Review Note

Let \(\rho, \tau\) be irreducible finite dimensional representations over \(\mathbb{C}\) of \(G\), \(|G| <\infty\), with characters \(\chi_\rho, \chi_\tau\). Then
\(\langle \chi_\rho, \chi_\tau \rangle = {{c1:: \begin{cases} 1, \qquad \rho \cong \tau \\ 0, \qquad \text{else} \end{cases} }}\)