Pairs of matrices that preserve the value of a generalized matrix function on the set of the upper triangular matrices (Q5962286)

Let \(X=\bigl(x_{ij}\bigr)_{ij}\) be an \(n\times n\) matrix over a field \(F\) of characteristic zero, and let \(\chi\) be an irreducible character of a subgroup \(H\leq \text{Sym}_n\), extended as a function from \(\text{Sym}_n\) to \(F\) by letting it vanish on \(\text{Sym}_n\backslash H\). The authors consider the generalized Schur function associated with \(\chi\) by NEWLINENEWLINE\[NEWLINE\text{d}_{\chi}(X)=\sum_{\sigma\in\text{Sym}_n}\chi(\sigma)\prod_i x_{i\sigma(i)}NEWLINE\]NEWLINE NEWLINE(it is implicit that only those characters \(\chi\) which take the values in \(F\) are considered). They proceed to classify pairs of \(n\)-by-\(n\) matrices \((A,B)\) such that the identity NEWLINENEWLINE\[NEWLINE\text{d}_{\chi}(AXB)=\text{d}_{\chi}(X)NEWLINE\]NEWLINE NEWLINEholds for all upper-triangular \(X\).NEWLINENEWLINESpecial attention is paid when \(H=\text{Sym}_n\) and \(\chi\) corresponds to the partition \(n=s+1^{n-s}\).