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This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following. Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...

Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$. Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$. For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...