Modified Macdonald polynomials and the multispecies zero range process: I

TL;DR

This paper introduces a new combinatorial formula for modified Macdonald polynomials involving queue inversions, linking them to multispecies zero-range particle systems and their stationary probabilities.

Contribution

It provides a novel tableau-based combinatorial formula for modified Macdonald polynomials and establishes a connection to the stationary distribution of the multispecies zero-range process.

Findings

New tableau formula for modified Macdonald polynomials

Connection between Macdonald polynomials and multispecies zero-range process

Stationary probabilities computed via queue inversion statistic

Abstract

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials $\widetilde{H}_\lambda(X;q,t)$, motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials $P_\lambda(X;q,t)$. In the case $q=1$ and $X=(1,1,\dots,1)$, that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary $X=(x_1,\dots, x_n)$; the $x_i$ play the role of site-dependent jump rates for the mTAZRP.