\begin{definition}[Definition:Character (Representation Theory)]
Let $\struct {G, \cdot}$ be a finite group.
Let $V$ be a finite dimensional  $k$-vector space.
Consider a linear representation $\rho: G \to \GL V$ of $G$.
{{explain|$\GL V$, presumably general linear group}}
Let $\map \tr {\map \rho g}$ denote the trace of $\map \rho g$.
The '''character associated with $\rho$''' is defined as:
:$\chi: G \to k$
where $\map \chi g = \map \tr {\map \rho g}$, the trace of $\map \rho g$; which is a linear automorphism of $V$.
{{explain|"which is a linear automorphism of $V$." What is, precisely? $\map \rho g$ or $\map \tr {\map \rho g}$?}}
\end{definition}