Lemma 5

Let $\pi$ be a unitary representation of G in a Hilbert space E of finite dimension. Let $(\varrho_i)_{i\in I}$ be a family of non-zero unitary representations of G and $(n_i)_{i\in I}$ a family of integers $\geqslant 1$ such that the representation $\overline{\pi}\otimes \pi$ of G (resp. the representation $\pi \otimes \pi$ ) is isomorphic to the direct sum $\bigoplus_{i\in I}\varrho^{n_i}_i$. Then one has

with equality if and only if the representations $\varrho_i$ are irreducible and pairwise non-isomorphic.

Let us denote by $\chi_i$ the character of $\varrho_i$ for $i\in I$. One has

$|\chi_{\pi}|^2=\chi_{\overline{\pi}\otimes\pi}=\sum_{i\in I}n_i\chi_i$, (resp. $\chi^2_{\pi}=\chi_{\pi\otimes\pi}=\sum_{i\in I}n_i\chi_i$).

From the definition and Cor. 3 of V, p. 466, there follows

whence, in both cases, the formula

$M_4(\pi ) =\sum_{i,j}n_in_j\int_G\chi_i\overline{\chi}_jd\mu=\sum_{i,j}n_in_j$ dim Hom$_G(\varrho_i, \varrho_j)$

(Cor. 4, b) of V, p. 459). Let $i\in I$. Since the representation $\varrho_i$ is not zero, the space Hom$_G(\varrho_i, \varrho_i)$ is non-zero, and one deduces the lower bound

Moreover, since $n_i\geqslant$ 1, there is equality if and only if one has dim Hom$_G(\varrho_i, \varrho_j) = 0$ if $i\not =j$ and dim Hom$_G(\varrho_i, \varrho_i) = 1$ for all $i$. The second condition holds if and only if the representations $\varrho_i$ are irreducible (loc. cit.). The first is then satisfied if and only if the representations $\varrho_i$ are pairwise non-isomorphic, by Schur’s Lemma (V, p. 387, Cor. 2).

If E is a module over a commutative ring A, the a-module $\mathsf{S}^2(E)$ (resp. $\wedge^2E$) will be called the symmetric square (resp. exterior square) of E.