What is the idempotent $e_\chi$ associated to a character $\chi$?

Question

Suppose $\chi\colon G\to k^\times$ is an (irreducible) character of a finite group $G$ into a field $k$. What is the definition of the corresponding idempotent $e_\chi$?

I know that over $\mathbb{C}$, the corresponding central idempotent is $$ e_\chi=\frac{1}{|G|}\sum_{g\in G}\overline{\chi(g)}g. $$

If we don't have a notion of conjugation, is the corresponding idempotent just $$ \frac{1}{|G|}\sum_{g\in G}\chi(g)^{-1}g? $$

Answer 1

It's

$$\frac{\chi(1)}{|G|} \sum_{g \in G} \chi(g^{-1}) g.$$

Note that $\chi(g)$ is not always invertible, and if it is, its inverse is usually not its conjugate. But $\chi(g^{-1}) = \overline{\chi(g)}$ if $k$ is a subfield of $\mathbb{C}$ closed under complex conjugation.

Answer 2

There is a different definition over $\mathbb{C}$ for the projector onto the irreducible representation $V$, which is $$\psi_V = \frac{\dim V}{|G|} \sum_{g \in G} \chi_V(g) g^{-1}$$

I believe this definition extends to any field $k$ whose characteristic does not divide $|G|$.