If $G$ is a finite group, and $\chi$ is an irreducible character, does the left $kG$-module $kGe_\chi$ always afford $\chi$ ($k$ a field), and serve as a representative of the $\chi$-isotypic component?
I'm wondering because many texts, like Isaac's, will just pick some representatives $M_i$ of the irreducible $\mathbb{C}G$ modules, but not write them more explicitly as $\mathbb{C}Ge_\chi$, if that is the case.
I was hoping to check the trace of $h\in G$ acting on an element $\sum_{g\in G}c_gge_\chi$, but I'm stumped as how to do so. My question is inspired by the answer here, saying one can take the module affording a $\chi$ character on a torus $T$ to just be $kTe_\chi$.
For an irreducible character $\chi$ of a finite group $G$, the $\mathbb CG$-module $\mathbb CGe_\chi$ affords the character $\chi(1)\chi$. See [James-Liebeck 2001, Proposition 14.26]. In other words, $\mathbb CGe_\chi$ does not affords the character $\chi$ unless it is linear. (cf. Every irreducible character of a finite abelian group is linear.)