Markov chains for promotion operators

TL;DR

This paper explores Markov chains based on promotion operators on linear extensions of posets, providing eigenvalue analysis, mixing time bounds, and generalizations to symmetric groups, extending previous work on rooted forests.

Contribution

It generalizes promotion-based Markov chains to arbitrary posets and symmetric groups, offering explicit eigenvalues, mixing time bounds, and conjectures for broader classes.

Findings

Eigenvalues of transition matrices for rooted forests are explicitly determined.

Explicit bounds on mixing times are provided.

Conjectured eigenvalue formulas for general posets are proposed.

Abstract

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset. This gives rise to a strongly connected graph on L. In earlier work (arXiv:1205.7074), we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of R-trivial monoids in an essential way to obtain explicitly the eigenvalues of the transition matrix in general when the poset is a rooted forest. We first survey these results and then present explicit bounds on the mixing time and conjecture eigenvalue formulas for more general posets. We also present a generalization of promotion to arbitrary subsets of the symmetric group.