PlanetMath: character

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character

(Definition)

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

$\displaystyle \chi_V(g) := \operatorname{Tr}(\rho(g))$

where $\operatorname{Tr}$ is the trace function.

$\chi_V(g) = \chi_V(h)$ if is conjugate to in . (Equivalently, a character is a class function on .)
If is finite, the characters of the irreducible representations of over the complex numbers form a basis of the vector space of all class functions on (with pointwise addition and scalar multiplication).
Over the complex numbers, the characters of the irreducible representations of are orthonormal under the inner product
$\displaystyle (\chi_1, \chi_2) := \frac{1}{\vert G\vert} \sum_{g \in G} \overline{\chi_1(g)} \chi_2(g)$

"character" is owned by djao.

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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
There are 25 references to this entry.

This is version 4 of character, born on 2002-02-07, modified 2003-02-10.
Object id is 1843, canonical name is Character.
Accessed 8686 times total.

Classification:

20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous)

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