Let M_n be the  algebra of all n-square complex matrices. I will characterize maps φ:M_n→M_n that satisfy:

dχ(φ(A)+λφ(B)) = dχ'(A+λB)

for all A,B ∈ M_n and every complex number λ; where χ and χ' are arbitrary irreducible complex characters of S_n.