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A combinatorial approach to the $$q,t$$-symmetry relation in Macdonald polynomials. (English) Zbl 1337.05109
Summary: Using the combinatorial formula for the transformed Macdonald polynomials of J. Haglund et al. [J. Am. Math. Soc. 18, No. 3, 735–761 (2005; Zbl 1061.05101)], 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.

##### MSC:
 05E10 Combinatorial aspects of representation theory 05E05 Symmetric functions and generalizations 33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
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