The Representation Theory of Somewhere-to-Below Shuffles

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\left\{ 1,2,\ldots,n\right\} $) in the group algebra $\mathbf{k}\left[ S_{n}\right] $ of the $n$-th symmetric group $S_{n}$. Their linear combinations are called the \emph{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 $S_{n}$.