Peter–Weyl theorem

E503519

theorem theorem in representation theory

The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.

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Statements (45)

Predicate	Object
instanceOf	theorem ⓘ theorem in representation theory ⓘ
appliesTo	compact groups ⓘ compact topological groups ⓘ
assumes	existence of a normalized Haar measure on a compact group ⓘ
concerns	L^2 spaces ⓘ finite-dimensional representations ⓘ harmonic analysis on groups ⓘ irreducible representations ⓘ regular representation ⓘ square-integrable functions ⓘ unitary representations ⓘ
context	noncommutative harmonic analysis ⓘ unitary representation theory ⓘ
describedIn	Hermann Weyl's work on representation theory of compact groups ⓘ
field	harmonic analysis ⓘ representation theory ⓘ topological groups ⓘ
generalizes	Fourier analysis on finite groups ⓘ Fourier series on the circle group ⓘ
hasConsequence	L^2(G) decomposes into isotypic components indexed by irreducible unitary representations ⓘ every continuous finite-dimensional unitary representation of a compact group is completely reducible ⓘ the set of equivalence classes of irreducible unitary representations of a compact group is countable ⓘ
holdsIn	Hilbert space L^2(G) ⓘ
implies	complete reducibility of unitary representations of compact groups ⓘ existence of an orthonormal basis of L^2(G) consisting of matrix coefficients of irreducible unitary representations ⓘ orthogonality relations for matrix coefficients of irreducible representations of compact groups ⓘ
namedAfter	Fritz Peter NERFINISHED ⓘ Hermann Weyl NERFINISHED ⓘ
originalAuthors	Fritz Peter NERFINISHED ⓘ Hermann Weyl NERFINISHED ⓘ
relatedTo	Fourier transform on compact groups ⓘ Pontryagin duality NERFINISHED ⓘ Tannaka–Krein duality NERFINISHED ⓘ representation theory of compact Lie groups ⓘ
statesThat	the irreducible unitary representations of a compact group occur with finite multiplicity in the regular representation ⓘ the matrix coefficients of irreducible unitary representations of a compact group are dense in the space of continuous functions on the group ⓘ the regular representation of a compact group decomposes as a Hilbert space direct sum of finite-dimensional irreducible unitary representations ⓘ
usedFor	analysis of convolution operators on compact groups ⓘ construction of characters of compact groups ⓘ decomposition of class functions on compact groups ⓘ harmonic analysis on compact Lie groups ⓘ spectral decomposition on compact groups ⓘ
uses	Haar measure NERFINISHED ⓘ
yearProved	1927 ⓘ

Referenced by (1)

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Weyl character formula → relatedTo → Peter–Weyl theorem ⓘ