On pairs of matrices that satisfy certain polynomial identities

Let FF be an arbitrary field, H be a subgroup of the symmetric group of degree m, SmSm, λλ be an irreducible FF-valued character of H and dλH be the generalized matrix function associated to H and λλ. In this note we present some properties of the pairs (A,B)(A,B) of m×mm×m matrices over FF that satisfydλH(AXB)=dλH(X)for all X .We also describe the pairs (A,B)(A,B) satisfyingdλH(AX)=dλH(XB)for all X.