Weyl dimension formula
E506993
The Weyl dimension formula is a fundamental result in representation theory that gives an explicit product expression for the dimension of each finite-dimensional irreducible representation of a semisimple Lie algebra or compact Lie group in terms of its highest weight and the root system.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weyl dimension formula canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5212024 — 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 dimension formula Context triple: [Weyl character formula, implies, Weyl dimension 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.
-
B.
Weyl denominator
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.
-
C.
Weyl vector
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.
-
D.
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.
-
E.
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.
- 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 dimension formula Target entity description: The Weyl dimension formula is a fundamental result in representation theory that gives an explicit product expression for the dimension of each finite-dimensional irreducible representation of a semisimple Lie algebra or compact Lie group in terms of its highest weight and the root system.
-
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 denominator
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.
-
C.
Weyl vector
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in representation theory ⓘ |
| appliesTo |
compact Lie groups
ⓘ
finite-dimensional irreducible representations ⓘ semisimple Lie algebras ⓘ |
| assumes | semisimple Lie algebra over complex numbers ⓘ |
| category |
theorems about Lie algebras
ⓘ
theorems about Lie groups ⓘ |
| contrastsWith | character formulas that give full weight multiplicities ⓘ |
| domain | finite-dimensional representations ⓘ |
| expressionType | product formula ⓘ |
| field |
Lie theory
ⓘ
representation theory ⓘ |
| generalizes | binomial coefficient dimension formulas for sl2 representations ⓘ |
| gives |
dimension as product over positive roots
ⓘ
dimension of irreducible representation ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor |
reductive Lie algebras with finite center
ⓘ
simple Lie algebras ⓘ |
| involvesOperation |
inner product on weight space
ⓘ
pairing of weights and coroots ⓘ |
| isPartOf | Weyl’s work on representation theory of Lie groups ⓘ |
| namedAfter | Hermann Weyl ⓘ |
| outputType | nonnegative integer ⓘ |
| relatedTo |
Borel–Weil theorem
ⓘ
Cartan subalgebra ⓘ
linked to:
Cartan subalgebras
Weyl character formula ⓘ Weyl group ⓘ highest weight theory ⓘ weight lattice ⓘ |
| requires |
choice of positive root system
ⓘ
dominant highest weight ⓘ |
| usedFor |
classifying irreducible representations
ⓘ
computing dimensions of representations ⓘ studying representation growth ⓘ |
| usedIn |
mathematical physics
ⓘ
particle physics ⓘ quantum mechanics ⓘ theory of algebraic groups ⓘ theory of compact Lie groups ⓘ |
| usesConcept |
Weyl vector
ⓘ
dominant integral weight ⓘ highest weight ⓘ positive roots ⓘ root system ⓘ |
| validFor | integrable highest weight modules of finite type ⓘ |
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 dimension formula Description of subject: The Weyl dimension formula is a fundamental result in representation theory that gives an explicit product expression for the dimension of each finite-dimensional irreducible representation of a semisimple Lie algebra or compact Lie group in terms of its highest weight and the root system.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.