TITLE: Alternate proof of Schur orthogonality relations
QUESTION [5 upvotes]: I am trying to find an alternate proof for Schur orthogonality relations along the following lines.
Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$.
Let $V$ denote the adjoint representation ($V = \langle \{e_g\}_{g\in G} \rangle$ with $h\cdot e_g = e_{hgh^{-1}}$). Also, let $W$ denote the representation given by $\bigoplus_{i=1}^{d} \text{End} (V_i)$.
The Schur orthogonality relations tell us these both have the same character, hence are isomorphic. But this requires one to assume the fact that $\{\chi_{V_i} \}_{i=1..d}$ forms a basis for the class functions on $G$.
I would like to show $V \cong W$ by some other means, maybe an explicit isomorphism. This would prove Schur orthogonality without knowing the fact about the basis for class functions. How might this be done?

REPLY [6 votes]: There is a natural map $\Bbb C[G]\to\bigoplus{\rm End}_{\Bbb C}(V)$ induced from the maps $G\to{\rm GL}(V)$. To see that it's an isomorphism, it suffices to check injectivity and equality of dimensions. Injectivity is easy: if any element $x\in{\Bbb C}[G]$ were mapped to $0$ then it would act as $0$ on any irrep, hence any rep, hence act trivially on $\Bbb C[G]$ itself, so $x\cdot e_G=0$ tells us $x=0$. Now for dimensions...
We have $\Bbb C[G]\cong\bigoplus V^{\oplus m(V)}$ for some unknown multiplicities $m(V)$. Convince yourself that
$$\hom_G(\Bbb C[G],W)\cong W$$
are isomorphic via the canonical map $\phi\mapsto\phi(1)$ for any rep $W$. Furthermore $\hom$ is distributive in both of its arguments (e.g. $\hom_G(A,B\oplus C)\cong\hom_G(A,B)\oplus\hom_G(A,C)$; what do you think the natural isomorphism will be?). Therefore we can compute (via Schur's on $\hom_G(V,W)$):
$$W  \cong \hom_G(\Bbb C[G],W) \cong\bigoplus_V \hom_G(V,W)^{\oplus m(V)} \cong \Bbb C^{\oplus m(W)} $$
as vector spaces, where the last $\cong$ sign assumes that $W$ is an irreducible representation (the rest holds for every $W$). Therefore $m(W)=\dim W$ for all irreps $W$, and now we compute
$$\begin{array}{lc} \Bbb C[G] & \cong\hom_G(\Bbb C[G],\Bbb C[G])  \cong\bigoplus_{V,W}\hom_G(V,W)^{\oplus m(V)\cdot m(W)} \\[4pt] & \cong\bigoplus_V \Bbb C^{\oplus m(V)^2}  \cong \bigoplus{\rm End}_{\Bbb C}(V) \end{array}$$
as vector spaces, yielding equality of dimensions. Thus $\Bbb C[G]\cong\bigoplus{\rm End}_{\Bbb C}(V)$ as algebras.
Fix an irrep $V$ and let $e_V\in{\Bbb C}[G]$ correspond to ${\rm Id}_V\in{\rm End}_{\Bbb C}(V)$ and $0$ in all of the other coordinates. Write $e_V=\sum c_gg$, "rotate" its coefficients as $e_Vh^{-1}=\sum c_g gh^{-1}$, then to both sides apply ${\rm tr}_{\Bbb C[G]}$ to get $m(V)\chi_V(h^{-1})=|G|c_h$, hence $e_V=\frac{\dim V}{|G|}\sum\chi_V(g^{-1})g$.
Now the orthogonality relations simply state $e_Ve_W=\delta_{VW}e_V$ (equate coefficients in $\Bbb C[G]$).