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A combinatorial approach to the q,t-symmetry in Macdonald polynomials

  • Author(s): Gillespie, Maria Monks
  • Advisor(s): Haiman, Mark D
  • et al.
Abstract

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H_mu(X;q,t)=H_{mu*}(X;t,q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q=0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization t=1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

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