BCn-symmetric polynomials

Abstract

We consider two important families of BCn-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.

Deformations of permutation representations of Coxeter groups

“…The motivation for the machinery developed in this paper arose when the first author was investigating certain conjectures involving Macdonald polynomials generalizing classical identities of Schur functions related to the representation theory of symmetric spaces [9]. These Schur function identities are closely related to Littlewood identities and thus via invariant theory [1, §8] to the action of S 2n by conjugation on fixed-point-free involutions.…”

Vanishing integrals for Hall–Littlewood polynomials

“…In this section, we prove four other vanishing identities from [20] and [18]. In all four cases, the structure of the partition that produces a nonvanishing integral is the same: opposite parts must add to zero (λ i + λ l+1−i = 0 for all 1 ≤ i ≤ l, where l is the total number of parts).…”

“…We find a two-parameter integral identity and, using a method of Rains [18], we show that in the limit n → ∞ it becomes Warnaar's identity. Thus, the following identity may be viewed as a finite-dimensional analog of Warnaar's summation result: Theorem 1.9.…”

“…. , z n , z −1 n ) In 2005, Rains [18] conjectured the existence of (q, t)-analogs of such restriction rules for Schur functions. That is, he conjectured choices of densities such that when one integrates a (suitably specialized) Macdonald polynomial against it over the n-torus, the result vanishes unless the indexing partition satisfies some explicit condition.…”

Macdonald Polynomials and Multivariable Basic Hypergeometric Series

“…The latter are specific 2 φ 1 series, see (7.1), which when principally specialized simplify to ratios of q-shifted factorials by virtue of the Chu-Vandermonde summation theorem, the terminating special case of the q-Gauß summation in (3.2). We mention that for the nonreduced irreducible root system BC n very-well-poised basic hypergeometric series identities involving Okounkov's [44] Macdonald interpolation polynomials or the more general Koornwinder-Macdonald polynomials (both which are of BC n type) have been established by Rains, see [45,Sec. 4] and [46,Sec.…”