Class function: Difference between revisions
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==Characters== | ==Characters== | ||
The 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|group ring]] ''K''[''G'']. Here a class function ''f'' is identified with the element <math> \sum_{g \in G} f(g) g</math>. | The character of a [[Linear representation|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|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 == | == 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|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 characters of {{mvar|G}} forms an [[Orthogonal basis|orthogonal basis]]. Further, if {{mvar|K}} is a [[Splitting field|splitting field]] for {{mvar|G}}{{--}}for instance, if {{mvar|K}} is algebraically closed, then the irreducible characters form an [[Orthonormal basis|orthonormal basis]]. | The set of class functions of a group {{mvar|G}} with values in a field {{mvar|K}} form a {{mvar|K}}-[[Vector space|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|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 characters of {{mvar|G}} forms an [[Orthogonal basis|orthogonal basis]]. Further, if {{mvar|K}} is a [[Splitting field|splitting field]] for {{mvar|G}}{{--}}for instance, if {{mvar|K}} is algebraically closed, then the irreducible characters form an [[Orthonormal basis|orthonormal basis]]. | ||
When {{mvar|G}} is a [[Compact group|compact group]] and {{math|''K'' {{=}} '''C'''}} is the field of [[Complex number|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|G}} is a [[Compact group|compact group]] and {{math|''K'' {{=}} '''C'''}} is the field of [[Complex number|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> In this setting, the irreducible characters form a Hilbert basis of the [[Hilbert space]] of square-integrable class functions, by the [[Peter–Weyl theorem]]. | ||
When {{mvar|K}} is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian [[Bilinear form|bilinear form]]. | When {{mvar|K}} is the real numbers or the complex numbers, the inner product is a non-degenerate [[Hermitian form|Hermitian]] [[Bilinear form|bilinear form]]. | ||
==See also== | ==See also== | ||
Latest revision as of 06:44, 15 April 2026
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes 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 of a linear representation of G over a field K is always a class function with values in K. The class functions form the center of the group ring K[G]. Here a class function f is identified with the element .
Inner products
The set of class functions of a group G with values in a field K form a K-vector space. If G is finite and the characteristic of the field does not divide the order of G, then there is an inner product defined on this space defined by where |G| denotes the order of G and the overbar denotes conjugation in the field K. The set of irreducible characters of G forms an orthogonal basis. Further, if K is a splitting field for G—for instance, if K is algebraically closed, then the irreducible characters form an orthonormal basis.
When G is a compact group and K = C is the field of complex numbers, the Haar measure can be applied to replace the finite sum above with an integral: In this setting, the irreducible characters form a Hilbert basis of the Hilbert space of square-integrable class functions, by the Peter–Weyl theorem.
When K is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.
See also
References
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, Berlin, 1977.
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