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character (Definition)

Let $\rho: G \longrightarrow \operatorname{GL}(V)$ be a finite dimensional representation of a group $G$ (i.e., $V$ is a finite dimensional vector space over its scalar field $K$ . The character of $\rho$ is the function $\chi_V: G \longrightarrow K$ defined by $$ \chi_V(g) := \operatorname{Tr}(\rho(g)) $$ where $\operatorname{Tr}$ is the trace function.

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Cross-references: inner product, orthonormal, multiplication, pointwise addition, basis, complex numbers, irreducible, finite, class function, conjugate, properties, trace, function, field, scalar, vector space, group, representation, finite dimensional
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This is version 4 of character, born on 2002-02-07, modified 2003-02-10.
Object id is 1843, canonical name is Character.
Accessed 10967 times total.

Classification:
AMS MSC20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous)

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