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A combinatorial formula for Macdonald polynomials. (English) Zbl 1061.05101
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 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 \(\widetilde{K}_{\lambda \mu}(q,t)\) in the case that \(\mu\) is a partition with parts \(\leq 2\).

MSC:
05E10 Combinatorial aspects of representation theory
05A30 \(q\)-calculus and related topics
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