Modified Macdonald polynomials and mu-Mahonian statistics

TL;DR

This paper proves a conjecture relating two Mahonian statistics, inv and quinv, on Young diagram fillings, confirming their equidistribution and invariance in the context of modified Macdonald polynomials.

Contribution

It establishes the equidistribution of inv and quinv statistics on Young diagram fillings, confirming a conjecture and advancing understanding of Macdonald polynomial combinatorics.

Findings

Proves the Ayyer--Mandelshtam--Martin conjecture.

Shows equidistribution of (inv, maj) and (quinv, maj) on row-equivalence classes.

Demonstrates distributional symmetry for rectangular fillings.

Abstract

The Haglund--Haiman--Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $$\tilde{H}_{\mu}(X;q,t)=\sum_{\sigma: \mu\rightarrow \mathbb{P}}x^{\sigma}t^{maj(\sigma)}q^{inv(\sigma)}.$$ Inspired by Martin's multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic $quinv$ and conjectured that the tableaux formula for $\tilde{H}_{\mu}(X;q,t)$ is invariant if the inversion statistic $inv$ is replaced by $quinv$. This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for $\tilde{H}_{\mu}(X;q,t)$. Our main result confirms this Ayyer--Mandelshtam--Martin conjecture. We establish an equidistribution between the pairs $(inv,maj)$ and $(quinv,maj)$ of $\mu$-Mahonian statistics on any row-equivalency class $[\tau]$, where $\tau$ is a filling of the Young diagram of $\mu$. As a byproduct of our approach, we show that if $\tau$ is a rectangular filling, the triples $(inv,quinv,maj)$ and $(quinv,inv,maj)$ have the same distribution over $[\tau]$.