Linear Immanant Converters on Skew-Symmetric Matrices of Order 4

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 d_χ' (T (A)) = d_χ(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.