Weyl character formula
E117655
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Weyl character formula canonical | 4 |
| Borel–Weil–Bott theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T990132 — 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 character formula Context triple: [Hermann Weyl, knownFor, Weyl character formula]
-
A.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
B.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
C.
Gell-Mann–Nishijima formula
The Gell-Mann–Nishijima formula is a key relation in particle physics that connects a particle’s electric charge to its isospin and hypercharge, helping classify hadrons within the quark model.
-
D.
Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
-
E.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
- 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 character formula Target entity description: 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.
-
A.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
B.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
C.
Gell-Mann–Nishijima formula
The Gell-Mann–Nishijima formula is a key relation in particle physics that connects a particle’s electric charge to its isospin and hypercharge, helping classify hadrons within the quark model.
-
D.
Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
-
E.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in representation theory ⓘ |
| appearsIn | Hermann Weyl's work on classical groups ⓘ |
| appliesTo |
compact Lie groups
ⓘ
semisimple Lie algebras ⓘ semisimple Lie groups ⓘ |
| assumes |
finite-dimensional representation
ⓘ
semisimple Lie algebra or group ⓘ |
| basedOn |
highest weight theory
ⓘ
structure theory of semisimple Lie algebras ⓘ |
| describes | characters of irreducible finite-dimensional representations ⓘ |
| expresses |
character as alternating sum over Weyl group
ⓘ
character as quotient of two Weyl group sums ⓘ |
| field |
Lie theory
ⓘ
representation theory ⓘ |
| generalizationOf | character formulas for sl(2,C) ⓘ |
| gives | explicit expression for irreducible characters ⓘ |
| hasConsequence |
integrality of weight multiplicities
ⓘ
orthogonality relations for characters ⓘ symmetry properties of characters ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor |
complex semisimple Lie algebras
ⓘ
connected compact Lie groups ⓘ |
| implies | Weyl dimension formula ⓘ |
| influenced |
algebraic group representation theory
ⓘ
modern representation theory of Lie groups ⓘ |
| introducedBy | Hermann Weyl ⓘ |
| involvesObject | exponential of weights ⓘ |
| involvesOperation | alternating sum over Weyl group elements ⓘ |
| namedAfter | Hermann Weyl ⓘ |
| relatedTo |
Borel–Weil theorem
ⓘ
Borel–Weil theorem ⓘ
surface form:
Borel–Weil–Bott theorem
Harish-Chandra character formula ⓘ Peter–Weyl theorem ⓘ |
| topicOf |
textbooks on Lie algebras
ⓘ
textbooks on representation theory ⓘ |
| usedFor |
classifying irreducible representations
ⓘ
computing characters of representations ⓘ computing dimensions of representations ⓘ computing weight multiplicities ⓘ |
| usesConcept |
Cartan subalgebras
ⓘ
surface form:
Cartan subalgebra
Weyl denominator ⓘ Weyl group ⓘ Weyl vector ⓘ dominant integral weights ⓘ highest weight ⓘ maximal torus ⓘ root system ⓘ weight lattice ⓘ |
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 character formula Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Borel–Weil–Bott theorem