On pairs of matrices that satisfy certain polynomial identities

Abstract

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