The Eigenvalues of Hyperoctahedral Descent Operators and Applications to Card-Shuffling

TL;DR

This paper generalizes descent operators to hyperoctahedral variants within Hopf algebras, computes their eigenvalues and eigenvectors, and applies these results to analyze complex card-shuffling Markov chains.

Contribution

It introduces a new algebraic framework for hyperoctahedral descent operators on Hopf algebras, extending prior work and providing spectral data for applications.

Findings

Eigenvalues and multiplicities of the new operators are explicitly determined.

A basis of eigenvectors is constructed for a subclass related to Adams operations.

Application to card-shuffling models with flips and rotations is demonstrated.

Abstract

We extend an algebra of Mantaci and Reutenauer, acting on the free associative algebra, to a vector space of operators acting on all graded connected Hopf algebras. These operators are convolution products of certain involutions, which we view as hyperoctahedral variants of Patras's descent operators. We obtain the eigenvalues and multiplicities of all our new operators, as well as a basis of eigenvectors for a subclass akin to Adams operations. We outline how to apply this eigendata to study Markov chains, and examine in detail the case of card-shuffles with flips or rotations.