Lumpings of Algebraic Markov Chains Arise from Subquotients

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Abstract

A function on the state space of a Markov chain is a “lumping” if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a “top-to-random-with-standardisation” chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto–Reutenauer algebra.

Keywords

Markov chain Random walks on groups Card shuffling Combinatorial Hopf algebras 

Mathematics Subject Classification (2010)

60J10 16T30 05E05 

Notes

Acknowledgements

I would like to thank Nathan Williams for a question that motivated this research, and Persi Diaconis, Jason Fulman and Franco Saliola for numerous helpful conversations, and Federico Ardila, Grégory Châtel, Mathieu Guay-Paquet, Simon Rubenstein-Salzedo, Yannic Vargas and Graham White for useful comments. SAGE computer software [65] was very useful, especially the combinatorial Hopf algebras coded by Aaron Lauve and Franco Saliola.

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Authors and Affiliations

  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontrealCanada
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloonHong Kong

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