Monomial expansions of modified Macdonald polynomials

Abstract

We discover a family $A$ of statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $${\tilde{H}}_{\lambda }(X;q,t)=\sum _{\sigma \in T(\lambda )}{x}^{\sigma }{t}^{maj(\sigma )}{q}^{\eta (\sigma )}$$ for each statistic $\eta \in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $${\tilde{H}}_{\lambda }(X;q,t)=\sum _{\sigma }{d}_{\epsilon }(\sigma )x^{\sigma }{t}^{maj(\sigma )}{q}^{\eta (\sigma )},$$ which are summed over all canonical or dual canonical fillings of a Young diagram, where $d_{\epsilon }(\sigma )$ is a product of $q$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula due to Garbali and Wheeler (2020), obtained by making extensive use of the Yang-Baxter integrability approach.

arXiv comment36 Pages, 3 Figures