Séminaire Lotharingien de Combinatoire, B41a (1998), 29pp.
Some Classical Expansions for Knop-Sahi and Macdonald Polynomials
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.
Received: August 12, 1998; Accepted: September 15, 1998.
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