$\DeclareMathOperator{\GL}{GL}$
Suppose that $T: G \rightarrow \GL(U)$ and $ S: G \rightarrow \GL(V)$ are two finite dimensional irreducible representations of some group $G$ .
I consider the tensor product representation $ T^*S : G \rightarrow \GL( U^* \otimes V)$ ,
where $T^*$ is the dual representation of $T$.
I know that $T^*S $ can be written as a finite sum of irreducible representations $ T^*S = \displaystyle \sum_{k} m_k T_k$.
I want to show that if for some $k$ , $T_k$ is the trivial representation, then $m_k \le 1$.
Can anyone give me a hint to get started please…
Best Answer
$\DeclareMathOperator{\Hom}{Hom}$ Hint: $U^*\otimes V\cong \Hom(U,V)$ with the usual action of $G$ on $\Hom(U,V)$. The trivial submodules here are exactly the subspaces of $\Hom_G(U,V)$. What does Schur's Lemma say about this? (I'm assuming you are working over a suitable algebraically closed field).