Peter–Weyl theorem
E503519
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Peter–Weyl theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5212028 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Peter–Weyl theorem Context triple: [Weyl character formula, relatedTo, Peter–Weyl theorem]
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A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
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B.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
-
C.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
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D.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
E.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Peter–Weyl theorem Target entity description: 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.
-
A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
-
B.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
-
C.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
-
D.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
E.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
- F. None of above. chosen
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
ⓘ
Hermann Weyl ⓘ |
| originalAuthors |
Fritz Peter
ⓘ
Hermann Weyl ⓘ |
| relatedTo |
Fourier transform on compact groups
ⓘ
Pontryagin duality ⓘ Tannaka–Krein duality ⓘ
linked to:
Tannakian categories
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 ⓘ |
| yearProved | 1927 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Peter–Weyl theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.