MULTIVARIABLE q-HAHN POLYNOMIALS AS COUPLING COEFFICIENTS FOR QUANTUM ALGEBRA REPRESENTATIONS

We study coupling coefficients for a multiple tensor product of highest weight representations of the SU(1,1) quantum group. These are multivariable generalizations of the q-Hahn polynomials. 2000 Mathematics Subject Classification. 17B37, 33D70, 33D80.


Introduction.
It is known that the Clebsch-Gordan coefficients for highest weight representations of SU(1, 1) may be identified with three classical systems of orthogonal polynomials, namely, the Jacobi polynomials, the Hahn polynomials, and the continuous Hahn polynomials [41].They correspond to the three conjugacy classes of one-parameter subgroups of SU (1,1).In our previous papers [29,31] we found that many multi-variable polynomials occurring in the literature have a similar interpretation, as coupling coefficients for an n-fold tensor product of highest weight representations.This interpretation leads to simple proofs of many properties of such polynomials.
The purpose of this paper is to generalize some of these results to the context of quantum groups and q-series.Several authors have found that q-Hahn polynomials appear as Clebsch-Gordan coefficients for the SU (2) and SU (1,1) quantum groups [16,19,22,34,39].More recently, Granovskiȋ and Zhedanov [11] and Koelink and Van der Jeugt [20] found similar interpretations of q-Racah polynomials and Askey-Wilson polynomials, respectively.In this paper, however, we only consider multivariable generalizations of q-Hahn polynomials.This case is more elementary and more similar to the Lie algebra case.
For comparison, we will summarize some of the basic facts of [29,31].Let, for ν > 0, Ꮽ ν be the Hilbert space of analytic functions on the complex unit disc with norm (1.1) Here is the Pochhammer symbol.
For ν > 1 one has the integral formula 2 1 −|z| 2 ν−2 dx dy. (1. 2) The norm of Ꮽ ν is invariant under the transformations which give rise to a unitary representation of a covering group of SU(1, 1) on each space Ꮽ ν .This representation is irreducible and has the highest weight vector 1 of weight −ν.The derived representation of the complexified Lie algebra sl (2, C) is given by the densely defined operators which satisfy the structure equations and the su(1, 1) reality conditions Consider a Hilbert tensor product Ꮽ ν 1 ⊗ ••• ⊗ Ꮽ νn of such spaces.It decomposes under the Lie algebra action as (where |ν| = ν i ), that is, Ꮽ |ν|+2s occurs with the multiplicity n+s−2 n−2 .A highest weight vector Q in Ꮽ ν 1 ⊗•••⊗Ꮽ νn of weight −(|ν|+2s) is an image of 1 under an intertwining embedding (1.8) In our realization this means that Q is a homogeneous polynomial of degree s which can be expressed as a function of the differences z i − z j of the coordinates.Equivalently, Q satisfies In agreement with (1.7), the space of such polynomials has dimension n+s−2 n−2 .Let, for each highest weight vector Q, Q be the intertwining operator (1.10) determined by * Q 1 = Q.In the present realization, Q is a differential operator.In fact, if (1.12) In particular, if n = 2 and (z). (1.13) These bilinear operators occur (although usually expressed in homogeneous coordinates (cf.Appendix A) and with ν i replaced by negative integers) in classical invariant theory and are called transvectants.They have also been used in the theory of modular forms, where they are called Rankin-Cohen brackets [26,43].It follows from (1.12) that, writing e m (z where T 1 Q is the polynomial Since the monomials are eigenfunctions of the rotations of the disc, this exhibits T 1 Q as a coupling coefficient, with respect to this subgroup, of our representation.If Q is another highest weight vector then, by an application of Schur's lemma to for k = 0, 1, 2,.... Thus, given an orthogonal system of highest weight vectors, the transform T 1 gives a corresponding system of discrete orthogonal polynomials.For fixed k one may eliminate one variable and view these as polynomials of n−1 variables.
In the case n = 2 one obtains in this way the Hahn polynomials.
Replacing the monomials by formal eigenvectors to other one-parameter subgroups gives multivariable Jacobi polynomials and multivariable continuous Hahn polynomials Many orthogonal and biorthogonal polynomial systems occurring in the literature may be obtained by applying the three transforms T i to specific bases in the space of highest weight vectors; confer [31] and the references given there.
In this paper, we will obtain a q-analogue of the transform T 1 , leading to multivariable q-Hahn polynomials.One may also consider more general coupling coefficients connected with the so-called twisted primitive elements of the quantum algebra.This leads to multivariable generalizations of Askey-Wilson polynomials and q-Racah polynomials; confer [32] for some further remarks.
The plan of the paper is as follows.Section 2 contains preliminaries on q-series and quantum algebra.In Section 3 we find a q-analogue of the expression (1.12).This is used in Section 4 to find analogues of the transform T 1 and of the orthogonality relations (1.16).In Section 5 we study q-analogues of the coupling kernels introduced in [31].These are the reproducing kernels for spaces of coupling coefficients.In Section 6 we discuss identities involving multivariable q-Hahn polynomials and coupling kernels which follow from the quantum algebraic interpretation.In Section 7 we consider explicit examples of orthogonal and biorthogonal systems which may be obtained as coupling coefficients.There are two appendices, where we use quasi-commuting homogeneous coordinates to give an algebraic description of the space of highest weight vectors.

Preliminaries
2.1.Notation.Throughout the paper, q will be a fixed number in the range 0 <q <1.It will be convenient to use the symmetric q-factorials and q-Pochhammer symbols defined by where This is related to the more standard notation of [10], Sometimes we will use the multi-index notation The basic hypergeometric series r φ s is defined by [10] r φ s a in particular, r +1 φ r q a 1 ,...,q a r +1 q b 1 ,...,q br ; q, z = where |x| = x i .The q-Hahn polynomials, introduced by Hahn [12], are given by Q n q −x ; α, β, N | q = 3 φ 2 q −n , αβq n+1 ,q −x αq, q −N ; q, q , n= 0, 1,...,N; (2.8) this is a polynomial of degree n in q −x .They form an orthogonal system with respect to a measure supported on {1,q,...,q −N }; confer [18].The dual orthogonality relation may be written as an orthogonality relation for a different system of polynomials, namely, the dual q-Hahn polynomials.We will write for the symmetric q-derivative.It satisfies the q-Leibniz rule (2.10) We will need the following simple lemma in a special case.
Lemma 2.1.For α ij scalars, f i formal power series and z i , w i formal variables, the following identity holds: (2.11) Proof.It suffices to take f i (z) = z m i .Since then the left-hand side is given by (2.13) Changing k to m − k gives the same expression with z and w interchanged and α ij replaced by α ji .This proves the lemma.
For any x ∈ R n , we will denote by x * the element in R n with coordinates defines a skew-symmetric form on R n .It appears naturally in connection with multivariable q-series; for instance, one has the multivariable q-Chu-Vandermonde formula (equivalent to [10, Exercise 1.3]) (2.16) 2.2.Quantum algebra.The quantum algebra (or quantized universal enveloping algebra) ᐁ = ᐁ q 1/2 (su(1, 1)) is the associative involutive algebra over the complex numbers defined by generators and involution here and below we write K + = K, K − = K −1 when convenient.We refer to [4] for an introduction to quantum group theory.A unitary representation of ᐁ is a representation by densely defined operators on a Hilbert space, such that the involution * coincides with the Hilbert space adjoint.
The algebra ᐁ has the additional structure of a Hopf algebra.We need to discuss only the coproduct, which is the map ∆ : ᐁ → ᐁ ⊗ ᐁ given on the generators by This means that, given two unitary representations H 1 and H 2 of ᐁ, another one is defined on the Hilbert tensor product H 1 ⊗ H 2 by The coproduct is non-cocommutative, that is, the flip f ⊗g g ⊗f is not an intertwining map It is, however, coassociative in the sense that there is only one way to repeat it to obtain a representation on a finite tensor product It is easily proved that where * is given by (2.14).
For ν > 0 and 0 < q < 1, we denote by Ꮽ ν q the Hilbert space of analytic functions with the norm where f (z) = f (k)z k .By Cauchy's q-binomial formula [10], Ꮽ ν q has the reproducing kernel (here we use that 0 < q < 1) and thus the natural domain of definition is the disc We write e k (z) = z k for monomials, so that There is a unitary representation of ᐁ q (su(1, 1)) on Ꮽ ν q , given by (2.28) In the limit q → 1, (2.29) so we recover the Lie algebra operators (1.4).
Though we will not use it, we remark that for ν > 1 there is an integral formula for the norm of Ꮽ ν q [17].In terms of Jackson's q-integral [10] it can be written as which is a q-analogue of (1.2).In contrast to the case q = 1, this formula extends to the case 0 < ν < 1, though the mass on the outer circle |z| = q (1/4)(ν−1) is then negative.

Quantum transvectants.
In this section we will obtain an expression for the multilinear transvectants in the present context.As in the case q = 1, there is a decomposition where |ν| = ν i .We define a transvectant of order s to be an intertwining map thus the transvectants of order s form a linear space of dimension n+s−2 n−2 .A highest weight vector is, by definition, a solution in These are mapped to constants by the transvectants of order s.The first equation means that they are homogeneous polynomials of degree s.For each highest weight vector Q, we denote by Q the transvectant of order s determined by * Q 1 = Q.This gives a one-to-one correspondence between highest weight vectors and transvectants.
We will now fix Q and seek a q-analogue of the expression (1.12) for Q .It will be convenient to write µ = |ν|+2s, For f a polynomial (to avoid questions of convergence) in where τ w is the q-translation operator As an operator on the subspace of polynomials in q , τ w is given by (3.9) Proof.The first expression follows from (2.23).The second one then follows from Lemma 2.1, in the case when (3.10) Plugging the last expression of the lemma into the equality gives the following explicit expression for the transvectant.
is a highest weight vector, then where * is defined by (2.14).
As an example, for n = 2 the space of highest weight vectors of degree s is onedimensional.It is easy to verify that it is spanned by (3.14) confer also Section 7 and Appendix B. The corresponding transvectant is which is a q-analogue of (1.13).

Coupling coefficients.
To each highest weight vector Q we associate the function where s is the degree of Q. Equivalently, This exhibits P as a coupling coefficient for the quantum algebra.If Q is given by (3.12), it follows from Theorem 3.2 that (4.3) Using that i x i x * i = 0, one may rewrite the exponent of q as where { , } is given by (2.15).Thus P = T Q, where T is the linear operator defined by Now let Q and Q be two highest weight vectors.Then, by an application of Schur's lemma to (4.6) This gives the following orthogonality property.q and k a nonnegative integer, where s is the degree of Q (or of Q ).
The function T Q is not a polynomial.To obtain orthogonal polynomials from Theorem 4.1, we introduce the new variables we obtain which is a polynomial in the variables x i of total degree |t|.Given a complete orthogonal (biorthogonal) family of highest weight vectors, Theorem 4.1 gives an orthogonality (biorthogonality) relation for the corresponding polynomials.For fixed k, one may view this as a system of polynomials in the n−1 variables x 1 ,...,x n−1 .By a dimension count, the latter system will be complete.
In particular, for n = 2 and Q as in (3.14) one has (4.10) Using transformation formulas from [10] one may check that where the right-hand side is a q-Hahn polynomial as defined in (2.8).Theorem 4.1 then gives the orthogonality relation for q-Hahn polynomials; confer [22,34].

Coupling kernels.
In [31] we introduced certain functions called coupling kernels.We expressed them explicitly as multivariable hypergeometric sums.The discrete coupling kernels, connected with the transform T 1 , were used in [30] to study Wigner 9j-symbols.The coupling kernels connected with T 2 were independently introduced by Xu [42], who used them to study Cesàro summability of multivariable Jacobi polynomial expansions.In this section we will generalize the explicit expression for discrete coupling kernels to the quantum algebra case.
Just as coupling coefficients are matrix elements for the intertwining maps Q : , coupling kernels are matrix elements for the intertwining projec- . More precisely, for any elements X, Y of the quantum algebra, consider the matrix element of Π s .Let (Q j ) j be an orthonormal basis of the space ᐂ s of highest weight vectors of degree s.Then, by Hilbert space arguments, and thus where We call the functions P s coupling kernels.Since is the reproducing kernel for the space ᐂ s , it follows from Theorem 4.1 that P s is the reproducing kernel for the corresponding space of coupling coefficients, with the reproducing property gives the addition formula for coupling kernels.Since, again by Hilbert space arguments, Π s e t ,e u z t wu , ( we may also write Π s e t ,e u q (1/8){ν+2l,ν+2t}+(1/8){ν+2m,ν+2u} . (5.9) We will use this identity to prove the following theorem.
Theorem 5.1.The coupling kernel P s is given by (5.10) with the convention 0/0 = 0, so that the sum is actually finite.
We remark that for n = 2 the space of coupling coefficients is one-dimensional, and therefore its reproducing kernel P s factors as a product of two q-Hahn polynomials.This gives a quantum algebraic proof of Rahman's Watson-type product formula for the q-Hahn polynomials [28].
The proof of Theorem 5.1 is similar to the case q = 1 treated in [31].It is based on the following projection formula.Lemma 5.2.Let Ᏼ s denote the subspace of n i=1 Ꮽ ν i q consisting of homogeneous polynomials of degree s.Then the restriction of Π s to Ᏼ s is given by (5.11) Since we failed to do so in [31], we give some history of this type of formulas.In the case q = 1 and the realization of su(1, 1) coming from spherical harmonics (X + = ∆, X − = i x 2 i generate a realization of (Ꮽ 1/2 ⊕Ꮽ 3/2 ) ⊗n on L 2 (R n )), Lemma 5.2 goes back to Clebsch [5]; confer also [7].For su(2), it was rediscovered and used by Löwdin [23] in the context of quantum physics.This was the starting point for projection operator methods in quantum physics and representation theory, developed by Ašerova, Smirnov and Tolstoy; confer [2,35,37].
We indicate the simple proof of Lemma 5.2.Using the fact that Ᏼ s is an eigenspace of K ± and the elementary identity (5.12) one proves by induction on k that We note that the restriction of Π s to Ᏼ s is the orthogonal projection onto the subspace ᐂ s of highest weight vectors.Denoting the operator in (5.11) by A, it follows from (5.13) that X + A = 0, so that AᏴ s ⊆ ᐂ s .On the other hand, the image of Id −A is in which is the orthogonal complement of ᐂ s in Ᏼ s .Thus A is indeed the orthogonal projection onto ᐂ s .Now let t and u be two multi-indices of length s.We want to compute the scalar product Π s e t ,e u occurring in (5.9).Since X * − = −X + , Lemma 5.2 gives (5.14) Now, by (2.23), which gives ( Inserting this expression in (5.9) gives × q (1/4){t−j,ν+2t}+(1/4){u−j,ν+2u}+(1/8){ν+2l,ν+2t}+(1/8){ν+2m,ν+2u} . (5.17) Replace t, u by t + j, u + j.The exponent of q may then be expressed as Thus, changing the order of summation, we obtain where the inner sums are of the form (2.16).This leads to the expression which may be rewritten as in Theorem 5.1.

Convolution and linearization formulas.
In the case q = 1, matrix elements for the group action in the basis of monomials are given by Meixner polynomials [18] where 2 F 1 is Gauss' hypergeometric function.The interpretation of the polynomials T 1 Q as coupling coefficients leads to the convolution formula and the linearization formula for Meixner polynomials, confer [31].In this section we will generalize these formulas to the present setting.Let τ λ be the q-translation operator occurring in Lemma 3.1.We will consider the matrix element which, by Lemma 3.1, equals Here we use the quantum algebra approach [8], considering q-exponentials of quantum algebra elements as generalized group elements.We remark that we would have obtained the same results considering instead the matrix elements where e q and E q are the q-exponential functions [10] e q (z) confer [9,14,44].Consider the identity (6.9) Inserting the expression (6.5) for matrix elements and letting c = 1/λµ we obtain the following proposition.
Proposition 6.1.For Q a highest weight vector in n i=1 Ꮽ ν i q of degree s, and for k a nonnegative integer, one has the identity (6.10) For n = 2 this is a degenerate case of identities proved in [20].We must point out that it is possible to deduce the general case from the case n = 2 by choosing a basis in the space of highest weight vectors constructed by binary coupling; confer Section 7. We also remark that for λ = 0, (6.8) takes the form This leads to the following proposition.Proposition 6.2.There is the identity Inserting the expression for coupling kernels given in Theorem 5.1, this is not hard to prove directly.It can be obtained as a special case of Verma's expansion formula [40], confer also [36].This gives an alternative proof of Theorem 5.1, since the expansion (6.13) determines the coefficients P s .For n = 2, factoring P s as a product of two Clebsch-Gordan coefficients, this linearization formula occurs in [14,34,39] in a similar context.
The special case λ = 0 of (6.12) is which, for |l| = |k|, reduces to For n = 2, factoring P s as a product of two terminating 3 φ 2 series, this is the orthogonality relation for dual q-Hahn polynomials.

Examples.
In this section we will consider some examples of orthogonal and biorthogonal multivariable Hahn polynomials which may be obtained by our method.
One way to construct orthogonal bases in the space of coupling coefficients is by binary coupling.This leads to multivariable polynomials which may be factored as products of q-Hahn polynomials.For q a power of a prime, such polynomials arise in connection with finite fields [6].In the limit q → 1, the corresponding polynomials are of importance in the quantum theory of angular momentum, and they have also been used in certain stochastic models in genetics [15].To obtain them as coupling coefficients, we first note that if q are highest weight vectors of degrees t 1 , t 2 and weights −µ 1 , −µ 2 , respectively, and Q s is a highest weight vector of degree s in Ꮽ q (unique up to a multiplicative constant), then the equation where m = (m 1 ,...,m k ), m = (m k+1 ,...,m n ).Moreover, one has the identity As an example, the polynomials form an orthogonal basis in the space of highest weight vectors of degree s in a threefold tensor product Ꮽ q .Applying the transform T gives a system of orthogonal polynomials which, by (7.2), factor as products of two q-Hahn polynomials.The orthogonality relation, as well as the convolution formula in Proposition 6.1, then follow from the corresponding identities for q-Hahn polynomials.The coefficients (1, 1) q , 1 r , 1,(1, 1) t u (7.5) are Racah coefficients, or Wigner 6j-symbols, for ᐁ q 1/2 (su(1, 1)), which may be identified with q-Racah polynomials [16].For an (n+1)-fold tensor product, there is a large number of ways to construct orthogonal bases by binary coupling, and the coefficients for a change between two such bases are given by Wigner 3nj-symbols.As a different example of coupling coefficients, we will construct q-analogues of certain biorthogonal multivariable Hahn polynomials introduced by Rahman [27] for n = 3 and by Tratnik [38] in general.For Jacobi-type weights, this kind of polynomials were first studied by Appell; confer [1].In the case q = 1, they correspond to highest weight vectors of the form and to the dual basis.
It will be convenient to write Ᏸ q (for Dirichlet, cf.Appendix A) for the space of polynomials in one variable, viewed as a ᐁ-module with the action which may be obtained by formally letting ν = 0 in (2.28).Write δ for the dilation operator δf (z) = f (qz), so that K = q −ν/4 δ −1/2 as an operator on Ꮽ ν q .Thus (7.8) where D q is the q-derivative (2.9), while Applying the commutation rule D q δ λ = q λ δ λ D q (7.10) gives the following lemma.
Lemma 7.1.The operators and X + | Ᏸ ⊗n q are connected by where In particular, this shows that if ᐂ s is the space of highest weight vectors of degree s in n i=1 Ꮽ ν i q , then Sᐂ s is, as a space of polynomials, independent of the parameters ν i .Lemma 7.2.Let j be an integer with 1 ≤ j ≤ n.If f and g are two elements in Ᏸ ⊗n q such that f depends only on the variables z 1 ,...,z j and g only on z j ,...,z n , then Proof.For n = 1 this is (2.10).To prove it in general, decompose Ᏸ ⊗n q as Ᏸ ⊗(j−1) q ⊗ Ᏸ q ⊗ Ᏸ ⊗(n−j) q . By linearity, one may assume The lemma then follows from the case n = 1, using the coproduct rule The lemma also follows from the results of Appendix B, where we describe a product • on Ᏸ ⊗n q which satisfies and which agrees with the usual product of polynomials under the assumptions of the lemma.
vectors.It follows from (5.9) that, in general, which gives We insert the explicit expression for P s from Theorem 5.1.Then the summation variable j n may be put equal to zero.Writing x = (x 1 ,...,x n−1 ) for x ∈ R n , we obtain (which should be interpreted as a finite sum, as in Theorem 5.1).Thus, as a consequence of Theorem 4.1, we have the following fact.
As indicated in Section 4, we may view this as a biorthogonality relation for a complete system of (n−1)-variable polynomials.It may be worth writing this out explicitly, in standard notation.To facilitate comparison with the one-variable case, we view the q ν i −1 as parameters and normalize the polynomials so that they take the value 1 at ; q ν 1 −1 ,...,q νn−1 , |m|; q , (7.33) and p t for the polynomials similarly obtained from P t .Replacing n by n + 1, and writing Φ 1:2 1:1 for the q-Kampé de Fériet function b n ; q kn c n ; q kn (q; q) kn e n ; q kn x k 1 ,...,a n bx n−1 q N+tn+2 , p t x 1 ,...,x n ; a 1 ,...,a n ,b,N; q In terms of these polynomials, Theorem 7.4 may be rewritten as × p t q −m 1 ,...,q −mn ; a 1 ,...,a n ,b,N; q p u q −m 1 ,...,q −mn ; a 1 ,...,a n ,b,N; q = δ tu (q; q) t 1 •••(q; q) tn a 1 q; q t 1 ••• a n q; q tn (bq; q) |t| q a j q t i . (7.36)

Appendices
A. Quasi-commuting homogeneous coordinates.In this appendix we will use quasi-commuting homogeneous coordinates to give a "geometric" motivation for the holomorphic realization, where geometric may be understood in the sense of noncommutative geometry (cf.[25]).First recall the classical case.If f is a function defined in the unit disc, one may (at least for integer ν) introduce a function F defined in the cone |x| < |y| in C 2 by The natural right action on F by SU(1, 1), then gives rise to the action (1.3) on f .ᐁ q 1/2 (su(1, 1))-module.Its completion is the quantum Dirichlet space, consisting of analytic functions on the disc |z| < q −1/2 , modulo constants, with the norm It is equivalent to the quantum Bergman space Ꮽ 2 q , an intertwining map being the q-derivative D q .
B. The algebra of highest weight vectors.In this section we will describe the kernel of X + in n i=1 Ꮽ ν i q .In the case q = 1, X + = − n i=1 ∂/∂z i satisfies the Leibniz rule X + (f g) = X + (f )g + f X + (g), so the kernel is formally an associative algebra.(The product is unbounded, but one can obtain a well-defined algebra for instance as the subspace of polynomials.)In this appendix we will find a product • on the space of polynomials in n i=1 Ꮽ ν i q which satisfies so that it gives an algebra structure to the space of polynomials annihilated by X + , that is, to the span of the highest weight vectors.First we recall the universal R-matrix R = q (1/4)(H⊗H) ∞ j=0 1 − q −1 j [j]! q (1/4)j(j−1) KX + j ⊗ K −1 X − j , ( where H satisfies q (1/4)H = K.It may be viewed as an element of a suitable extension of ᐁ ⊗ ᐁ.For "nice" ᐁ-modules V 1 and V 2 , is intertwining, where σ is the flip σ (f ⊗ g) = g ⊗ f .We will only need this when There is a canonical way to give a module algebra structure to a tensor product of module algebras, known as the braided tensor product [24].Namely, if A 1 and A 2 are ᐁ-module algebras with products then A 1 ⊗ A 2 is another one with the product This construction is associative in the sense that regardless of how it is iterated to define a module algebra structure on a finite tensor product A 1 ⊗ ••• ⊗ A n , the result is the same.Consider the space Ᏸ q ⊗ Ᏸ q , consisting of polynomials in two variables z 1 and z 2 , and view it as a sub-module algebra of the braided tensor product Ᏺ q ⊗ Ᏺ q .Thus we write z 1 = q 1/4 xy −1 ⊗ 1, z 2 = 1 ⊗ q 1/4 xy −1 .(B.6) Denoting by • the product defined by (B.5), we have where the product on the right-hand side is the usual product of polynomials, while z 2 • z 1 = Ψ q 1/4 xy −1 ⊗ q 1/4 xy −1 = q 3/2 xy −1 ⊗ xy −1 + (1 − q)x 2 y −2 ⊗ 1 By associativity, this determines the product • uniquely.One may prove by induction that, more generally, the product on Ᏸ ⊗n q considered as a sub-module algebra of Ᏺ ⊗n q is determined by  z i z j , i ≤ j, qz j z i + (1 − q)z 2 j , i > j. (B.9)With this product, the kernel of X + is a graded algebra KerX + | Ᏸ ⊗n q = ∞ s=0 ᐂs , where ᐂs consists of homogeneous polynomials of degree s.It is generated by ᐂ0 , which is the space of constants, together with ᐂ1 = {a 1 z 1 +•••+a n z n ; a 1 +•••+a n = 0}.By Lemma 7.1, these observations carry over to the space n i=1 Ꮽ ν i .The span of all highest weight vectors is a graded algebra generated by 1 and ᐂ 1 = S −1 ᐂ1 , with the product given on z i by (1/4)(ν * i −ν * j ) z 2 j , i > j, (B.11)where ν * i − ν * j = ν j + 2(ν j+1 +•••+ν i−1 ) + ν i .For n = 2, the space ᐂs is one-dimensional and generated by (z 1 − z 2 ) •s .Now, by a generalized binomial formula due to Benaoum [3], (cf.also [33]), one has for variables satisfying z 2 z 1 = qz 1 z 2 + (1 − q)z 2 1 .However, for commuting variables, the right-hand side of this equation equals

Theorem 4 . 1 .
For Q and Q two highest weight vectors in n i=1 Ꮽ ν i