Card-Shuffling via Convolutions of Projections on Combinatorial Hopf Algebras

TL;DR

This paper generalizes Markov chains on combinatorial Hopf algebras, including card shuffling models, by replacing coproducts with convolutions of projections, enabling detailed spectral analysis and control over piece sizes.

Contribution

It introduces a new class of Markov chains based on convolutions of projections, extending previous Hopf algebra models of card shuffling with explicit spectral properties.

Findings

Explicit stationary distributions derived

Eigenvalues and multiplicities computed

Eigenfunctions calculable in some cases

Abstract

Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of the previous chains. An important example is removing one "vertex" and reattaching it, in analogy with top-to-random shuffling. This larger family of Markov chains all admit analysis by Hopf-algebraic techniques. There are simple combinatorial expressions for their stationary distributions and for their eigenvalues and multiplicities and, in some cases, the eigenfunctions are also calculable.