Properties of the Promotion Markov Chain on Linear Extensions

The Tsetlin library is a very well studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. This result has been generalized in different ways by various people. In this work we investigate one of the generalizations given by the extended promotion Markov Chain on linear extensions of a poset $P$ introduced by Ayyer, Klee, and Schilling in 2014. They showed that if the poset $P$ is a rooted forest, the transition matrix of this Markov chain has eigenvalues that are linear in the transition probabilities and described their multiplicities. We show that the same property holds for a larger class of posets for which we also derive convergence to stationarity results.