A combinatorial formula for Macdonald polynomials

Haglund, J.¹ ; Haiman, M.² ; Loehr, N.^1, 3

¹ Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
² Department of Mathematics, University of California, Berkeley, California 97420-3840
³ Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187-8795

Journal of the American Mathematical Society, Tome 18 (2005) no. 3, pp. 735-761

Résumé

We prove a combinatorial formula for the Macdonald polynomial $\tilde {H}_{\mu }(x;q,t)$ which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of $\tilde {H}_{\mu }(x;q,t)$ in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schützenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi’s combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients $\tilde {K}_{\lambda \mu }(q,t)$ in the case that $\mu$ is a partition with parts $\leq 2$.

Détail

DOI : 10.1090/S0894-0347-05-00485-6

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