The Combinatorics of Macdonald Polynomials
by Dr Michael Wheeler
Abstract: The Macdonald polynomials are a family of multivariate, symmetric functions introduced by Ian Macdonald in the 1980s. They mutually generalize a number of important subfamilies, including the Schur, Hall-Littlewood and Jack polynomials. Since their discovery, Macdonald polynomials have been very well studied by mathematicians, prized for their deep and beautiful connections with representation theory and algebraic geometry. In the last few years, they have also come to play a key role both in theoretical physics and exactly-solvable stochastic systems. In the opening half of this talk, I will review some of these developments.
In the latter half of the talk, I will outline two combinatorial developments in which I have recently been involved. The first of these is an explicit formula for Macdonald polynomials, allowing them to be written as the trace of a certain infinite-dimensional matrix. This formula makes visible some of the finer combinatorial structure of the polynomials, and is closely related with steady-state probabilities of the asymmetric simple exclusion process (ASEP). The second is a new, positive formula for the Kostka-Foulkes polynomials, which are the transition coefficients between the Schur and Hall-Littlewood bases. This formula can be seen as a q-deformation of puzzles originally introduced by Allen Knutson and Terence Tao.