Hecke algebra

E911207

associative algebra deformation of group algebra object in algebraic combinatorics object in quantum algebra object in representation theory

A Hecke algebra is a deformation of a group algebra (often of a Coxeter or Weyl group) that plays a central role in representation theory, algebraic combinatorics, and the study of quantum groups and integrable systems.

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Observed surface forms (1)

Surface form	Occurrences
Hecke algebras	1

Statements (49)

Predicate	Object
instanceOf	associative algebra ⓘ deformation of group algebra ⓘ object in algebraic combinatorics ⓘ object in quantum algebra ⓘ object in representation theory ⓘ
actsOn	cohomology of arithmetic varieties ⓘ spaces of modular forms ⓘ
definedOver	commutative ring ⓘ field ⓘ
fieldOfStudy	algebraic combinatorics ⓘ geometric representation theory ⓘ harmonic analysis ⓘ number theory ⓘ representation theory ⓘ
generalizes	group algebra ⓘ
hasBasis	elements indexed by a Coxeter group ⓘ
hasParameter	deformation parameter ⓘ q ⓘ
hasVariant	0-Hecke algebra NERFINISHED ⓘ Iwahori–Hecke algebra NERFINISHED ⓘ affine Hecke algebra ⓘ cyclotomic Hecke algebra ⓘ double affine Hecke algebra NERFINISHED ⓘ
namedAfter	Erich Hecke NERFINISHED ⓘ
relatedTo	Coxeter group NERFINISHED ⓘ Hecke operator NERFINISHED ⓘ Iwahori–Hecke algebra NERFINISHED ⓘ Kazhdan–Lusztig polynomial NERFINISHED ⓘ Weyl group NERFINISHED ⓘ Young tableau ⓘ affine Hecke algebra NERFINISHED ⓘ affine Weyl group NERFINISHED ⓘ automorphic form ⓘ double affine Hecke algebra NERFINISHED ⓘ modular form ⓘ p-adic group ⓘ quantum group ⓘ symmetric group ⓘ
satisfiesRelation	braid relations ⓘ quadratic relation depending on q ⓘ
specializationCondition	q = 1 ⓘ
specializesTo	group algebra ⓘ
usedIn	Schur–Weyl duality NERFINISHED ⓘ classification of representations of p-adic groups ⓘ construction of Kazhdan–Lusztig theory ⓘ integrable systems ⓘ knot invariants ⓘ study of canonical bases ⓘ study of quantum groups ⓘ

Referenced by (2)

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

Yang–Baxter equation → relatedTo → Hecke algebra ⓘ

Deligne–Lusztig theory → studies → Hecke algebra ⓘ

this entity surface form: Hecke algebras