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

2015/03/06 by Maria Gillespie, Gillespie, Maria Monks
Mathematics · #05A19 #05E05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1503.02109

openalex publication_date 2015/03/06 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Using the combinatorial formula for the transformed Macdonald polynomials of\nHaglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry\nrelation widetildeH_\μ(\x;q,t) =\n widetildeH_\μ^\∗(\x;t,q). We provide a purely combinatorial\nproof of the relation in the case of Hall-Littlewood polynomials (q=0) when\n\μ is a partition with at most three rows, and for the coefficients of the\nsquare-free monomials in \x for all shapes \μ. We also provide a\nproof for the full relation in the case when \μ is a hook shape, and for all\nshapes at the specialization t=1. Our work in the Hall-Littlewood case\nreveals a new recursive structure for the cocharge statistic on words.\n

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