Schur’s lemma
E924214
Schur’s lemma is a fundamental result in representation theory stating that any homomorphism between irreducible representations is either zero or an isomorphism, and that endomorphisms of an irreducible representation over an algebraically closed field are scalar multiples of the identity.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Schur’s lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11412035 — 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: Schur’s lemma Context triple: [Methods of Representation Theory, covers, Schur’s lemma]
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A.
Schur
Schur is a surname most prominently associated with Michael Schur, the American television writer and producer known for creating and co-creating acclaimed comedy series such as Parks and Recreation and The Good Place.
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B.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
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C.
Peter–Weyl theorem
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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D.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
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E.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
- 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: Schur’s lemma Target entity description: Schur’s lemma is a fundamental result in representation theory stating that any homomorphism between irreducible representations is either zero or an isomorphism, and that endomorphisms of an irreducible representation over an algebraically closed field are scalar multiples of the identity.
-
A.
Schur
Schur is a surname most prominently associated with Michael Schur, the American television writer and producer known for creating and co-creating acclaimed comedy series such as Parks and Recreation and The Good Place.
-
B.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
C.
Peter–Weyl theorem
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.
-
D.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
E.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in representation theory ⓘ |
| appearsIn |
courses on group representations and character theory
ⓘ
standard graduate texts on representation theory ⓘ |
| appliesTo |
modules over rings
ⓘ
representations of algebras ⓘ representations of groups ⓘ |
| assumes | homomorphism of representations ⓘ |
| concludes |
any homomorphism between nonisomorphic irreducible representations is zero
ⓘ
any nonzero homomorphism between irreducible representations is an isomorphism ⓘ commutant of an irreducible representation over an algebraically closed field is the scalar operators ⓘ endomorphism ring of an irreducible representation over an algebraically closed field is isomorphic to the base field ⓘ endomorphisms of an irreducible representation over an algebraically closed field are scalar multiples of the identity ⓘ |
| domain |
irreducible representations
ⓘ
simple modules ⓘ |
| field |
abstract algebra
ⓘ
group representation theory ⓘ representation theory ⓘ |
| firstSchurLemmaStatement | if V and W are irreducible representations and T:V→W is a nonzero homomorphism then T is an isomorphism GENERATED ⓘ |
| formalizes | rigidity of irreducible representations ⓘ |
| generalizationOf | fact that homomorphisms between simple modules are either zero or isomorphisms ⓘ |
| hasGeneralization |
Jacobson density theorem
NERFINISHED
ⓘ
double commutant theorem in representation theory ⓘ |
| holdsOver | any field for the first part ⓘ |
| implies |
commuting algebra of an irreducible representation is commutative and one-dimensional over an algebraically closed field
ⓘ
irreducible representations have simple endomorphism rings over algebraically closed fields ⓘ |
| namedAfter | Issai Schur NERFINISHED ⓘ |
| relatedTo |
Burnside’s theorem
NERFINISHED
ⓘ
Maschke’s theorem NERFINISHED ⓘ Wedderburn’s theorem NERFINISHED ⓘ |
| requires | algebraically closed field for the scalar endomorphism conclusion ⓘ |
| secondSchurLemmaStatement | if V is an irreducible representation over an algebraically closed field then End(V) is one-dimensional over that field ⓘ |
| typicalContext |
finite-dimensional representations
ⓘ
unitary representations of groups ⓘ |
| usedIn |
character theory of finite groups
ⓘ
classification of irreducible representations ⓘ construction of central idempotents in group algebras ⓘ proof of complete reducibility theorems ⓘ proofs involving Maschke’s theorem ⓘ quantum mechanics via irreducible unitary representations ⓘ representation theory of compact groups ⓘ representation theory of semisimple Lie algebras ⓘ |
| version |
first Schur’s lemma
NERFINISHED
ⓘ
second Schur’s lemma ⓘ |
| yearIntroducedApprox | early 20th century ⓘ |
How these facts were elicited
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Input
Subject: Schur’s lemma Description of subject: Schur’s lemma is a fundamental result in representation theory stating that any homomorphism between irreducible representations is either zero or an isomorphism, and that endomorphisms of an irreducible representation over an algebraically closed field are scalar multiples of the identity.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.