The class-function inner product $⟨{\chi }_V,{\chi }_W⟩$ equals $\dim {\mathrm{Hom}}_G(W,V)$

Given: Finite-dimensional complex representations $V$ and $W$ of a finite group $G$, with characters ${\chi }_V$ and ${\chi }_W$.

The inner product of class functions is $⟨\phi ,\psi ⟩=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{g\in G}\phi (g)\overline{\psi (g)}$ (The standard inner product on $\mathrm{c}\mathrm{f}(G)$).

For a finite-dimensional complex representation $U$, the averaging operator $P_U=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g{\rho }_U(g)$ has image $U^G$ and trace $\dim U^G$ (The averaging operator projects onto the fixed subspace).

The fixed points of the diagonal representation on $W^{\ast }\otimes V$ are exactly the intertwiners $W\to V$, so $(W^{\ast }\otimes V{)}^G\cong {\mathrm{Hom}}_G(W,V)$ (For finite-dimensional complex $V$, the intertwiners $V\to W$ are exactly the fixed points of $V^{\ast }\otimes W$).

Characters add on direct sums, multiply on tensor products, and conjugate on duals (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

Trace is linear on finite square matrices (Trace is a linear functional on $M_n(F)$).

The character of a representation is by definition the trace of its action operator, ${\chi }_U(g)=\mathrm{tr}{\rho }_U(g)$.