Schur’s lemma

E924214

mathematical theorem result in representation theory

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.

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All labels observed (3)

Label	Occurrences
Schur’s lemma canonical	2
Schur's lemma	1
first Schur’s lemma	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]

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
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 ⓘ
formalizes	rigidity of irreducible representations ⓘ
generalizationOf	fact that homomorphisms between simple modules are either zero or isomorphisms ⓘ
hasGeneralization	Jacobson density theorem ⓘ 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 ⓘ
relatedTo	Burnside’s theorem ⓘ linked to: Burnside's lemma Maschke’s theorem ⓘ Wedderburn’s theorem ⓘ linked to: Artin–Wedderburn theorem
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 ⓘ linked to: Schur’s lemma second Schur’s lemma ⓘ
yearIntroducedApprox	early 20th century ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Methods of Representation Theory → covers → Schur’s lemma ⓘ

Representations of groups → usesConcept → Schur's lemma ⓘ

linked to: Schur’s lemma

Maschke’s theorem → relatedTo → Schur’s lemma ⓘ

Schur’s lemma → version → first Schur’s lemma ⓘ