imported>Howtonotwin: /* Inner products */ proper formatting, slight reword

2025-02-27T09:10:49Z

Inner products: proper formatting, slight reword

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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]] on a [[group (mathematics)|group]] ''G'' that is constant on the [[conjugacy class]]es of ''G''. In other words, it 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 (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)},</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]].

When {{mvar|G}} is a [[compact group]] and {{math|''K'' {{=}} '''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]]

Class function - Revision history

imported>Howtonotwin: /* Inner products */ proper formatting, slight reword