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.
All labels observed (10)
| Label | Occurrences |
|---|---|
| Hecke algebras | 4 |
| Hecke algebra canonical | 3 |
| Hecke algebra of type A | 2 |
| Iwahori–Hecke algebra | 2 |
| double affine Hecke algebras | 2 |
| 0-Hecke algebra | 1 |
| Hecke algebra of GL(2) | 1 |
| Hecke algebra of GL(n) | 1 |
| affine Hecke algebra | 1 |
| affine Hecke algebras | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11205457 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hecke algebra Context triple: [Yang–Baxter equation, relatedTo, Hecke algebra]
-
A.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
-
D.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
-
E.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hecke algebra Target entity description: 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.
-
A.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
-
D.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
-
E.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
- F. None of above. chosen
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
ⓘ
linked to:
Hecke algebra
Iwahori–Hecke algebra ⓘ
linked to:
Hecke algebra
affine Hecke algebra ⓘ cyclotomic Hecke algebra ⓘ double affine Hecke algebra ⓘ |
| namedAfter | Erich Hecke ⓘ |
| relatedTo |
Coxeter group
ⓘ
Hecke operator ⓘ
linked to:
Hecke operators
Iwahori–Hecke algebra ⓘ
linked to:
Hecke algebra
Kazhdan–Lusztig polynomial ⓘ
linked to:
Kazhdan–Lusztig theory
Weyl group ⓘ Young tableau ⓘ affine Hecke algebra ⓘ
linked to:
Hecke algebra
affine Weyl group ⓘ automorphic form ⓘ double affine Hecke algebra ⓘ 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
ⓘ
classification of representations of p-adic groups ⓘ construction of Kazhdan–Lusztig theory ⓘ integrable systems ⓘ knot invariants ⓘ study of canonical bases ⓘ study of quantum groups ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hecke algebra Description of subject: 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.
Referenced by (18)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra
linked to: Hecke algebra