Question

Show that the First Orthogonality Relation can be generalized to
(sum over g∈G) χ_i(g^−1)χ_j(hg) = δ_ij |G|χ_i(h)/n_i,
where h is any element in G, and n_i = χ_i(1).

Answer 1

Here $\chi_{i}$ are the characters of the simple left $kG$-modules, where $k$ is a splitting field for $G$ of characteristic not dividing $|G|$. The proof depends on the expressions for the central idempotents in $kG$ giving the Wedderburn decomposition of $kG$ into its simple components. These central idempotents are given by

$e_i = |G|^{-1}n_i\sum_{g\in G}\chi_i(g^{-1})g$ . Now use the equations $e_i e_j = \delta_{ij} e_i$, where the $\delta_{ij}$ 's are the Kronecker deltas.

The coefficient of $h^{-1}$ on the RHS is $\delta_{ij}|G|^{-1}n_{i}\chi_i(h)$, and the coefficient of $h^{-1}$ on the LHS is

$n_i n_j |G|^{-2} \sum_{g \in G} \chi_i(g^{-1}) \chi_j(hg)$ . Therefore, the desired formula follows.