Basis change of $Z(\mathbb C [G])$ Let $G$ be a finite group with . Consider the two bases of $Z(\mathbb C [G])$: One is the elements of the form $e_{g} =\Sigma_{x\in c_g} x$, where $c_g$ is a conjugacy class, and another is elements of the form $e_E = \frac {dim E} {|G|} \Sigma_{g\in G}\chi_E (g^{-1}) g$ for an irreducible representation $E$. I'm trying to understand the relations between the bases. On one direction, denote $I$ to be representatives of conjugacy classes. Then I can easily write:
$$e_E =  \frac {dim E} {|G|} \Sigma_{g\in G}\chi_E (g^{-1})g =  \frac {dim E} {|G|} \Sigma_{g\in I}\chi_E (g^{-1})e_g $$
I'm trying to make the opposite direction as well - express the elements $e_g$ in the basis of the form $e_E$. How can it be done?
 A: I will use the notations
$$ e_{\small U}=\frac{\dim U}{|G|}\sum_{g\in G}\chi_{\small U}(g^{-1})g, \qquad \sigma_C=\sum_{g\in C} g $$
Note $e_{\small U}$ is the $U$-isotypical projector. Together, these projectors form the primitive central orthogonal idempotents of the group algebra $\mathbb{C}[G]$, and the coefficients of $\sigma_C$s (the Kronecker delta functions $\delta_C(g)$ over conjugacy classes $C\subseteq G$) can similarly be thought of as the primitive central orthogonal idempotents of the algebra $\mathcal{C}(G)$ of $\mathbb{C}$-valued functions on $G$.
Solution. Note that $\rho_{\small U}(\sigma_C)$ is an intertwining operator, so by Schur's lemma (assume $U$ is an irrep) must be a scalar multiple of the identity operator, $\lambda\,\mathrm{Id}_U$. Take traces to find $\lambda$:
$$ \mathrm{tr}\,\rho_{\small U}(\sigma_C) = |C|\chi_{\small U}(C)=(\dim U)\lambda \quad \Rightarrow \quad \lambda = \frac{|C|}{\dim U} \chi_{\small U}(C). $$
By Artin-Wedderburn, $\mathbb{C}[G]\cong\bigoplus \mathrm{End}(U)$ as algebras (and $\mathbb{C}[G]$-$\mathbb{C}[G]$-bimodules), so we conclude our element $\sigma_C$ is a sum of the projectors $e_{\small U}$ (corresponding to $\mathrm{Id}_U$s):
$$ \sigma_C = \sum_{\mathrm{Irr}\,G} \left[\frac{|C|}{\dim U}\chi_{\small U}(C)\right] e_{\small U} \tag{$\ast$}$$
Corollary. If you rewrite the $e_{\small U}$s above in terms of $\sigma_C$s and then compare coefficients of $\sigma_C$s you get the Schur orthogonality relations for the columns of $G$'s character table.
You can also use the Schur row orthogonality relations if you have them as a given:
Solution II. Write $\sigma_C$ in terms of $e_{\small V}$s with unknown coefficients $\mu_{\small V}$:
$$ \sigma_C = \sum_{\mathrm{Irr}\,G} \mu_{\small U} e_{\small U} = \sum_{\mathrm{Irr}\,G}\mu_{\small U} \left[ \frac{\dim U}{|G|}\sum_{g\in G}\chi_{\small U}(g^{-1})g \right] $$
Write both sides as sums of group elements $g$:
$$ \sigma_C=\sum_{g\in G}\delta_C(g)g = \sum_{g\in G}\left[\sum_{\mathrm{Irr}\, G} \mu_{\small U}\frac{\dim U}{|G|}\chi_{\small U}(g^{-1})\right]g $$
Equate coefficients:
$$ \delta_C(g) = \sum_{\mathrm{Irr}\,G}\mu_{\small U}\frac{\dim U}{|G|}\chi_{\small U}(g^{-1}) $$
Multiply by $\chi_{\small V}(g)$ and average over $G$:
$$ \frac{1}{|G|}\sum_{g\in G}\delta_C(g)\chi_{\small V}(g) = \sum_{\mathrm{Irr}\,G} \mu_{\small U}\frac{\dim U}{|G|}\langle \chi_U,\chi_V\rangle $$
where $\langle f_1,f_2\rangle=\frac{1}{|G|}\sum_{g\in G}\overline{f_1(g)}f_2(g)$ is the usual complex inner product on $\mathcal{C}(G)$ (note for complex characters $\chi_{\small U}(g^{-1})=\overline{\chi_{\small U}(g)}$; over other fields use a bilinear form with $f_1(g^{-1})$ instead).
The summands on the left are constant, and on the right we use orthogonality relations:
$$ \frac{|C|}{|G|}\chi_{\small V}(C) = \mu_{\small V}\frac{\dim V}{|G|} $$
from which we solve $\mu_{\small V}=\frac{|C|}{\dim V}\chi_{\small V}(C)$, as in $(\ast)$.
