10 - Combinatorial Aspects of Macdonald and Related Polynomials

Summary

This chapter contains a survey of known results and open problems connected to the combinatorics of (type A) Macdonald polynomials. Macdonald polynomials are symmetric functions in a set of variables which depend on two extra parameters q,t. They include most of the commonly studied bases for the ring of symmetric functions, such as Schur functions and Hall-Littlewood polynomials, as special cases. Macdonald polynomials have geometric interpretations which make them important to algebraic geometry and mathematical physics, and are also fundamental to the study of special functions. Their combinatorial properties are rather mysterious, although a lot of progress has been made on the type A case in the past 20 years, in conjunction with the study of the representation theory of the ring of diagonal coinvariants. This survey shows how to express Macdonald polynomials, and other important objects such as the bigraded Hilbert series of the diagonal coinvariant ring, in terms of popular combinatorial structures including tableaux, Dyck paths, and parking functions.