On Combinatorial Formulas for Macdonald Polynomials

Macdonald polynomials are generalizations of the irreducible characters of semisimple Lie algebras depending on two parameters; they appear in statistical physics, as eigenfunctions of a certain family of commuting differential operators. Haglund, Haiman and Loehr exhibited a combinatorial formula for the type A Macdonald polynomials in terms of a pair of statistics on fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary Lie type in terms of the corresponding affine Weyl group. In this talk, I relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms; in this context, the statistics on Young diagrams mentioned above follow naturally from more general concepts. I also explain how this work extends to types B and C, where no analog of the Haglund-Haiman-Loehr formula exists. The talk is largely self-contained.