Minimal output entropy for the channels that add or remove a box of a Young diagram, and a Pauli principle for every permutation symmetry

Two irreducible representations of $U(d)$ whose Young diagrams differ by a single box are connected by four covariant quantum channels: the box can be removed or added, and one keeps either the new diagram or the box ($\mathbb{C}^d$ or its dual). We prove that for each of these channels the output o