anton Gerasimov, Dimitri Lebedev, and Sergey Oblezin. On q -deformed gl l +1 -Whittaker func-tion. I. Comm. Math. Phys. , 294(1):97–119, 2010.[GLO11] Anton Gerasimov, Dimitri Lebedev, and Sergey Oblezin. On q -deformed gl ℓ +1 -Whittaker func-tion III. Lett. Math. Phys. , 97(1):1–24, 2011.[Ked08] Rinat Kedem. Q -systems as cluster algebras. J. Phys. A , 41(19):194011, 14, 2008.[KN99] A. N. Kirillov and M. Noumi. q -difference raising operators for Macdonald polynomials and theintegrality of transition coefficients. In Algebraic methods and q -special functions (Montr´eal, QC,1996) , volume 22 of CRM Proc. Lecture Notes , pages 227–243. Amer. Math. Soc., Providence,RI, 1999.[KR87] A. N. Kirillov and N. Yu. Reshetikhin. Representations of Yangians and multiplicities of theinclusion of the irreducible components of the tensor product of representations of simple Liealgebras. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , 160(Anal. Teor.Chisel i Teor. Funktsii. 8):211–221, 301, 1987.[Mac95] I. G. Macdonald. Symmetric functions and Hall polynomials . Oxford Mathematical Mono-graphs. The Clarendon Press, Oxford University Press, New York, second edition, 1995. Withcontributions by A. Zelevinsky, Oxford Science Publications.[OSS01] Masato Okado, Anne Schilling, and Mark Shimozono. Crystal bases and q -identities. In q -serieswith applications to combinatorics, number theory, and physics (Urbana, IL, 2000) , volume 291of Contemp. Math. , pages 29–53. Amer. Math. Soc., Providence, RI, 2001.[SW99] Anne Schilling and S. Ole Warnaar. Inhomogeneous lattice paths, generalized Kostka polyno-mials and A n − supernomials. Comm. Math. Phys. , 202(2):359–401, 1999.