On pairs of matrices that satisfy certain polynomial identities (Q765170)

Let \(H\) be a subgroup of the symmetric group of degree \(m\) and let \(\lambda\) be an irreducible character of \(H\) with values in a field \(F\). The generalized matrix function \(d_\lambda^H\) associated to \(H\) and \(\lambda\) arises by changing the usual definition of an \(m\times m\) matrix over \(F\) as follows: The symmetric group \(S_m\) is replaced by \(H\) and the sign of \(\sigma\in S_m\) is replaced by \(\lambda(\sigma)\). The authors present properties of pairs of \(m\times m\) matrices \((A,B)\) over \(F\) that satisfy one of the following conditions for all \(X\): (i) \(d_\lambda^H(AXB)=d_\lambda^H(X)\); (ii) \(d_\lambda^H(AX)=d_\lambda^H(XB)\)