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In [[mathematics]], aespecially '''classin function'''the infields 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]].
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 we identify 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 finite group ''G'' with values in a field ''K'' form a ''K''-[[vector space]]. There If the [[characteristic (algebra)|characteristic]] of the field does not divide the order of ''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) \psi(g^{-1}) </math> where |''G''| denotes the order of ''G''. The set of [[Character theory|irreducible characters]] of ''G'' forms an [[orthogonal basis]], and if ''K'' is a splitting field for ''G'', for instance if ''K'' is [[algebraically closed]], then the irreducible characters form an [[orthonormal basis]].
<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]].
In the case of a [[compact group]] and ''K''='''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>.
<math> \langle \phi, \psi \rangle = \int_G \phi(t) \psi(t^{-1})\, dt </math>.
When restricted to real linear combinations of characters, the inner product is a [[degenerate form|non-degenerate]] [[Hermitian form|Hermitian]] [[bilinear form]].
== References ==
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