Symmetry, Integrability and Geometry: Methods and Applications Polynomial Relations for q-Characters via the ODE/IM Correspondence

Let $U_q(\mathfrak{b})$ be the Borel subalgebra of a quantum affine algebra of type $X^{(1)}_n$ ($X=A,B,C,D$). Guided by the ODE/IM correspondence in quantum integrable models, we propose conjectural polynomial relations among the $q$-characters of certain representations of $U_q(\mathfrak{b})$.


Introduction
Let U q (g) be a quantum affine algebra of type X (1) n (X = A, B, C, D), and let U q (b) be its Borel subalgebra. In this paper we shall consider the problem of finding polynomial relations satisfied by the q-characters of the fundamental modules in the sense of [20] and related modules. This problem is intimately related with that of functional equations for Baxter's Q-operators in quantum integrable models. In order to motivate the present study let us review this connection.
In quantum integrable systems, one is interested in the spectra of a commutative family of transfer matrices. The latter are constructed from the universal R matrix of a quantum affine algebra, by taking the trace of the first component over some finite-dimensional representation called 'auxiliary space'. When the auxiliary spaces are the Kirillov-Reshetikhin (KR) modules, the corresponding transfer matrices satisfy an important family of polynomial identities known as the T -system [24,26]. (For a recent survey on this topic, see [25].) Subsequently the Tsystem has been formulated and proved [19,28] as identities of q-characters. It has been shown further [21] that the T -system is actually the defining relations of the Grothendieck ring, which is a polynomial ring [17], of finite-dimensional modules of quantum affine algebras. Notice that, from the construction by trace, the q-characters and the transfer matrices are both ring homomorphisms defined on the Grothendieck ring. Since the q-characters are injective [16,17], identities for q-characters imply the same identities for transfer matrices.
Baxter's Q-operators were first introduced in the study of the 8-vertex model [1]. Since then they have been recognized as a key tool in classical and quantum integrable systems, and there is now a vast literature on this subject. In the seminal paper [4], Bazhanov, Lukyanov and Zamolodchikov revealed that the Q-operators can also be obtained from the universal R matrix, provided the auxiliary space is chosen to be a (generically infinite-dimensional) representation of the Borel subalgebra. The work [4] for U q ( sl 2 ) has been extended by several authors [2,3,5,23,29] to higher rank and supersymmetric cases.
In view of the results mentioned above, it is natural to ask whether one can find polynomial relations, analogous to the T -system, for the q-characters of U q (b) (see definitions in [20]) (and hence for the Q-operators as well). The goal of this paper is to propose candidates of such identities.
Our idea is to use the so-called ODE/IM correspondence which relates the eigenvalues of Q-operators and certain ordinary linear differential equations. The reader is referred e.g. to the review [12] on this topic. In [11], the correspondence is discussed for general non-twisted affine Lie algebras using scalar (pseudo-)differential operators. In this paper we reformulate their results in terms of first-order systems. The general setting (to be explained in Section 3) is as follows.
Let L g denote the Langlands dual Lie algebra of g. Consider the following L g-valued firstorder linear differential operator (1.1) explained in Remark 5.9.) Also the relations for the Q (a) J,z 's or R (a) ε,z 's are not exactly the same as those for Q (a) J (E)'s, but it is necessary to fine-tune the coefficients by some power functions independent of z. The details will be given in Section 5 below. That it is natural to consider all Q (a) J,z corresponding to general weights of V (a) is a viewpoint suggested by the work [29] for type A algebras. Now let us come to the content of the present work. As a first step, we give explicit candidates for Q (1) i,z associated with each weight of the vector representation V (1) . This is done by taking suitable limits of the known q-characters of KR modules given by tableaux sums. It is known [20] that for the highest and lowest weights this procedure indeed gives the irreducible q-characters. As the next step, we define the formal series Q (a) J,z for a general node a of the Dynkin diagram. We define them by Casorati determinants whose entries are Q (1) i,z with suitable shifts of parameters. We then give candidates of polynomial relations among the Q (a) J,z expected from the ψ-system. In the cases related to spin representations, however, we do not have explicit candidates the R For the type A algebras, one can check that Q (1) i,z (not necessarily the highest or lowest ones) are indeed irreducible q-characters of modules given in [23]. The polynomial relations corresponding to the ψ-system can be summarized as a single identity det Q (1) ν,q −2µ+n+2 z x −µ+ν ν n+1 µ,ν=1 = 1, (1.2) where x ν = e ν , { ν } n+1 ν=1 being an orthonormal basis related to the simple roots by α ν = ν − ν+1 . This relation (as an identity for Q-operators) has been known to experts under the name 'Wronskian identity'. We shall give a direct proof that (1.2) is satisfied for q-characters in Appendix B.
For the other types of algebras, the situation is less satisfactory. At the moment we do not know the irreducibility of modules corresponding to the Q (1) i,z given by our procedure (except those corresponding to the highest or lowest vectors). For the spin representations of C (1) n and D (1) n , explicit formulas for the R (a) ε,z 's are missing. More seriously, we have not been able to prove the proposed identities for q-characters by computational methods. Instead, we support our working hypothesis by performing the following checks: n , proof in the limit to ordinary characters.
The main results of the present paper consist in formulating the conjectured relations, and in performing the checks mentioned above.
The text is organized as follows. In Section 2, we prepare basic definitions concerning the Borel subalgebra U q (b) of a quantum affine algebra U q (g), and collect necessary facts about their representations. In Section 3, we give an account of the ψ-system in the ODE/IM correspondence and indicate how to derive them using the formulation by first-order systems. We note that in [11] the ψ-system for algebras other than A type is mentioned as conjectures. In Section 4, we introduce the series Q (1) i,z as limits of the q-characters of KR modules. We also define these series for the other nodes of the Dynkin diagram. Section 5 is devoted to the proposals for polynomial relations. By comparing with the relations for the connection coefficients, we write down the relations for each type of algebras A (1) n , B (1) n , C (1) n and D (1) n . In Section 6, we give a summary of our work.
The text is followed by four appendices. Appendix A gives a list of realizations of the dual Lie algebras L g. In Appendix B we give a proof of the Wronskian identity for type A (1) n . In Appendix C, we prove the identities for type B (1) n in the limit to ordinary characters. In Appendix D the same is done for type C (1) n and D (1) n .

Preliminaries
In this section we introduce our notation on quantum affine algebras and their Borel subalgebras, and collect necessary facts that will be used later. Throughout this paper, we assume that q is a nonzero complex number which is not a root of unity.

Quantum Borel algebras
Let g be an affine Lie algebra associated with a generalized Cartan matrix C = (c ij ) 0≤i,j≤n of non-twisted type. Let D = diag(d 0 , . . . , d n ) be the unique diagonal matrix such that DC is symmetric and d 0 = 1. Set I = {1, 2, . . . , n}, and letg denote the simple Lie algebra with the Cartan matrix (c ij ) i,j∈I . Let {α i } i∈I , {α ∨ i } i∈I and {ω i } i∈I be the simple roots, simple coroots and the fundamental weights ofg, respectively. We set P = ⊕ i∈I Zω i , Q = ⊕ i∈I Zα i .
Set q i = q d i . We shall use the standard notation The quantum affine algebra U q (g) is the C-algebra defined by generators E i , F i , K ±1 i (i = 0, . . . , n) and the relations We do not write the formulas defining the Hopf structure on U q (g) since we are not going to use them.
As is well known [6,13], U q (g) is isomorphic to the C-algebra with generators x ± i,r (i ∈ I, r ∈ Z), k ±1 i (i ∈ I), h i,r (i ∈ I, r ∈ Z\{0}) and central elements c ±1/2 , with the following defining relations for all integers r j , where S m is the symmetric group on m letters, and the φ ± i,r are given by By definition, the Borel subalgebra U q (b) is the Hopf subalgebra of U q (g) generated by and the defining relations In this subsection we recall basics about U q (b)-modules in category O. For more details see [20].
A series of complex numbers Ψ = (Ψ i,r ) i∈I,r∈Z ≥0 is called an -weight if Ψ i,0 = 0 for all i ∈ I. We denote by t * the set of -weights. For a U q (b)-module V and Ψ ∈ t * , the subspace For each Ψ ∈ t * there exists a unique simple U q (b)-module of highest -weight Ψ. We denote it by L(Ψ).
A highest -weight module is of type 1 if its highest -weight Ψ satisfies For any non-zero complex numbers c i ∈ C × , the map gives rise to an automorphism of U q (b). After twisting by such an automorphism, any highest -weight module can be brought to one satisfying the condition (2.1). We denote by t * ,P the set of -weights satisfying (2.1).
Set D(λ) = λ − Q + , Q + = i∈I Z ≥0 α i . A U q (b)-module V of type 1 is said to be an object in category O if 1. for all λ ∈ P we have dim V λ < ∞, 2. there exist a finite number of elements λ 1 , . . . , λ s ∈ P such that the weights of V are contained in j=1,...,s In what follows we shall identify Ψ ∈ t * with their generating series, Simple objects in category O are classified by the following theorem. 20]). Suppose that Ψ ∈ t * ,P . Then the simple module L(Ψ) is an object in category O if and only if Ψ i (u) is a rational function of u for any i ∈ I.
In particular, for i ∈ I and z ∈ C × , the simple modules L ± i,z = L(Ψ) defined by the highest -weight are objects in category O. These modules are called the fundamental representations [20]. It is known [7,8] that finite-dimensional simple U q (g)-modules remain simple when restricted to U q (b). According to the classification of the former [9,10], the simple module L(Ψ) is finitedimensional if its highest -weight has the form where P i (u) is a polynomial such that P i (0) = 1. In the case where with some i ∈ I, m ∈ Z >0 and z ∈ C × , the module L(Ψ) is called a Kirillov-Reshetikhin (KR) module. We denote it by W (i) m,z .

Characters and q-characters
We recall the definition of q-characters (see [20]) and characters of representations of U q (b). Let Similarly let Z P denote the set of maps P → Z, and define e λ ∈ Z P by e λ (µ) = δ λ,µ . The ordinary character χ(V ) is an element of Z P , We have a natural map : Z t * ,P → Z P which sends [Ψ] to e λ such that Ψ i,0 = q . Under the q-character specializes to the ordinary character,

ψ-system
In this section, we reformulate the ψ-systems given in [11].

L g-connection
From now on, let g be an affine Lie algebra of type X (1) n (X = A, B, C, D), and let L g denote its Langlands dual algebra. Let h ∨ be the dual Coxeter number of g (see Table 1). Table 1.
Denote by e j , f j , h j (0 ≤ j ≤ n) the Chevalley generators of L g. We set e = n j=1 e j . Fix also an element ∈ L h from the Cartan subalgebra of L g, and let ζ ∈ C × . We consider the following L g-valued connection (cf. e.g. [15]): where k ∈ C and On any finite-dimensional L g-module V , (3.1) defines a first-order system of differential equations L(x, E; 1)φ(x, E) = 0. Quite generally, for a L g-module V , we denote by V k the L g-module obtained by twisting V by the automorphism e j → exp(2πikδ j,0 )e j , f j → exp(−2πikδ j,0 )f j . The operator L(x, E; e 2πik ) represents the action of L(x, Then the symmetry (3.3) implies L(x, E; e 2πik )φ k (x, E) = 0.
With each node a of the Dynkin diagram of Lg is associated a fundamental module V (a) of L g. We summarize our convention about them and some facts which will be used later. We leave the proofs to the readers (see Subsection 5.3 for an example).
Remark 3.1. In [11], scalar (pseudo-)differential equations are considered. Using the realization of L g given in Appendix A and rewriting the equation Lφ = 0 for the highest component of φ, one obtains the formulas [11, (3.18)-(3.21)] (for simplicity we have taken K = 1 there).

J. Sun
The module V (1) is called the vector representation of L g. Its explicit realization is given in Appendix A. If k is an integer, it is obvious that V k = V for any V . The vector representation for g = C (1) n has the additional property For general a, we distinguish the following two cases, We shall refer to them as the non-spin case and the spin case, respectively. Here and after we set In the non-spin case, we have In the case g = B (1) where V (a) for a > n stands for the right-hand side of (3.7).
In the case (g, a) = (C n . Let us consider the solutions at the irregular singularity x = ∞. It is convenient to use a gauge transformed form of L, where Λ = e + e 0 . Let µ (a) be the eigenvalue of Λ on V (a) which has the largest real part. This eigenvalue is multiplicity free, and is given explicitly as follows: Let u (a) be an eigenvector of Λ corresponding to µ (a) . From the representation (3.9) it follows that there is a unique V (a) -valued solution ψ (a) (x, E) which satisfies the following in a sector containing the positive real axis x > 0: We call ψ ψ ψ (a) (x, E) the canonical solution. In view of the relation (3.7) and the formula for µ (a) given above, we have for (NS). (3.10) Here ψ ψ ψ

ψ-system
In this subsection we state the ψ-system for the canonical solutions ψ ψ ψ (a) (x, E) introduced above. Let us consider them case by case.
n ). For a = 1, . . . , n, we have the embedding of L g-modules where The explicit expression of the embedding is given in Subsection 5.1.
satisfy the equation Lφ = 0 and have the behavior in a sector containing x > 0. Since such a solution is unique upto a constant multiple, we conclude that (after adjusting the constant multiple) We call the relations (3.12), (3.13) the ψ-system for A (1) n . The ψ-system for the other types can be deduced by the same argument, using the relevant embeddings of representations. We obtain relations of the following form.
Here ι stands for the embedding of L g (3.11) or , which follows from (3.11) and (3.8).
Here we have set , J. Sun and ι stands for an analog of (3.11) or the embeddings where (3.6) is taken into account.
We have set and ι stands for the embedding (3.11) or

Connection coef f icients
Now we introduce the connection coefficients Q  j (x, E) characterized by the expansion at the origin, From the symmetry (3.3) of L(x, E) we find that where we have set . For a sequence J = (j 1 , . . . , j a ), introduce the notation It follows from (3.10) that in the non-spin case where the sum is taken over all J = (j 1 , . . . , j a ), 1 ≤ j 1 < · · · < j a ≤ N . One can similarly define the connection coefficients in the spin case as well.
It has been shown in [20] that the fundamental modules L ± i,z of the Borel subalgebra arise as certain limits of the KR modules. In this section we follow the same procedure to obtain a family of formal power series Q (1) i,z associated with each weight space of the vector representation of L g. Up to simple overall multipliers, those corresponding to the highest or lowest weights are the irreducible q-characters χ q (L ± i,z ). We expect that in general the Q Below we shall use the following elements of Z t * ,P : Highest -weights are monomials in Y ±1 i,z and e ±ω i . Abusing the notation, for a monomial M = [Ψ] we shall also write L(M ) for L(Ψ).

The limiting procedure
Let us illustrate on examples the procedure for taking the limit.
2 ). We consider first the case g = A (1) 2 . Following [26,27], we write where the k-th box from the right carries the parameter q m+1−2k z. This can be rewritten further as Let us consider the limit m → ∞. There are three possibilities to obtain meaningful answers, Writing e α i = x i /x i+1 and defining for i = 1, 2, 3 we obtain the result Up to simple multipliers, they are the irreducible q-characters Using the explicit construction of modules [23], it can be checked that the second one is the qcharacter of the simple module whose highest -weight corresponds to the monomial Y −1 1,q 2 z Y 2,qz . We give two more examples.
Here we set K1 = k1, K2 = k2 + k1 and K 2 = k 2 + k2 + k1. We have Here we set K1 = k1, K2 = k2 + k1 and K 2 = k 2 + 2k 0 + k2 + k1. Note that the weight 0 has multiplicity 2. Correspondingly Q 0,z is a sum of two terms Q 0,z and Q 0,z . These terms cannot be separated in the process of taking the limit. Since they have highest -weights whose ratio is not a monomial of the A i,z 's, Q 0,z cannot be an irreducible q-character. We have

Series Q
(1) i,z In order to discuss the general case, let us prepare some notation. For g = X n (X = A, B, C, D), introduce a parametrization of {α i , ω i } by orthonormal vectors { i }, and an index set J with a partial ordering ≺ as follows.
A (1) n : . . , n,n, . . . ,1}, 1 ≺ · · · ≺ n ≺n ≺ · · · ≺1, . . , n, 0,n, . . . ,1}, 1 ≺ · · · ≺ n ≺ 0 ≺n ≺ · · · ≺1, Define also x i and f j,k for j, k ∈ J by In the following, in the sum of the form k i ,··· ,k j , unless mentioned explicitly, k i , . . . , k j run over all non-negative integers, and we use the abbreviation K l = j µ=l k µ for i ≺ l ≺ j. In the case g = C (1) n and l ≺ 0, we set K l = n j=l k j + 2k 0 + n j=1 kj. We give below the formula for Q i,z = Φī ,z n : Remark 4.4. As in [26,27], it is known that the q-character of KR module W  n , n). We recall that q 1 = q 1/2 for C (1) n and q 1 = q in the other cases. We fix our convention about the indices as follows. For J = (j 1 , . . . , j a ) ∈ J a , we denote by J ∼ the underlying set {j 1 , . . . , j a } ⊆ J . Set further J = (j a , . . . , j 1 ), (4.12) We say J is increasing if j 1 ≺ · · · ≺ j a . For an element J = (j 1 , . . . , j a ) ∈ J a , we define Q (4.14) We set Q For an increasing element J = (j 1 , . . . , j a ) ∈ J a , set where Φ j,z are defined in (4.4)-(4.11) with Y i,z being given in (4.1). Let L(Φ J,z ) be the unique irreducible U q (b)-module with the highest -weight Φ J,z . Except in the case g = C n , we expect that the q-character of L(Φ J,z ) is given by In particular, from (4.15) we expect that n .
In the case of g = C n , we expect the same to be true for Q

Series R (n)
ε,z for the spin node: case C (1) n In this subsection, we introduce another set of series R (n) ε,z for the spin node. Unlike the series Q (a) J,z , in general we do not know the explicit formulas for them. We define R (n) ε,z by the q-characters of the irreducible U q (b) module L(M ε,z ), and give a rule to determine the highest -weight M ε,z which is a monomial in C[Y ±1 i,z ] 1≤i≤n,z∈C × . Exhibiting the n-dependence explicitly, let P n be the weight lattice of the simple Lie algebra of type C n . Set We define two weight functions Borrowing an idea from [26], we introduce monomials M ε,z in C[Y ±1 i,z ] 1≤i≤n,z∈C × inductively as follows. Define two operators τ Y , τ z c by For n = 1 we set M +,z = Y 1,z and M −,z = Y −1 1,q 2 z . In the general case we set where ξ ∈ E n−2 . Now let ε ∈ E n with w 1 (ε) = 1 2 s k=1 ϕ k − t k=1 ψ k , and consider the simple module L(M ε,z ). We define R (n) ε,z andR (n) ε through the q-character and the character as follows.
For the latter we have the following guess.
Conjecture 4.5. For ε = (ε 1 , . . . , ε n ) ∈ E n , we have In the special case n = 2, one can obtain the series for C 2 from that of B i,z (1 ↔ 2) stand for the series for B (1) 2 with a = 1 given in Example 4.2, wherein we interchange Y 1,z with Y 2,z and A 1,z with A 2,z . Then we have 1,z (1 ↔ 2).
n , we follow the procedure for C (1) n . We use the same weight functions (4.17), (4.18) as for C (1) n . We also introduce two subsets of E n , We define monomials M ε,z inductively by (4.24) with ξ ∈ E n−2 and the initial values Conjecture 4.6. For ε = (ε 1 , . . . , ε n ) ∈ E n,ς (ς = 0, 1), we have Although we do not have formulas for the series R (n−ς) ε,z (ς = 0, 1) for general n, in the special case D ε,z (resp. R (4) ε,z ) as certain limits of the q-characters of the KR modules W and so on.

Polynomial relations
In this section, we shall give the main results of this paper. We propose polynomial relations among the series defined in Section 4, which are expected from the ψ-system obtained in Section 3. We prepare some notation which we will use below. For an element j ∈ J , an element J = (j 1 , . . . , j a ) ∈ J a and a positive integer k with 1 ≤ k ≤ a, we set

Plücker-type relations
We begin by writing down the ψ-system and the corresponding relations for Q (a) J,z for all nodes excepting the last (or the second last for D (1) n ). In fact they are just the consequence of the fact that the Q (a) J,z are defined by determinants of the Q (1) i,z 's. So they are rather definitions, and we write them down just for uniformity reasons. As we shall see later, the only non-trivial relations come from the last (or the second last) node.

Wronskian identity for
n , the non-trivial relation is the following 'Wronskian identity'.  n . We give conjectural relations which correspond to the last identity (3.15) of the ψ-system. We first give identities for the connection coefficients Q (a) J (E). In this subsection, for an element j ∈ J and J = (j 1 , . . . , j a ) ∈ J a we define We also denote by σ(J, J * ) the signature of the permutation (1, . . . , n,n, . . . ,1) → (J, J * ). The bilinear form , From this it is easy to see that we have the following L g-module isomorphism For a given element i ∈ J and J ∈ J a , let The counterpart of (5.4) for the series Q (a) J,z is given by the following conjecture. Conjecture 5.3. Let J 1 = (i 1 , . . . , i n ), J 2 = (j 1 , . . . , j n ) ∈ J n be increasing. Then we have Q (n) J * When n = 2, we have verified the above conjecture by direct computations. To save space we do not write the proofs here. We have also checked them for n = 3 by Mathematica 5.0 up to certain degree, counting the degree of A a,q k z to be k. For n ≥ 4, although at the moment we do not have a proof of these identities, in Appendix C, we show that the identities hold when specialized to the characters.
Conjecture 5.5. Suppose J ∈ J n−1 is increasing and 0 / ∈ J ∼ . Then we have where the sum is taken over all ε, ε ∈ E n satisfying Condition C J where γ γ γ, δ δ δ are determined there.
Conjecture 5.6. Suppose J ∈ J n−1 is increasing and 0 ∈ J ∼ . Then we have where the sum is taken over all ε, ε ∈ E n satisfying Condition C J where γ γ γ, δ δ δ are determined there.
Conjecture 5.7. Suppose J ∈ J n is increasing and 0 / ∈ J ∼ . Then we have where the sum is taken over all ε, ε ∈ E n satisfying Condition C J where γ γ γ, δ δ δ are determined there.
Conjecture 5.8. Suppose J ∈ J n is increasing and 0 ∈ J ∼ . Then we have where the sum is taken over all ε, ε ∈ E n satisfying Condition C J where γ γ γ, δ δ δ are determined there.
For n = 2 one can verify these relations using the relation between C 0,z separately. The ψ-system also suggests that there are identities involving them separately. This is indeed the case for n = 2 where we have, for example, On the other hand, if we define Q    Let J ∈ J a be increasing. We introduce the following conditions for ε, ε ∈ E n : Condition D J,ς : δ k , then σ σ σ ⊆ γ γ γ, and t ≡ r + ς (mod 2).
Conjecture 5.10. Let J be an increasing element of J n−2 , and let ς = 0, 1. Then we have where the sum is taken over all ε, ε ∈ E n,ς satisfying Condition D J,ς , where γ γ γ and δ δ δ are determined there.
Conjecture 5.11. Let J be an increasing element of J n−1 . Then we have where the sum is taken over all ε ∈ E n,1 , ε ∈ E n,0 satisfying Condition D J , where γ γ γ and δ δ δ are determined there.
For n = 4, we have checked by Mathematica 5.0 the conjectures hold up to certain degree. For general n, we prove in Appendix D the conjectures specialized to the ordinary charactersQ

Conclusion
In this section, we give a summary of this paper. We have done the following things: 1. To each weight of the fundamental representation V (a) , we associated a formal series Q (a) J,z or R (a) ε,z . We expect that with some simple factors the formal series are q-characters of certain irreducible modules of U q (b).
2. Under suitable identifications, using relations for the connection coefficients implied by the ψ-system, we proposed the following conjecture relations for the series Q  For the last three cases, we support our conjectures by checking the following. For the special cases g = B 4 , we checked the conjectures up to some degrees by Mathematica. When specialized to characters, the conjectured relations hold in all cases. This will be proved in Appendix C or D.
We hope we have presented reasonable grounds to suggest that the correspondence between the connection coefficients of certain differential equations and the q-characters of the Borel subalgebra U q (b) supplies an effective way to find polynomial relations. Certainly this is only the first step, and more serious checks are desirable along with attempts toward proving these identities.
A Vector representation of L g Following [14, Appendix 2], we give an explicit realization of the vector representation of L g.
The symbol E i,j stands for the matrix unit δ i,a δ j,b a,b=1,...,N where N = dim V (1) .

B Proof of Theorem 5.2
In this section we give a proof of Theorem 5.2. Let g = A n , and set One can rewrite Q i,z as For i ∈ J , we define x n+2−a x i , a = 1, . . . , n.
By the above definition and the formula of Q i,z , we have the following lemma.
Lemma B.1. For i ∈ J and 0 ≤ a ≤ n, we have Proposition B.2. For an element J = (j 1 , j 2 , . . . , j a ) ∈ J a , we have Proof . From the definition we have For k = 2, . . . , a, we subtract the (k − 1)-th row multiplied by x n+1 from the k-th row and then extract the factor A −1 n,q n−a−1+2k z . Thus we get Repeating the above steps in the order b = 2, . . . , a − 1, by replacing x n+1 and A −1 The proof is over by noting that

J. Sun
Now we give a proof of Theorem 5.2.
Similarly, in the case of g = D n , by (4.15) and Conjecture 4.6 the relevant identities read as follows.
In order to show (D.7), we prove it in a slightly more general form. (1 − x i z).
Let us rewrite R (m,n) as follows:  which is 0 by the induction hypothesis. This shows that R (m,n) is divisible by L (m,n) . The total degree of L (m,n) is l (m,n) = m(m − 1) + n(3n + 1)/2, while that of R (m,n) is at most r (m,n) = n 2 /2 + mn + (m + n) 2 /4 − m/2. Notice that l (m,n) > r (m,n) for n > m and l (n,n) = r (n,n) . From this we conclude that if n > m then R (m,n) = 0, and if n = m then R (n,n) is a constant multiple of L (n,n) . The constant is shown to be 1 by setting x = 0 and using the Weyl denominator formula of type C n , (1 − y i y j )(y j − y i ). , which is 0 by the induction hypothesis. The total degree of K (m,n) is k (m,n) = m(m + 1) + 3n(n − 1)/2, while that of R (m,n) is at most r (m,n) = mn+n(n−1)/2+(m+n) 2 /4−δ/4, where δ = 0, 1 is determined by δ ≡ m−n (mod 2).
Then k (m,n) > r (m,n) if m ≤ n − 2, and k (n−1,n) = r (n−1,n) . Hence we find that R (m,n) = 0 if m ≤ n − 2, and R (n−1,n) is a constant multiple of K (n−1,n) . The constant can be found by setting x = 0 and using the Weyl denominator formula of type D n , (1 − y i y j )(y j − y i ).