Modified Macdonald polynomials and mu-Mahonian statistics

The Haglund–Haiman–Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $${\tilde{H}}_{\mu }(X;q,t)=\sum _{\sigma :\mu \to \mathbb{P}}{x}^{\sigma }{t}^{maj(\sigma )}{q}^{inv(\sigma )}\mathrm{.}$$ 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 $[\tau ]$.