Decomposable numerical ranges on orthonormal tensors

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∈Hχ(σ)∏j=1mbj,σ(j), and the decomposable numerical range of an n×n matrix A on orthonormal tensors associated with χ is defined byWχ⊥(A)={dχ(X*AX):Xisann×mmatrixsuchthatX*X=Im}.We study relations between the geometrical properties of Wχ⊥(A) and the algebraic properties of A, and determine the structure of those linear operators L on n×n complex matrices that satisfy Wχ⊥(L(A))=Wχ⊥(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.