Cyclotomic shuffles

TL;DR

This paper introduces and analyzes analogues of 1-shuffle elements for complex reflection groups of type G(m,1,n), providing geometric interpretations, eigenvalue computations, and convergence estimates for associated random walks.

Contribution

It presents new 1-shuffle analogues for G(m,1,n), computes their eigenvalues, and studies their spectral properties and convergence in related algebraic structures.

Findings

Eigenvalues and multiplicities of 1-shuffle elements computed

Geometric interpretation in terms of polygonal rotations provided

Convergence rates of the associated Markov chains estimated

Abstract

Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra of the group $G(m,1,n)$. Considering shuffling as a random walk on the group $G(m,1,n)$, we estimate the rate of convergence to randomness of the corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue in the cyclotomic Hecke algebra $H(m,1,n)$ for $m=2$ and small $n$.