每日定理15

Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(2.8)

Every class function $\varphi$ of $G$ can be uniquely expressed in the form

$$\varphi=\sum_{\chi\in Irr(G)}a_{\chi}\chi,$$

where $a_{\chi}\in\mathbb{C}$. Furthermore, $\varphi$ is a character iff all of the $a_\chi$ are nonnegtive integers and $\varphi\neq0$.

Pf:

• Class functions of $G$ form a vector space whose dimension is the number of classes of $G$.
• $Irr(G)$ is a basis for this space. Consider $\sum a_i\chi_i=0$ at $e_i$.