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A combinatorial formula for Macdonald polynomials

2005; American Mathematical Society; Volume: 18; Issue: 3 Linguagem: Inglês

10.1090/s0894-0347-05-00485-6

1088-6834

J. Haglund, Mark Haiman, Nicholas A. Loehr,

Advanced Algebra and Geometry

We prove a combinatorial formula for the Macdonald polynomial H ~ μ ( x ; q , t ) \tilde {H}_{\mu }(x;q,t) which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of H ~ μ ( x ; q , t ) \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 K ~ λ μ ( q , t ) \tilde {K}_{\lambda \mu }(q,t) in the case that μ \mu is a partition with parts ≤ 2 \leq 2 .