The University of Michigan Combinatorics Seminar
Fall 2008
October 10, 4:10-5:00, 3866 East Hall

On combinatorial formulas for Macdonald polynomials

Cristian Lenart

SUNY at Albany


Macdonald polynomials (of type A) are generalizations of Schur polynomials depending on two parameters. Haglund, Haiman and Loehr exhibited a combinatorial formula for these 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.