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 :‎H\rightarrow \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 $K(T)\in‎ ‎\text{End}(V_{\chi}(H))$ be the operator induced by $T\in‎ ‎\text{End}(V)$‎. ‎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: ‎$$‎ ‎W_{\chi}(T)=\{(K(T)x^{\ast }‎, ‎x^{\ast}):x^{\ast }\ \text{is a‎ ‎decomposable unit tensor}\}‎. ‎$$‎ ‎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, ‎{\em Linear Algebra Appl.} {308}‎ ‎(2000), no, 1-3, 139--152] ‎and ‎[‎‎C‎. ‎K‎. ‎Li and A‎. ‎Zaharia‎, ‎Induced operators on symmetry classes‎ ‎of tensors‎, ‎{\em Trans‎. ‎Amer‎. ‎Math‎. ‎Soc.} {354} (2002), no. 2, 807--836]‎, ‎to non-linear irreducible characters‎.