A class of representations of the full linear group. II

Abstract:

Let $V$ be an $n$-dimensional vector space over complex numbers $C$. Let $W$ be the $m$th tensor product of $V$. If $T \in {\operatorname {Hom} _C}(V,V)$, let ${ \otimes ^m}T \in {\operatorname {Hom} _C}(W,W)$ be the $m$th tensor product of $T$. The homomorphism $T \to { \otimes ^m}T$ is a representation of the full linear group ${\text {G}}{{\text {L}}_n}(C)$. If $H$ is a subgroup of the symmetric group ${S_m}$, and $\chi$ a linear character on $H$, let $V_\chi ^m(G)$ be the subspace of $W$ consisting of all tensors symmetric with respect to $H$ and $\chi$. Then $V_\chi ^m(H)$ is invariant under ${ \otimes ^m}T$. Let $K(T)$ be the restriction of ${ \otimes ^m}T$ to $V_\chi ^m(H)$. For $n$ large compared with $m$ and for $H$ transitive, we determine all cases when the representation $T \to K(T)$ is irreducible.