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{{distinguish|text=a [[Class (set theory)#Classes in formal set theories|class function]] in set theory}}

In [[mathematics]], a '''class function''' in [[group theory]] 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]].

TheIn [[charactermathematics]], (especially in the fields of [[group theory)|character]] of aand [[lineargroup representation|representation theory of groups]], ofa ''G'class function''' overis a [[fieldfunction (mathematics)|fieldfunction]] on a [[group (mathematics)|group]] ''KG'' that is alwaysconstant aon classthe function[[conjugacy withclass]]es values inof ''KG''. TheIn classother functionswords, formit theis [[centerinvariant (algebra)|center]] ofunder the [[groupconjugation ringmap]] ''K''[on&nbsp;''G'']. Here weSuch identifyfunctions play a classbasic functionrole ''f'' with the element <math> \sum_{g \in G}[[representation f(g) g</math>theory]].

==Characters==

== Inner Products ==

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 (ring theory)|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>.

The set of class functions of a finite group ''G'' with values in a field ''K'' form a ''K''-[[vector space]]. There is an [[inner product]] defined on this space defined by

<math> \langle \phi , \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \phi(g) \psi(g^{-1}) </math>

where |''G''| denotes the order of ''G''. In the case that ''K'' has [[characteristic (algebra)|characteristic]] 0, the set of [[Character theory|irreducible characters]] of ''G'' forms an [[orthogonal basis]]. If ''K'' is [[algebraically closed]], we can say further that they form an [[orthonormal basis]].

== Inner Productsproducts ==

In the case of a [[compact group]], the notion of [[Haar measure]] allows one to replace the finite sum above with an integral:

The set of class functions of a group {{mvar|G}} with values in a field {{mvar|K}} form a {{mvar|K}}-[[vector space]]. If {{mvar|G}} is finite and the [[characteristic (algebra)|characteristic]] of the field does not divide the order of {{mvar|G}}, then there is an [[inner product]] defined on this space defined by <math>\langle \phi , \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \phi(g) \overline{\psi(g)},</math> where {{math|{{!}}''G''{{!}}}} denotes the order of {{mvar|G}} and the overbar denotes conjugation in the field {{mvar|K}}. The set of [[irreducible character]]s of {{mvar|G}} forms an [[orthogonal basis]]. Further, if {{mvar|K}} is a [[splitting field]] for {{mvar|G}}{{--}}for instance, if {{mvar|K}} is [[algebraically closed]], then the irreducible characters form an [[orthonormal basis]].

<math> \langle \phi, \psi \rangle = \int_G \phi(t) \psi(t^{-1})\, dt </math>.

When {{mvar|G}} is a [[compact group]] and {{math|''K''&nbsp;{{=}}&nbsp;'''C'''}} is the field of [[complex number]]s, the [[Haar measure]] can be applied to replace the finite sum above with an integral: <math>\langle \phi, \psi \rangle = \int_G \phi(t) \overline{\psi(t)}\, dt.</math>

When {{mvar|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.

[[Category:Group theory]]

[[fr:Fonction centrale d'un groupe fini]]

[[ja:類関数]]

[[zh:類函數]]