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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.

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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