Semilinear preservers of the immanants in the set of the doubly stochastic matrices

Let S-n denote the symmetric group of degree n and M-n denote the set of all n-by-n matrices over the complex field, C. Let chi : S-n -> C be an irreducible character of degree greater than 1 of S-n.The immanant d chi : M-n -> C associated with chi is defined by [GRAPHICS] Let Omega(n) be the set of all n-by-n doubly stochastic matrices, that is, matrices with nonnegative real entries and each row and column sum is one. We say that a map T from Omega(n) into Omega(n) is semi linear if T (lambda S-1 + (1 - lambda)S2) = lambda T(S-1) + (1 - lambda)T(S-2) for all S-1, S-2,S- is an element of Omega(n) and for all real number lambda such that 0 <= lambda <= 1; preserves d chi if d chi (T(S)) = d chi (S) for all S is an element of Omega(n). We characterize the semilinear surjective maps T from Omega(n) into Omega(n) that preserve d(chi,)when the degree of chi is greater than one.