A combinatorial formula for Macdonald polynomials

Haglund, J.; Haiman, M.; Loehr, N.

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

A combinatorial formula for Macdonald polynomials
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by J. Haglund, M. Haiman and N. Loehr

J. Amer. Math. Soc. 18 (2005), 735-761

DOI: https://doi.org/10.1090/S0894-0347-05-00485-6

Published electronically: April 8, 2005

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Abstract:

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 $\leq 2$.

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