The lumpability property for a family of Markov chains on poset block structures

TL;DR

This paper explores the lumpability property of Markov chains structured by posets, providing combinatorial methods to classify all lumpings and linking them to group actions, with spectral analysis insights for homogeneous spaces.

Contribution

It characterizes when all lumpings arise from group actions in poset-structured Markov chains and connects spectral properties to group representation theory.

Findings

All lumpings are from subgroup actions if the poset is totally ordered and the Markov operator is uniform.

Lumping can be fully characterized by the group action on the poset block structure.

Spectral analysis of lumped chains relates to the decomposition into irreducible modules.

Abstract

We construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We use essentially combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associated with a Gelfand pair, the spectral analysis of the corresponding lumped Markov chain is completely determined by the decomposition of the group action into irreducible submodules.