Schur–Weyl duality
E503508
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Schur–Weyl duality canonical | 2 |
| Schur–Weyl duality for Hecke algebras | 1 |
| Schur–Weyl duality for quantum groups | 1 |
| Schur–Weyl reciprocity | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5211789 — 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–Weyl duality Context triple: [The Classical Groups: Their Invariants and Representations, topic, Schur–Weyl duality]
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A.
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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B.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
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C.
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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D.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
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E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- 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–Weyl duality Target entity description: 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.
-
A.
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.
-
B.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
-
C.
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.
-
D.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in representation theory ⓘ |
| appliesTo | tensor power V^{\otimes n} of a vector space V ⓘ |
| characteristicAssumption | often formulated over fields of characteristic zero ⓘ |
| connects |
Specht modules of the symmetric group
ⓘ
irreducible polynomial representations of GL(V) ⓘ |
| context | finite-dimensional vector spaces ⓘ |
| describes | bimodule structure of V^{\otimes n} ⓘ |
| extendsTo | fields of positive characteristic with modifications ⓘ |
| field |
algebra
ⓘ
group theory ⓘ representation theory ⓘ |
| formalism | bimodule decomposition ⓘ |
| generalizedBy | Howe duality ⓘ |
| gives | decomposition of V^{\otimes n} into irreducible GL(V)-modules and S_n-modules ⓘ |
| hasVariant |
Schur–Weyl duality for Hecke algebras
ⓘ
linked to:
Schur–Weyl duality
Schur–Weyl duality for quantum groups ⓘ
linked to:
Schur–Weyl duality
q-Schur–Weyl duality ⓘ |
| historicalPeriod | early 20th century ⓘ |
| implies |
V^{\otimes n} \cong \bigoplus_{\lambda} S^{\lambda}(V) \otimes M^{\lambda}
ⓘ
double centralizer property for GL(V) and S_n ⓘ |
| involves |
group algebra of the symmetric group
ⓘ
representation of GL(V) ⓘ representation of S_n ⓘ |
| namedAfter |
Hermann Weyl
ⓘ
Issai Schur ⓘ |
| relates |
centralizer algebra of GL(V) on V^{\otimes n}
ⓘ
centralizer algebra of S_n on V^{\otimes n} ⓘ general linear group ⓘ
linked to:
general linear group GL(n,R)
symmetric group ⓘ tensor powers of a vector space ⓘ |
| requires | dimension of V at least n for full correspondence ⓘ |
| statesThat |
actions of GL(V) and S_n on V^{\otimes n} commute
ⓘ
images of GL(V) and S_n actions on V^{\otimes n} are mutual centralizers ⓘ |
| typeOf | duality between groups and centralizer algebras ⓘ |
| usedIn |
Schur–Weyl reciprocity
ⓘ
linked to:
Schur–Weyl duality
algebraic combinatorics ⓘ categorification ⓘ construction of Schur algebras ⓘ invariant theory ⓘ representation theory of GL_n ⓘ representation theory of S_n ⓘ theory of symmetric functions ⓘ |
| uses |
Schur functors
ⓘ
Young diagrams ⓘ commuting group actions ⓘ partitions of n ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Schur–Weyl duality Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Schur–Weyl duality
linked to: Schur–Weyl duality
linked to: Schur–Weyl duality