Multiplicities of \(S_ n\)-modules and the index and the charge of tableaux (Q1337917)

Let \(A\) and \(B\) be skew diagrams of the same order \(n\), and let \(T_A\) and \(T_B\) be the corresponding representations of the symmetric group \(S_n\) (over a fixed field of characteristic zero). There are several combinatorial interpretations of the intertwining number \(\nu(A,B)=\dim\text{Hom}_{S_n}(T_A,T_B)\). We give a new combinatorial interpretation of the number \(\nu(A,B)\) as the set of integral solutions of a system of linear inequalities, the inequalities being symmetric with respect to \(A\) and \(B\) and the relation \(\nu(A,B)=\nu(B,A)\) thus being self-evident. Furthermore, it turns out that the deviations from these inequalities can be characterized by numbers which are generalizations of the index and the charge of a tableau.