A compact formula for the symmetric Macdonald polynomials

Abstract: We give a compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ in terms of a queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle. Moreover, the statistics in our formula are naturally related to the dynamics of the ASEP. Our tableaux are in bijection with Martin's multiline queues (arXiv:1810.10650), from which we obtain an alternative multiline queue formula for $P_{\lambda}$.