The representation theory of somewhere-to-below shuffles

TL;DR

This paper investigates the algebraic structure of somewhere-to-below shuffles in the symmetric group, determining their eigenvalues on Specht modules, thus advancing understanding of their representation theory.

Contribution

It provides a complete characterization of the eigenvalues of one-sided cycle shuffles on all Specht modules, a novel result in the representation theory of these shuffles.

Findings

Eigenvalues of any one-sided cycle shuffle on Specht modules are explicitly determined.

The results unify and extend previous partial analyses of these shuffles.

The work offers new insights into the spectral properties of certain group algebra elements.

Abstract

The *somewhere-to-below shuffles* are the elements \[ t_{\ell} := \operatorname{cyc}_{\ell}+\operatorname{cyc}_{\ell,\ell+1}+\operatorname{cyc}_{\ell,\ell+1,\ell+2}+\cdots+\operatorname{cyc}_{\ell,\ell+1,\ldots,n} \] (for $\ell \in \{1,2,\dots,n\}$) in the group algebra $\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$. Their linear combinations are called the *one-sided cycle shuffles*. We determine the eigenvalues of the action of any one-sided cycle shuffle on any Specht module $\mathcal{S}^{\lambda}$ of $S_n$.