Converting immanants on skew-symmetric matrices

Abstract

Let Qn denote the space of all n×n skew-symmetric matrices over the complex field C and T:Qn→Qn be a map satisfying the condition dχ′(T(A)+zT(B))=dχ(A+zB) for all matrices A,B∈Qn and all constants z∈C. Here χ and χ′ are irreducible characters of the permutation group Sn and dχ(C) denotes the immanant of the matrix C associated with the character χ. Let Pn 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 Pn. The second is to characterize such maps T if χ and χ′ are proportional on Pn. In particular, we prove that T is bijective and linear. Observe that the general problem of the characterization for bijective linear maps on Qn, 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.