On the decomposable numerical range of operators

Let $V$ be an $n$-dimensional complex inner product space. Suppose $H$ is a subgroup of the symmetric group of degree $m$, and $chi: Hrightarrow mathbb{C} $ is an irreducible character (not necessarily linear). Denote by $V_{chi}(H)$ the symmetry class of tensors associated with $H$ and $chi$. Let $K(T)in (V_{chi}(H))$ be the operator induced by $Tin text{End}(V)$. The decomposable numerical range $W_{chi}(T)$ of $T$ is a subset of the classical numerical range $W(K(T))$ of $K(T)$ defined as: $$ W_{chi}(T)={(K(T)x^{ast}, x^{ast}):x^{ast} is a decomposable unit tensor}. $$ In this paper, we study the interplay between the geometric properties of $W_{chi}(T)$ and the algebraic properties of $T$. In fact, we extend some of the results of [C. K. Li and A. Zaharia, Decomposable numerical range on orthonormal decomposable tensors, Linear Algebra Appl. 308 (2000), no, 1-3, 139--152] and [C. K. Li and A. Zaharia, Induced operators on symmetry classes of tensors, Trans. Amer. Math. Soc. 354 (2002), no. 2, 807--836], to non-linear irreducible characters.