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arXiv · 1503.02109

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

Maria Monks Gillespie
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Abstract

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t) = \widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. 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 $\mathbf{x}$ 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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Maria Monks Gillespie. 2015-03-06. A combinatorial approach to the q,t-symmetry relation in Macdonald polynomials. https://arxiv.org/abs/1503.02109

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