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

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]].

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 aSuch classfunctions functionplay ''f''a isbasic identifiedrole with the element <math> \sum_{g \in G}[[representation f(g) g</math>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 (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>.

== Inner products ==

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^{-1}) },</math> where {{math|{{!}}''G''|{{!}}}} denotes the order of ''{{mvar|G''}} and the overbar denotes conjugation in the field {{mvar|K}}. The set of [[Character theory|irreducible characterscharacter]]s of ''{{mvar|G''}} forms an [[orthogonal basis]],. andFurther, 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]].

InWhen the{{mvar|G}} case ofis a [[compact group]] and {{math|''K''&nbsp;{{=}}&nbsp;'''C'''}} is the field of [[complex number]]s, the notion of [[Haar measure]] allowscan onebe applied to replace the finite sum above with an integral: <math> \langle \phi, \psi \rangle = \int_G \phi(t) \overline{\psi(t^{-1})}\, 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]].

In the case of a [[compact group]] and ''K''&nbsp;=&nbsp;'''C''' the field of [[complex number]]s, the notion of [[Haar measure]] allows one to replace the finite sum above with an integral: <math> \langle \phi, \psi \rangle = \int_G \phi(t) \psi(t^{-1})\, dt. </math>

==See also==

When ''K'' is the real numbers or the complex numbers, the inner product is a [[degenerate form|non-degenerate]] [[Hermitian form|Hermitian]] [[bilinear form]].

*[[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]]