The monomial expansions of modified Macdonald polynomials

TL;DR

This paper introduces a new combinatorial framework with sixteen statistics to derive explicit and compact formulas for modified Macdonald polynomials, enhancing understanding of their monomial expansions.

Contribution

It presents a novel family of statistics on Young diagram fillings and establishes new combinatorial formulas for modified Macdonald polynomials, including explicit monomial expansion expressions.

Findings

New combinatorial formulas for modified Macdonald polynomials

Four compact formulas involving canonical and dual canonical fillings

Explicit monomial expansion formulas, including one matching Garbali and Wheeler (2020)

Abstract

We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in T(\lambda)}x^{\sigma}q^{maj(\sigma)}t^{\eta(\sigma)}$$ for each statistic $\eta\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma}d_{\varepsilon}(\sigma)x^{\sigma}q^{maj(\sigma)}t^{\eta(\sigma)}$$ which is summed over all canonical or dual canonical fillings of a Young diagram and $d_{\varepsilon}(\sigma)$ is a product of $t$-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 given by Garbali and Wheeler (2020).