# Definition:Character (Representation Theory)

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## Definition

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 $G$.

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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 $V$.

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## Note

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If $G$ is abelian, then this definition for character is equivalent to the one of character.