Modified Macdonald polynomials and mu-Mahonian statistics

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 $[\tau]$.