Author/Authors :
Zamani، Yousef نويسنده Sahand University of Technology , , Ahsani، Sima نويسنده Sahand University of Technology ,
Abstract :
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.
