Inner Product

The inner product of two characters (Definition Character) of representations of a finite group $G$ (Definition Group Representation is \[ \langle \chi_1, \chi_2 \rangle = \frac{1}{|G|}\sum_{g \in G} \chi_1 (g) \overline{\chi_2(g)} \] \begin{remark} The inner product of two characters (Definition Inner Product) for a finite group is a sort of "averaging" over character values for a finite group. Keep this in mind when we define a similar notion for compact Lie groups using the Haar measure. Intuitively, averaging over continuous symmetries require making the sum finer and finer, which turns into an integral! \end{remark}