  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.
