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

Maria Monks Gillespie¹

¹University of California, Berkeley

The electronic journal of combinatorics, Tome 23 (2016) no. 2

Résumé

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.

Détail

DOI : 10.37236/5350

Classification : 05E10, 05E05, 33D52
Mots-clés : Macdonald polynomials, Hall-Littlewood polynomials, Young tableaux, cocharge, Garsia-Procesi modules, Mahonian statistics