Séminaire Lotharingien de Combinatoire, 82B.97 (2019), 12 pp.

Sylvie Corteel, Olya Mandelshtam, and Lauren Williams

From multiline queues to Macdonald polynomials via the exclusion process

Abstract. Recently James Martin introduced \emph{multiline queues}, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion exclusion process (ASEP) on a circle. The ASEP is a model of particles hopping on a one-dimensional lattice, which has been extensively studied in statistical mechanics, probability, and combinatorics. In this article we give an independent proof of Martin's result, and we show that by introducing additional statistics on multiline queues, we can use them to give a new combinatorial formula for both the symmetric Macdonald polynomials Pλ(x;q,t), and the nonsymmetric Macdonald polynomials Eλ(x;q,t), where λ is a partition. This formula is rather different from others that have appeared in the literature (Haglund-Haiman-Loehr'05, Ram-Yip'11, and Lenart'09). Our proof uses results of Cantini, de Gier, and Wheeler who recently linked the multispecies ASEP on a circle to Macdonald polynomials.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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