Weyl denominator

E503517

invariant under root lattice translations up to sign mathematical concept object in Lie theory

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.

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Statements (50)

Predicate	Object
instanceOf	invariant under root lattice translations up to sign ⓘ mathematical concept ⓘ object in Lie theory ⓘ
appearsIn	Weyl character formula NERFINISHED ⓘ theory of Macdonald identities and affine root systems ⓘ
context	compact connected Lie groups ⓘ reductive algebraic groups ⓘ semisimple Lie algebras ⓘ
coreDefinition	alternating sum over the Weyl group of exponentials of the Weyl vector ⓘ product over positive roots of (e^{α/2} − e^{−α/2}) ⓘ
definedOn	Cartan subalgebra of a semisimple Lie algebra ⓘ maximal torus of a compact Lie group ⓘ
dependsOn	choice of positive root system ⓘ
encodes	alternating sum over the Weyl group ⓘ
expressedAs	determinant of a matrix of exponentials in type A ⓘ product over positive roots of 2 sinh(α/2) ⓘ
field	Lie theory NERFINISHED ⓘ algebra ⓘ mathematical physics ⓘ representation theory ⓘ
generalizationOf	classical trigonometric product identities for SU(2) and SU(n) ⓘ
historicalOrigin	work of Hermann Weyl on representation theory of compact Lie groups ⓘ
invariantUnder	Weyl group up to sign ⓘ
property	Weyl anti-invariant ⓘ changes sign under simple reflections ⓘ holomorphic on the complexified torus away from root hyperplanes ⓘ vanishes on root hyperplanes ⓘ
relatedTo	Weyl character formula NERFINISHED ⓘ Weyl integration formula NERFINISHED ⓘ Weyl numerator NERFINISHED ⓘ Weyl vector NERFINISHED ⓘ discriminant of a root system ⓘ
roleIn	Harish-Chandra’s formulae for characters NERFINISHED ⓘ Plancherel formula for compact Lie groups NERFINISHED ⓘ denominator of the Weyl character formula ⓘ description of characters of irreducible highest-weight representations ⓘ harmonic analysis on compact Lie groups ⓘ
specialCase	Vandermonde determinant for type A root systems ⓘ sin-product formula for SU(2) ⓘ
symbol	Δ ⓘ δ (in some conventions) ⓘ
usedFor	computing characters of finite-dimensional irreducible representations ⓘ computing weight multiplicities via Weyl character formula ⓘ expressing partition functions in some conformal field theories ⓘ formulating Kac–Weyl character formula in affine Kac–Moody theory ⓘ formulating Weyl’s dimension formula ⓘ
uses	Weyl group NERFINISHED ⓘ half-sum of positive roots ⓘ root system ⓘ set of positive roots ⓘ

Referenced by (1)

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

Weyl character formula → usesConcept → Weyl denominator ⓘ