# Definition:Character (Representation Theory)

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## Definition

Let $\left({G, \cdot}\right)$ be a finite group.

Let $V$ be a finite dimensional $k$-vector space.

Consider a linear representation $\rho: G \to \operatorname{GL}\left({V}\right)$ of $G$.

The **character associated with $\rho$** is defined as:

- $\chi:G \to k$

where $\chi \left({g}\right) = \operatorname{Tr} \left({\rho\left({g}\right)}\right)$, the trace of $\rho\left({g}\right)$; which is a linear automorphism of $V$.

## Note

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