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ISSN 1088-6834(e) ISSN 0894-0347(p)

A combinatorial formula for Macdonald polynomials

Author(s): J. Haglund; M. Haiman; N. Loehr
Journal: J. Amer. Math. Soc. 18 (2005), 735-761.
MSC (2000): Primary 05E10; Secondary 05A30
Posted: April 8, 2005
MathSciNet review: 2138143
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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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