Weyl group

E117654

finite reflection group mathematical concept

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.

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All labels observed (13)

Label	Occurrences
Weyl group canonical	6
Weyl groups	3
Coxeter groups	1
Weyl group (for semisimple Lie groups)	1
Weyl group of type A_n	1
Weyl group of type B_n	1
Weyl group of type C_n	1
Weyl group of type D_n	1
Weyl group of type E6 via lines configuration	1
Weyl group of type E_6	1
Weyl group of type E_8	1
Weyl group of type F_4	1
Weyl group of type G_2	1

Statements (50)

Predicate	Object
instanceOf	finite reflection group ⓘ mathematical concept ⓘ
actsOn	Cartan subalgebra ⓘ Cartan subalgebra dual ⓘ root system ⓘ weight lattice ⓘ
appearsIn	theory of Kac–Moody algebras ⓘ theory of algebraic groups ⓘ
associatedWith	Coxeter group ⓘ root system ⓘ semisimple Lie algebra ⓘ semisimple Lie group ⓘ
correspondsTo	isomorphism class of root system ⓘ
definedAs	group generated by reflections in hyperplanes orthogonal to roots ⓘ
encodes	structure of semisimple Lie algebra ⓘ structure of semisimple Lie group ⓘ symmetries of Dynkin diagram ⓘ symmetries of root system ⓘ
field	Lie theory ⓘ algebraic geometry ⓘ combinatorics ⓘ differential geometry ⓘ representation theory ⓘ
hasExample	Weyl group self-linksurface differs ⓘ surface form: Weyl group of type A_n Weyl group self-linksurface differs ⓘ surface form: Weyl group of type B_n Weyl group self-linksurface differs ⓘ surface form: Weyl group of type C_n Weyl group self-linksurface differs ⓘ surface form: Weyl group of type D_n Weyl group self-linksurface differs ⓘ surface form: Weyl group of type E_6 Weyl group of type E_7 ⓘ Weyl group self-linksurface differs ⓘ surface form: Weyl group of type E_8 Weyl group self-linksurface differs ⓘ surface form: Weyl group of type F_4 Weyl group self-linksurface differs ⓘ surface form: Weyl group of type G_2
hasOrder	finite integer depending on root system ⓘ
hasProperty	acts on Euclidean space ⓘ crystallographic ⓘ finite ⓘ generated by reflections ⓘ
namedAfter	Hermann Weyl ⓘ
relatedTo	Cartan matrix ⓘ Coxeter–Dynkin diagrams ⓘ surface form: Dynkin diagram
subclassOf	Coxeter group ⓘ crystallographic reflection group ⓘ reflection group ⓘ
usedIn	Weyl character formula ⓘ surface form: Borel–Weil–Bott theorem Kazhdan–Lusztig theory ⓘ Weyl character formula ⓘ classification of semisimple Lie algebras ⓘ classification of simple Lie groups ⓘ representation theory of Lie algebras ⓘ representation theory of Lie groups ⓘ

Referenced by (20)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hermann Weyl → knownFor → Weyl group ⓘ

H. S. M. Coxeter → knownFor → Weyl group ⓘ

this entity surface form: Coxeter groups

Weyl → knownFor → Weyl group ⓘ

subject surface form: Hermann Weyl