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In [[mathematics]], especially in the fields of [[group theory]] and [[group representation|representation theory of groups]], a '''class function''' is a [[function (mathematics)|function]] ''f'' on a [[group (mathematics)|group]] ''G'', such that ''f'' is constant on the [[conjugacy class]]es of ''G''. In other words, ''f'' is invariant under the [[conjugation map]] on ''G''. Such functions play a basic role in [[representation theory]].
==Characters==
The [[character (group theory)|character]] of a [[linear representation]] of ''G'' over a [[field (mathematics)|field]] ''K'' is always a class function with values in ''K''. The class functions form the [[center (algebra)|center]] of the [[group ring]] ''K''[''G'']. Here a class function ''f'' is identified with the element <math> \sum_{g \in G} f(g) g</math>.
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When ''K'' is the real numbers or the complex numbers, the inner product is a [[degenerate form|non-degenerate]] [[Hermitian form|Hermitian]] [[bilinear form]].
==See also==
*[[Brauer's theorem on induced characters]]
== References ==
* [[Jean-Pierre Serre]], ''Linear representations of finite groups'', Graduate Texts in Mathematics '''42''', Springer-Verlag, Berlin, 1977.
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