Some Classical Expansions for Knop-Sahi and Macdonald Polynomials

Jennifer Morse

Séminaire lotharingien de combinatoire, Tome 41 (1998)

Résumé

In recent simultaneous work, Knop and Sahi introduced a non-homogeneous non-symmetric family of polynomials whose highest homogeneous component gives the non-symmetric Macdonald polynomials. It has been shown that an appropriate Hecke algebra symmetrization of these non-symmetric polynomials yields the symmetric Macdonald polynomials. In the original papers, all these polynomials are only shown to exist. No explicit expressions are given relating them to the more classical bases. Our basic discovery here is that the Knop-Sahi polynomials appear to have surprisingly elegant expansions in terms of a q-shifted monomial basis. In this paper we present the first results obtained in the problem of determining the connection coefficients relating the Knop-Sahi basis to the q-shifted monomial basis. In particular we give a solution to the two variable case. Our proofs rely heavily on the theory of basic hypergeometric series and reveal a connection between this classical subject and the theory of Macdonald polynomials.