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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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 ⓘ

Referenced by (1)

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

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