The depolarising channel and Horns problem

We investigate the action of the depolarising (qubit) channel on permutation invariant input states. More specifically, we raise the question on which invariant subspaces the output of the depolarising channel, given such special input, is supported. An answer is given for equidistributed states on isotypical subspaces, also called symmetric Werner states. Horns problem and two of the corresponding inequalities are invoked as a method of proof.