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.
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 NERFINISHED ⓘ |
| outputType | nonnegative integer ⓘ |
| relatedTo |
Borel–Weil theorem
NERFINISHED
ⓘ
Cartan subalgebra NERFINISHED ⓘ Weyl character formula NERFINISHED ⓘ Weyl group NERFINISHED ⓘ 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
NERFINISHED
ⓘ
dominant integral weight ⓘ highest weight ⓘ positive roots ⓘ root system ⓘ |
| validFor | integrable highest weight modules of finite type ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.