Combinatorial theory of Macdonald polynomials I: Proof of Haglund's formula

^*Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395; and ^‡Department of Mathematics, University of California, Berkeley, CA 94720-6395

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved December 7, 2004 (received for review November 15, 2004)

Abstract

Haglund recently proposed a combinatorial interpretation of the modified Macdonald polynomials H̃ _μ. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of H̃ _μ. As corollaries, we obtain the cocharge formula of Lascoux and Schützenberger for Hall–Littlewood polynomials, a formula of Sahi and Knop for Jack's symmetric functions, a generalization of this result to the integral Macdonald polynomials J _μ, a formula for H̃ _μ in terms of Lascoux–Leclerc–Thibon polynomials, and combinatorial expressions for the Kostka–Macdonald coefficients K̃ _λ,μ when μ is a two-column shape.

Footnotes

↵ † To whom correspondence should be addressed. E-mail: jhaglund{at}math.upenn.edu.
This paper was submitted directly (Track II) to the PNAS office.
Abbreviations: iff, if and only if; RSK, Robinson–Schensted-Knuth; LLT, Lascoux–Leclerc–Thibon.