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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