Let $\rho: G \longrightarrow \operatorname{GL}(V)$ be a finite dimensional representation of a group $G$ (i.e., $V$ is a finite dimensional vector space over its scalar field $K$ . The character of $\rho$ is the function $\chi_V: G \longrightarrow K$ defined by $$ \chi_V(g) := \operatorname{Tr}(\rho(g)) $$ where $\operatorname{Tr}$ is the trace function.

Properties:

• $\chi_V(g) = \chi_V(h)$ if $g$ is conjugate to $h$ in $G$ (Equivalently, a character is a class function on $G$ )
• If $G$ is finite, the characters of the irreducible representations of $G$ over the complex numbers form a basis of the vector space of all class functions on $G$ (with pointwise addition and scalar multiplication).
• Over the complex numbers, the characters of the irreducible representations of $G$ are orthonormal under the inner product $$ (\chi_1, \chi_2) := \frac{1}{|G|} \sum_{g \in G} \overline{\chi_1(g)} \chi_2(g) $$