Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials

Abstract

We prove a new tableaux formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ that has considerably fewer terms and simpler weights than previously existing formulas. Our formula is a sum over certain sorted non-attacking tableaux, weighted by the queue inversion statistic quinv. The quinv statistic originates from a formula for the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ due to Ayyer, Martin, and the author (2022), and is naturally related to the dynamics of the asymmetric simple exclusion process (ASEP) on a circle. We prove our results by introducing probabilistic operators that act on non-attacking tableaux to generate a set of tableaux whose weighted sum equals $P_{\lambda}(X;q,t)$. These operators are a modification of the inversion flip operators of Loehr and Niese (2012), which yield an involution on tableaux that preserves the major index statistic, but fails to preserve the non-attacking condition. Our tableaux are in bijection with the multiline queues introduced by Martin (2020), allowing us to derive an alternative multiline queue formula for $P_{\lambda}(X;q,t)$. Finally, our formula recovers an alternative formula for the Jack polynomials $J_{\lambda}(X;\alpha)$ due to Knop and Sahi (1996) using the same queue inversion statistic.