Linear Immanant Converters on Skew-Symmetric Matrices of Order 4

Abstract

Let Qn denote the space of all n × n skew-symmetric matrices over the complex field ℂ. It is proved that for n = 4, there are no linear maps T : Q4 → Q4 satisfying the condition dχ' (T (A)) = dχ(A) for all matrices A ∈ Q4, where χ, χ' ∈ {1, ∈, [2, 2]} are two distinct irreducible characters of S4. In the case χ = χ' = 1, a complete characterization of the linear maps T : Q4 → Q4 preserving the permanent is obtained. This case is the only one corresponding to equal characters and remaining uninvestigated so far.