Breakthroughs in the theory of Macdonald polynomials

Abstract

In 1998, I. G. Macdonald (1) introduced a remarkable new basis for the space of symmetric functions. The elements of this basis are denoted , where λ is a partition and p, q are two free parameters. The 's, which are now called “Macdonald polynomials,” specialize to many of the well known bases for the symmetric functions, by suitable choices of the parameters q and t. In fact, we can obtain in this manner the Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the zonal symmetric functions, the zonal spherical functions, and the elementary and monomial symmetric functions.