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.

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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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Correspondence to C. Y. Amy Pang.

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Pang, C.Y.A. Lumpings of Algebraic Markov Chains Arise from Subquotients. J Theor Probab 32, 1804–1844 (2019). https://doi.org/10.1007/s10959-018-0834-0

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Keywords

  • Markov chain
  • Random walks on groups
  • Card shuffling
  • Combinatorial Hopf algebras

Mathematics Subject Classification (2010)

  • 60J10
  • 16T30
  • 05E05