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 $\ell \in {1,2,\dots,n}$) 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 $S_n$.