Converting immanants on singular symmetric matrices

Abstract

Let Σ_n(F) denote the space of all n×n symmetricmatrices over the complex field F, and χ be an irreducible character of S_n and d_χ the immanant associated with χ. The main objective of this paper is to prove that the maps Φ: Σ_n(F) → Σ_n(F) satisfying d_χ(Φ(A) + αΦ(B)) = det(A + αB) for all singular matrices A, B ∈ Σ_n(F) and all scalars α ∈ F are linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on the set of all symmetric matrices.