Weyl vector
E503518
The Weyl vector is a distinguished element in the weight space of a semisimple Lie algebra, defined as half the sum of all positive roots and playing a central role in representation theory and the Weyl character formula.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weyl vector canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T5212021 — 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: Weyl vector Context triple: [Weyl character formula, usesConcept, Weyl vector]
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A.
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.
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B.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
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C.
Weyl
Weyl is a surname most famously associated with Hermann Weyl, a prominent 20th-century mathematician and theoretical physicist known for major contributions to group theory, quantum mechanics, and the foundations of mathematics.
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D.
Weyl tensor
The Weyl tensor is the traceless part of the Riemann curvature tensor in differential geometry and general relativity, encoding the purely shape-distorting (conformal) aspects of spacetime curvature independent of matter content.
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E.
Weil representation
The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
- 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: Weyl vector Target entity description: The Weyl vector is a distinguished element in the weight space of a semisimple Lie algebra, defined as half the sum of all positive roots and playing a central role in representation theory and the Weyl character formula.
-
A.
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.
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B.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
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C.
Weyl
Weyl is a surname most famously associated with Hermann Weyl, a prominent 20th-century mathematician and theoretical physicist known for major contributions to group theory, quantum mechanics, and the foundations of mathematics.
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D.
Weyl tensor
The Weyl tensor is the traceless part of the Riemann curvature tensor in differential geometry and general relativity, encoding the purely shape-distorting (conformal) aspects of spacetime curvature independent of matter content.
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E.
Weil representation
The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
element of weight space
ⓘ
mathematical object ⓘ weight ⓘ |
| alsoKnownAs | Weyl weight ⓘ |
| appearsIn |
Kac–Moody algebra representation theory
ⓘ
affine Lie algebra theory ⓘ algebraic group representation theory ⓘ theory of Verma modules ⓘ theory of compact Lie groups ⓘ theory of complex semisimple Lie groups ⓘ |
| associatedWith |
Cartan subalgebra of a semisimple Lie algebra
ⓘ
root lattice ⓘ weight lattice ⓘ |
| componentOf |
shift λ+ρ in the Weyl character formula
ⓘ
shift λ+ρ in the Weyl dimension formula ⓘ |
| constructedFrom | positive roots ⓘ |
| definedInContextOf |
root system
ⓘ
semisimple Lie algebra ⓘ |
| definition | half the sum of all positive roots of a root system ⓘ |
| dependsOn | choice of positive root system ⓘ |
| field |
Lie algebras
ⓘ
Lie theory ⓘ representation theory ⓘ |
| generalizationOf | half-sum of positive roots in finite root systems to Kac–Moody root systems ⓘ |
| invariantUnder | Weyl group of the root system up to sign change under simple reflections ⓘ |
| liesIn |
Cartan subalgebra dual
ⓘ
weight space ⓘ |
| namedAfter | Hermann Weyl ⓘ |
| property |
has strictly positive pairing with all simple coroots
ⓘ
is a regular weight ⓘ lies in the interior of the dominant Weyl chamber ⓘ |
| relatedTo |
dominant weight
ⓘ
fundamental weights ⓘ highest weight ⓘ simple roots ⓘ |
| role |
appears in exponents of characters of irreducible highest weight modules
ⓘ
defines the ρ-shift in representation theory ⓘ shifts highest weights in the Weyl character formula ⓘ |
| symbol | ρ ⓘ |
| usedFor |
computing multiplicities in highest weight representations
ⓘ
defining the dot action of the Weyl group on weights ⓘ formulating the Kazhdan–Lusztig conjecture ⓘ formulation of the Harish-Chandra character formula ⓘ |
| usedIn |
Borel–Weil–Bott theorem
ⓘ
linked to:
Borel–Weil theorem
Harish-Chandra isomorphism ⓘ Weyl character formula ⓘ Weyl dimension formula ⓘ |
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: Weyl vector Description of subject: The Weyl vector is a distinguished element in the weight space of a semisimple Lie algebra, defined as half the sum of all positive roots and playing a central role in representation theory and the Weyl character formula.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.