## 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.

Original language | English |
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Pages (from-to) | 1804-1844 |

Number of pages | 41 |

Journal | Journal of Theoretical Probability |

Volume | 32 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

## Scopus Subject Areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

## User-Defined Keywords

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