Lumpings of Algebraic Markov Chains arise from Subquotients

TL;DR

This paper establishes general conditions under which algebraically-defined Markov chains can be lumped, with applications to chains on permutations and partitions derived from combinatorial Hopf algebras.

Contribution

It introduces broad criteria for lumpings of algebraic Markov chains, linking them to subquotients of Hopf algebras, and provides explicit examples involving permutation and partition chains.

Findings

Derived lumping conditions for algebraic Markov chains

Constructed a permutation chain that lumps to a partition chain

Connected algebraic structures to Markov chain simplifications

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.