Jennifer Morse, University of Pennsylvania, U.S.A.
e-mail: morsej@math.upenn.edu

Expressions for Macdonald polynomials and the generalization of Macdonald's positivity conjecture

Abstract

We give a determinantal expression for the Macdonald basis which immediately produces efficient recursions for the Macdonald, Jack, and Schur polynomials (Kostka numbers).

We then discuss the long standing open problem of determining statistics for certain expansion coefficients for the Macdonald polynomials. We present recent progress that has been made using tableaux combinatorics and in particular, give a generalization of the Macdonald positivity conjecture.