Converting immanants on skew-symmetric matrices

Let Q_n denote the space of all n×n skew-symmetric matrices over the complex field C and T:Q_n→Q_n be a map satisfying the condition d_χ^′(T(A)+zT(B))=d_χ(A+zB) for all matrices A,B∈Q_n and all constants z∈C. Here χ and χ^′ are irreducible characters of the permutation group S_n and d_χ(C) denotes the immanant of the matrix C associated with the character χ. Let P_n be the set of permutations with no cycles of odd length in the decomposition into the product of independent cycles. The main goal of this paper is twofold. The first one is to show that there are no maps T satisfying the above conditions if n≥6 is even and χ and χ^′ are not proportional on P_n. The second is to characterize such maps T if χ and χ^′ are proportional on P_n. In particular, we prove that T is bijective and linear. Observe that the general problem of the characterization for bijective linear maps on Q_n, that convert one immanant into another, remained an open problem with the exception for determinant and permanent. Our results include all known characterizations for immanant preserving and converting maps and provide the complete solution of this problem.