The representation theory of somewhere-to-below shuffles

The somewhere-to-below shuffles are the elements $$t_{\mathrm{\ell }}:={\mathrm{cyc}}_{\mathrm{\ell }}+{\mathrm{cyc}}_{\mathrm{\ell },\mathrm{\ell }+1}+{\mathrm{cyc}}_{\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\ell }+2}+\cdots +{\mathrm{cyc}}_{\mathrm{\ell },\mathrm{\ell }+1,\dots ,n}$$ (for $\mathrm{\ell }\in \{1,2,\dots ,n\}$) 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 $S_n$.