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

TL;DR

This paper explores the combinatorial symmetry of Macdonald polynomials, providing new proofs and revealing recursive structures in special cases, advancing understanding of their algebraic and combinatorial properties.

Contribution

It offers a purely combinatorial proof of the q,t-symmetry for specific shapes and specializations, introducing new recursive structures in the cocharge statistic.

Findings

Proof of symmetry for Hall-Littlewood polynomials with up to three rows

Symmetry proof for hook-shaped partitions at all parameters

Discovery of a new recursive structure in the cocharge statistic

Abstract

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.