Again something from Fulton and Harris I'm having trouble with:
Exercise 2.33 (c). If $U$, $V$, and $W$ Are irreducible representations, show that $U$ appears in $V \otimes W$ if and only if $W$ occurs in $V^{*} \otimes U$. Deduce that this cannot occur unless $\dim U \geq \dim W / \dim V$.
The hint suggests to look at the fact that $\mathrm{Hom}_{G}(V \otimes W, U) = \mathrm{Hom}_{G}(W, V^{*} \otimes U)$...