Character

Definition

Let $(V,\rho )$ be a representation of a finite group $G$. The character of $\rho$ is the complex valued function ${\chi }_{\rho ,V}:G\to \mathbb{C}$ defined by ${\chi }_{\rho ,V}(g)=\mathrm{Tr}(\rho (g)$ We usually write${\chi }_V$ or ${\chi }_{\rho }$rather than ${\chi }_{\rho ,V}$ .

Observation The following properties are immediate from the definition. (1) The character of a representation is a class function: gxg 1 = x , x g G. (2) If V W then V = W. (3) V W = V + W. (4) V 1 =dimV.