an:02172232
Zbl 1061.05101
Haglund, J.; Haiman, M.; Loehr, N.
A combinatorial formula for Macdonald polynomials
EN
J. Am. Math. Soc. 18, No. 3, 735-761 (2005).
00116261
2005
j
05E10 05A30
charge formula; Hall-Littlewood polynomials; Jack polynomials; Kostka-Macdonald coefficients
Summary: We prove a combinatorial formula for the Macdonald polynomial \(\widetilde{H}_{\mu}(x;q,t)\) which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of \(\widetilde{H}_{\mu}(x;q,t)\) in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Sch??tzenberger 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 \(\widetilde{K}_{\lambda \mu}(q,t)\) in the case that \(\mu\) is a partition with parts \(\leq 2\).