New formulas for Macdonald polynomials via the multispecies exclusion and zero range processes

TL;DR

This paper uncovers new formulas for Macdonald polynomials by connecting them to multispecies exclusion and zero range particle processes, revealing combinatorial structures and plethystic relationships.

Contribution

It introduces novel formulas for Macdonald polynomials derived from particle models and explains their combinatorial and plethystic connections.

Findings

New formula for $P_{oldsymbol{ u}}$ using queue inversion statistic

Connection between multiline queues and tableaux with queue inversion

Relation of plethystic transformation to integrable systems fusion

Abstract

We describe some recently discovered connections between one-dimensional interacting particle models and Macdonald polynomials. The first such model is the multispecies asymmetric simple exclusion process (ASEP) on a ring, linked to the symmetric Macdonald polynomial $P_{\lambda}(X;q,t)$ through its partition function. Through this connection, a new formula was found for $P_{\lambda}$ by generalizing multiline queues, which were introduced by Martin in 2018 to compute stationary probabilities of the ASEP. The second particle model is the multispecies totally asymmetric zero range process (TAZRP) on a ring, which was very recently found to have an analogous connection to the modified Macdonald polynomial $\widetilde{H}_{\lambda}(X;q,t)$ through its partition function. This discovery coincided with a new formula for $\widetilde{H}_{\lambda}$, this time in terms of tableaux with a queue inversion statistic, which also compute stationary probabilities of the TAZRP. We explain the plethystic relationship between multiline queues and queue inversion tableaux, and along the way, derive a new formula for $P_{\lambda}$ using the queue inversion statistic. This plethystic correspondence is closely related to fusion in the setting of integrable systems.