Séminaire Lotharingien de Combinatoire, 85B.28 (2021), 12 pp.

Donghyun Kim and Lauren Williams

Schubert Polynomials and the Inhomogeneous TASEP on a Ring

Abstract. Consider a lattice of n sites arranged around a ring, with the n sites occupied by particles of weights {1,2,...,n}; the possible arrangements of particles in sites thus corresponds to the n! permutations in Sn. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which two adjacent particles of weights i<j swap places at rate xi-yn+1-j if the particle of weight j is to the right of the particle of weight i. (Otherwise nothing happens.) In the case that yi=0 for all i, the stationary distribution was conjecturally linked to Schubert polynomials by Lam-Williams, and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues by Ayyer-Linusson and Arita-Mallick. In the case of general yi, Cantini showed that n of the n! states have probabilities proportional to double Schubert polynomials. In this paper we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns 2413, 4132, 4213 and 3214. We show that there are (1/2) (2+21/2)n-1 + (2-21/2)n-1) evil-avoiding permutations in Sn, and for each evil-avoiding permutation w, we give an explicit formula for the steady state probability ψw as a product of double Schubert polynomials. We also show that the Schubert polynomials that arise in these formulas are flagged Schur functions, and give a bijection in this case between multiline queues and semistandard Young tableaux.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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