The representation theory of somewhere-to-below shuffles

The somewhere-to-below shuffles are the elements te := cyce+ cyce,e+1 + cyce,e+1,e+2 + & centerdot;& centerdot;& centerdot; + cyce,e+1,...,n (for & ell; is an element of {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 lambda of Sn.