A combinatorial approach to the \(q,t\)-symmetry relation in Macdonald polynomials (Q289979)

Summary: Using the combinatorial formula for the transformed Macdonald polynomials of \textit{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.