Converting immanants on singular symmetric matrices

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 ) + α Φ( 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.