Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(2.3)

1. Similar $F$-representations of $G$ afford equal characters.
2. Characters are constant on conjugacy classes of a group.

Pf:

• $tr(P^{-1}AP)=tr(A)$
• $\mathfrak{X}(h^{-1}gh)=\mathfrak{X}(h)^{-1}\mathfrak{X}(g)\mathfrak{X}(h)$, and hence $tr(\mathfrak{X}(h^{-1}gh))=tr(\mathfrak{X}(g))$