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 | 1 |
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Harish-Chandra isomorphism NERFINISHED ⓘ Weyl character formula NERFINISHED ⓘ Weyl dimension formula NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.