Borel–Weil theorem
E504917
The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Borel–Weil theorem canonical | 3 |
| Borel–Weil–Bott theorem | 3 |
| Borel–Bott–Weil theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5212025 — 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: Borel–Weil theorem Context triple: [Weyl character formula, relatedTo, Borel–Weil theorem]
-
A.
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.
-
B.
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.
-
C.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
D.
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.
-
E.
Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
- 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: Borel–Weil theorem Target entity description: The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
-
A.
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.
-
B.
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.
-
C.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
D.
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.
-
E.
Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in representation theory ⓘ |
| appliesTo |
compact Lie groups
ⓘ
complex semisimple Lie groups ⓘ |
| assumes | compactness of the Lie group in its classical form ⓘ |
| characterizes | irreducible representations by highest weights ⓘ |
| codomain | holomorphic sections of line bundles ⓘ |
| concerns |
compact connected Lie groups
ⓘ
complex semisimple algebraic groups ⓘ irreducible finite-dimensional representations ⓘ |
| constructionMethod | global holomorphic sections of an equivariant line bundle ⓘ |
| constructs | irreducible representation from a dominant weight ⓘ |
| context |
complex analytic geometry on homogeneous spaces
ⓘ
representation theory of compact connected Lie groups ⓘ |
| describes | irreducible representations of compact Lie groups ⓘ |
| domain |
representation theory of Lie algebras
ⓘ
representation theory of Lie groups ⓘ |
| field |
Lie theory
ⓘ
algebraic geometry ⓘ representation theory ⓘ |
| generalizedBy | Borel–Weil–Bott theorem ⓘ |
| hasVariant | algebraic version for complex reductive groups ⓘ |
| implies |
existence of all irreducible finite-dimensional representations
ⓘ
uniqueness of irreducible representation for each dominant integral weight ⓘ |
| namedAfter |
André Weil
ⓘ
Armand Borel ⓘ |
| realizesAs | spaces of holomorphic sections of line bundles ⓘ |
| relatedTo |
Borel–Weil–Bott theorem
ⓘ
Bott–Borel–Weil theory ⓘ
linked to:
Borel–Weil–Bott theorem
Peter–Weyl theorem ⓘ highest weight classification ⓘ |
| relates |
Lie group representations and line bundles on flag varieties
ⓘ
representation theory and complex geometry ⓘ |
| toolFor |
classification of irreducible representations of compact Lie groups
ⓘ
geometric representation theory ⓘ |
| typicalDomainObject |
maximal torus of a compact Lie group
ⓘ
weight lattice of a Lie group ⓘ |
| typicalGeometricObject |
complex flag manifold
ⓘ
projective homogeneous variety ⓘ |
| usedIn |
modern geometric representation theory
ⓘ
theory of automorphic forms ⓘ |
| usesConcept |
Borel subgroup
ⓘ
dominant integral weights ⓘ flag manifolds ⓘ highest weight theory ⓘ holomorphic line bundles ⓘ |
| usesObject |
flag variety G/B
ⓘ
homogeneous space of a Lie group ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Borel–Weil theorem Description of subject: The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Borel–Weil theorem
linked to: Borel–Weil theorem
linked to: Borel–Weil theorem
linked to: Borel–Weil theorem