Weyl denominator
E503517
The Weyl denominator is a key product expression in Lie theory that appears in the Weyl character formula, encoding the alternating sum over the Weyl group and playing a central role in describing characters of irreducible representations of semisimple Lie algebras.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weyl denominator canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5212020 — 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 denominator Context triple: [Weyl character formula, usesConcept, Weyl denominator]
-
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.
-
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.
-
C.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
D.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
-
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 denominator Target entity description: The Weyl denominator is a key product expression in Lie theory that appears in the Weyl character formula, encoding the alternating sum over the Weyl group and playing a central role in describing characters of irreducible representations of semisimple Lie algebras.
-
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.
-
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.
-
C.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
D.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
-
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 (50)
| Predicate | Object |
|---|---|
| instanceOf |
invariant under root lattice translations up to sign
ⓘ
mathematical concept ⓘ object in Lie theory ⓘ |
| appearsIn |
Weyl character formula
ⓘ
theory of Macdonald identities and affine root systems ⓘ |
| context |
compact connected Lie groups
ⓘ
reductive algebraic groups ⓘ semisimple Lie algebras ⓘ |
| coreDefinition |
alternating sum over the Weyl group of exponentials of the Weyl vector
ⓘ
product over positive roots of (e^{α/2} − e^{−α/2}) ⓘ |
| definedOn |
Cartan subalgebra of a semisimple Lie algebra
ⓘ
maximal torus of a compact Lie group ⓘ |
| dependsOn | choice of positive root system ⓘ |
| encodes | alternating sum over the Weyl group ⓘ |
| expressedAs |
determinant of a matrix of exponentials in type A
ⓘ
product over positive roots of 2 sinh(α/2) ⓘ |
| field |
Lie theory
ⓘ
algebra ⓘ mathematical physics ⓘ representation theory ⓘ |
| generalizationOf | classical trigonometric product identities for SU(2) and SU(n) ⓘ |
| historicalOrigin | work of Hermann Weyl on representation theory of compact Lie groups ⓘ |
| invariantUnder | Weyl group up to sign ⓘ |
| property |
Weyl anti-invariant
ⓘ
changes sign under simple reflections ⓘ holomorphic on the complexified torus away from root hyperplanes ⓘ vanishes on root hyperplanes ⓘ |
| relatedTo |
Weyl character formula
ⓘ
Weyl integration formula ⓘ Weyl numerator ⓘ Weyl vector ⓘ discriminant of a root system ⓘ |
| roleIn |
Harish-Chandra’s formulae for characters
ⓘ
linked to:
Harish-Chandra character formula
Plancherel formula for compact Lie groups ⓘ denominator of the Weyl character formula ⓘ description of characters of irreducible highest-weight representations ⓘ harmonic analysis on compact Lie groups ⓘ |
| specialCase |
Vandermonde determinant for type A root systems
ⓘ
sin-product formula for SU(2) ⓘ |
| symbol |
Δ
ⓘ
δ (in some conventions) ⓘ |
| usedFor |
computing characters of finite-dimensional irreducible representations
ⓘ
computing weight multiplicities via Weyl character formula ⓘ expressing partition functions in some conformal field theories ⓘ formulating Kac–Weyl character formula in affine Kac–Moody theory ⓘ formulating Weyl’s dimension formula ⓘ |
| uses |
Weyl group
ⓘ
half-sum of positive roots ⓘ root system ⓘ set of positive roots ⓘ |
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 denominator Description of subject: The Weyl denominator is a key product expression in Lie theory that appears in the Weyl character formula, encoding the alternating sum over the Weyl group and playing a central role in describing characters of irreducible representations of semisimple Lie algebras.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.