Hopf Algebras and Markov Chains

TL;DR

This thesis develops a framework for constructing and analyzing Markov chains from Hopf algebras, providing explicit formulas, algorithms, and insights into their behavior, exemplified by the riffle-shuffle model.

Contribution

It introduces the concept of Hopf-power Markov chains, offering new methods for analyzing their properties using algebraic structures and presenting algorithms for eigenbasis computation.

Findings

Explicit stationary distribution formula (Theorem 4.5.1)

Markov chains derived from quotient algebras preserve certain statistics (Theorem 4.7.1)

Algorithms for eigenbasis in common Hopf algebra cases (Theorem 2.5.1)

Abstract

This thesis introduces a way to build Markov chains out of Hopf algebras. The transition matrix of a "Hopf-power Markov chain" is (the transpose of) the matrix of the coproduct-then-product operator on a combinatorial Hopf algebra with respect to a suitable basis. These chains describe the breaking-then-recombining of the combinatorial objects in the Hopf algebra. The motivating example is the famous Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards, which arises in this manner from the shuffle algebra. The primary reason for constructing Hopf-power Markov chains, or for rephrasing familiar chains through this lens, is that much information about them comes simply from translating well-known facts on the underlying Hopf algebra. For example, there is an explicit formula for the stationary distribution (Theorem 4.5.1), and constructing quotient algebras show that certain statistics on a Hopf-power Markov chain are themselves Markov chains (Theorem 4.7.1). Perhaps the pinnacle is Theorem 2.5.1, a collection of algorithms for a full left and right eigenbasis in many common cases where the underlying Hopf algebra is commutative or cocommutative. This arises from a cocktail of the Poincare-Birkhoff-Witt theorem, the Cartier-Milnor-Moore theorem, Reutenauer's structure theory of the free Lie algebra, and Patras's Eulerian idempotent theory. Since Hopf-power Markov chains can exhibit very different behaviour depending on the structure of the underlying Hopf algebra and its distinguished basis, one must restrict attention to certain styles of Hopf algebras in order to obtain stronger results. This thesis will focus respectively on a free-commutative basis, which produces "independent breaking" chains, and a cofree basis; there will be both general statements and in-depth examples.