Abstract
We consider generalizations of Schützenberger’s promotion operator on the set \(\mathcal{L}\) of linear extensions of a finite poset. This gives rise to a strongly connected graph on \(\mathcal{L}\). In earlier work (Ayyer et al., J. Algebraic Combinatorics 39(4), 853–881 (2014)), we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of \(\mathcal{R}\)-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.
Dedicated to Mohan Putcha and Lex Renner on the occasion of their 60th birthdays
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Acknowledgements
A.A. would like to acknowledge support from MSRI, where part of this work was done. S.K. was supported by NSF VIGRE grant DMS–0636297. A.S. was supported by NSF grant DMS–1001256 and OCI–1147247.
We would like to thank the organizers Mahir Can, Zhenheng Li, Benjamin Steinberg, and Qiang Wang of the workshop on “Algebraic monoids, group embeddings and algebraic combinatorics” held July 3–6, 2012 at the Fields Institute at Toronto for giving us the opportunity to present this work. We would like to thank Nicolas M. Thiéry for discussions.
The Markov chains presented in this paper are implemented in a Maple package by the first author (AA) available from his home page and in Sage [29, 32] by the third author (AS). Many of the pictures presented here were created with Sage.
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Ayyer, A., Klee, S., Schilling, A. (2014). Markov Chains for Promotion Operators. In: Can, M., Li, Z., Steinberg, B., Wang, Q. (eds) Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol 71. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0938-4_13
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