The representation theory of somewhere-to-below shuffles

The *somewhere-to-below shuffles* are the elements \[ t_{} := cyc_{}+cyc,+1+cyc,+1,+2+⋯+cyc,+1,…,n \] (for ∈ \1,2,,n\) in the group algebra k[Sₙ] of the n-th symmetric group Sₙ. 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 S^λ of Sₙ.