Class function: Difference between revisions

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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]].

In fact theThe [[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'' 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 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]].

In the case of a [[compact group]], 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>.

== References ==

* [[Jean-Pierre Serre]], ''Linear representations of finite groups'', Graduate Texts in Mathematics '''42''', Springer-Verlag, Berlin, 1977.

[[Category:Group theory]]