On the equality of generalized matrix functions

Let H and K be arbitrary subgroups of the symmetric group Sn and let φ and ψ be irreducible characters of H and K, respectively. The main result of this paper is that the two generalized matrix functions and are equal on the set of singular matrices if and only if φH∩K=ψH∩K and both of φ and ψ vanish outside of H∩K.