Real 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

Let $G$ be a group. A character $\chi$ of a complex linear representation of $G$ is called real if $\chi (g)\in \mathbb {R}$ for every $g\in G$. Equivalently, $\chi (g)={\overline {\chi (g)}}$ for every $g\in G$.

Facts

• Number of real conjugacy classes is equal to number of irreducible real characters