  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.
