A combinatorial formula for Macdonald polynomials
by
Mark Haiman
U.C. Berkeley
Coauthors: Jim Haglund and Nick Loehr

We prove a a combinatorial formula conjectured by Haglund for the Macdonald polynomial [H\tilde]_m(x;q, t). Such a combinatorial formula had been sought since Macdonald introduced his polynomials in 1988. The new formula has various pleasant consequences, including the expansion of Macdonald polynomials in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, and a new proof (and more general version) of Knop and Sahi's combinatorial formula for Jack polynomials. In general, our formula doesn't yet give a new proof of the positivity theorem for Macdonald polynomials, because it expresses them in terms of monomials, rather than Schur functions. However, it does yield a new combinatorial expression for the Schur function expansion when the partition has parts less than or equal to 2, and there is hope to extend this result.

Paper reference: arXiv:math.CO/0409538

Date received: February 24, 2005