# Character of a linear representation

This term makes sense in the context of a linear representation of a group, viz an action of the group as linear automorphisms of a vector space

This article gives a basic definition in the following area: linear representation theory
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## Definition

### Definition in terms of linear representation as a homomorphism

Let $G$ be a group and $\rho: G \to GL(V)$ be a finite-dimensional linear representation over a field $k$. Then, the character of $\rho$ is the composite $Tr \circ \rho$ where $Tr$ is the trace map from $GL(V)$ to $k$.