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Markov Chains for Promotion Operators

  • Arvind Ayyer
  • Steven Klee
  • Anne SchillingEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 71)

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.

Keywords

Posets Linear extensions Promotion Markov chains Tsetlin library \(\mathcal{R}\)-trivial monoids 

Subject Classifications

Primary 06A07 20M32 20M30 60J27 Secondary 47D03 

Notes

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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUC DavisDavisUSA
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  3. 3.Department of MathematicsSeattle UniversitySeattleUSA

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