每日定理16

Isaacs, $\textit{Character Theory of Finite Groups}$, Corollary(2.9)

Let $\mathfrak{X}$ and $\mathfrak{Y}$ be $\mathbb{C}$-representations of a group. Then $\mathfrak{X}$ and $\mathfrak{Y}$ are similar iff they afford equal characters.

Pf:

• $(\Rightarrow)$ Obviously by Lemma(2.3).

•  $(\Leftarrow)$ $\mathfrak{X}$ and $\mathfrak{Y}$ correspond to $V$ and $W$, respectively. Then $\mathfrak{X}$ affords $\sum n_{M_i}(V)\chi_i$ and $\mathfrak{Y}$ affords $\sum n_{M_i}(W)\chi_i$.