Corollary

Let $\pi$ and $\pi '$ be finite-dimensional linear representations of G. In the field K, we have

(29) $\langle \chi_{\pi}, \chi_{\pi'}\rangle_G=$ (dim$_K$ Hom$_G(\pi , \pi ')$)$\cdot 1$.

We first suppose that $\pi$ and $\pi '$ are simple representations. The vector space Hom$_G(\pi , \pi ')$ has dimension 1 or 0 according to whether or not $\pi$ and $\pi '$ are isomorphic (Schur’s lemma, VIII, p. 47, Proposition 2). In this case, formula (29) follows from Proposition 4.

In the general case, the representation $\pi$ (resp. $\pi '$) is the direct sum of simple representations $\pi_1, . . . , \pi_m$ (resp. $\pi '_1, . . . , \pi '_n$). The space Hom$_G(\pi , \pi ')$ is isomorphic to the direct sum of the spaces Hom$_G(\pi_i, \pi '_j)$ for $1\leqslant i\leqslant m$, $1\leqslant j\leqslant n$, and we have

By linearity, the proof of formula (29) is reduced to the case of simple representations.