Number of times an irreducible representation occurs in the tensor product of two irreducible representations

Question

Let $G$ be a finite group. The motivation of my question is to prove that if $S\subseteq \hat{G}$ is a set of irreducible representations of $G$ and
$$(\bigoplus_{\rho_1\in S}d_{\rho_1}\rho_1) \otimes (\bigoplus_{\rho_2\in S}d_{\rho_2}\rho_2^*)=\bigoplus_{\pi \in \hat{G}}n_\pi \pi$$
then $n_\pi\leq \sum_{\pi \in S} d_\pi^2$. (\rho_2^* is the contragradient representation) I am reading https://link.springer.com/article/10.1007/BF02511813, and the latter is neccesary to prove Theorem 5. In the article they point out that $\chi_\pi \chi_{reg}=d_{\pi}\chi_{reg}$ (where $\chi_{reg}$ is the character of the regular representation ), which is obviously true. What I don't understand is why this implies that

for any $\rho_1\in S$, as $\rho_2$ runs through $S$ a given
irreducible representation $\pi$ can occur in all the
$\rho_1\otimes\rho_2^*$ at most $d_{\pi}$ times.

Answer 1

Let $\chi$, $\phi$ and $\psi$ be irreducible characters of a finite group $G$. Then $\chi$ appears $d_\chi$ many times in $\chi_{reg}$, so $$ \langle \chi\phi,\psi\rangle\leq \frac{1}{d_\chi}\langle \chi_{reg}\phi,\psi\rangle=\frac{1}{d_\chi}\langle d_\phi\chi_{reg},\psi\rangle=\frac{d_\psi d_\phi}{d_\chi}.$$

So now set $\phi=\pi$ in your situation, and choose $\chi$ and $\phi$ to be $\rho_1$ and $\rho_2^*$, chosen so that $\chi(1)\geq \phi(1)$. Then the inner product is at most $d_\pi$ (and can only reach $d_\pi$ if $\chi(1)=\phi(1)$, as it happens).