Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ d^\rho_1,\ldots,d^\rho_{k_\rho}\}$, each of which is defined by choosing a standard tableaux on $Y_\rho$ and summing the descents. The $d_i^\rho$ are sometimes referred to as the degrees of $\rho$ and $k_\rho$ is the hook length of $Y_\rho$ (or the dimension of $\rho$).
The question is in two parts. First, for a fixed $\rho$ is the minimal degree unique? Perhaps there is some algorithm that produces a tableaux with minimal major index?
Secondly, can two non-isomorphic irreducibles, $\rho$ and $\mu$ say, have the same minimal degrees?
Thank you in advance for any help.