$\begin{aligned}
&1/|G| (\sum_g \in G Tr(\rho(g))) \\
&= 1/|G| (\sum_g \in G \chi_\rho(g) ) \\
&\chi \rho \cdot \chi_1
\end{aligned}$

Where we have:
- $\chi_1$ is the charcter of the trivial representation $g \mapsto 1$
- The inner product $\langle \cdot , \cdot \rangle$ is the $G$-average inner product over $G$-functions $G \rightarrow \mathbb C$:

$\langle f , f' \rangle \equiv \sum_{g \in G} f(g) \overline{f'(g)}$

So, we need to show that the number of orbits $|S/G|$ is equal to the
multiplicity of the trivial representation $1$ in the current representation
$\rho$, given by the inner product of their characters $\chi_1 \cdot \chi_\rho$.
let $s* in S$ whose orbit we wish to inspect. Build
the subspace spanned by the vector $v[s*] \equiv \sum_{g \in G} \rho(g) v[s]$.
This is invariant under $G$ and is 1-dimensional. Hence, it corresponds
to a 1D subrepresentation for all the elements in the orbit of $s*$.
(TODO: why is it the