Definition:Character (Representation Theory)
In particular: Linear automorphism.
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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 $G$.
$\GL V$, presumably general linear group
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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 $V$.
"which is a linear automorphism of $V$." What is, precisely? $\map \rho g$ or $\map \tr {\map \rho g}$?
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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.