Combinatorial Markov chains on linear extensions

TL;DR

This paper explores generalized Markov chains on linear extensions of posets, revealing new stationary distributions, explicit eigenvalues for rooted forests, and extending classical results in combinatorics and algebra.

Contribution

It introduces new Markov chains based on promotion operators on linear extensions, providing explicit eigenvalues and stationary distributions, and generalizes known results to broader poset classes.

Findings

Stationary distribution for one chain is uniform

Stationary distribution for the other has a product formula

Eigenvalues explicitly computed for rooted forests

Abstract

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.