A combinatorial formula for Macdonald polynomials

Authors:
  • JHJ. Haglund
    J. Haglund
  • MHM. Haiman
    M. Haiman
  • NLN. Loehr
    N. Loehr
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Abstract
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 .

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