A Markov chain on the symmetric group which is Schubert positive?

TL;DR

This paper investigates a multivariate Markov chain on the symmetric group with unique enumerative properties and conjectures a positive Schubert polynomial-based stationary distribution, generalizing previous affine Weyl group studies.

Contribution

It introduces a new multivariate Markov chain on the symmetric group and conjectures a novel Schubert polynomial-based stationary distribution.

Findings

Enumerative properties of the chain are remarkable

Conjecture of Schubert positivity in the stationary distribution

Generalization of affine Weyl group Markov chain

Abstract

We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.