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.