and

Let 1 ≤ m ≤ n, and let χ: H → C be a degree 1 character on a subgroup H of the symmetric group of degree m. The generalized matrix function on an m × m matrix B = (bij) associated with χ is defined by dχ(B) = � σ∈H χ(σ) � m j=1 bj,σ(j), and the decomposable numerical range of an n × n matrix A on orthonormal tensors associated with χ is defined by W ⊥ χ (A) = {dχ(X ∗ AX) : X is an n × m matrix such that X ∗ X = Im}. We study relations between the geometrical properties of W ⊥ χ (A) and the algebraic properties of A, and determine the structure of those linear operators L on n×n complex matrices that satisfy W ⊥ χ (L(A)) = W ⊥ χ (A) for all n × n matrices A. These results extend those of other researchers who treat the special cases of χ such as the principal or alternate character