Exercise 8, § 4

Let G be a compact group, and let $\varrho_1$ and $\varrho_2$ be irreducible unitary representations of G. Let $n\geqslant 2$ be an integer. If $\mathsf{S}^n\varrho_1$ is isomorphic to $\mathsf{S}^n\varrho_2$, then there exists a subgroup H of finite index of G such that the restriction of $\varrho_1$ to H is isomorphic to the restriction of $\varrho_2$ to H.