Triple

T5211990
Position Surface form Disambiguated ID Type / Status
Subject Weyl group E117654 entity
Predicate usedIn P98 FINISHED
Object 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.
E503515 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Kazhdan–Lusztig theory | Statement: [Weyl group, usedIn, Kazhdan–Lusztig theory]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Kazhdan–Lusztig theory
Context triple: [Weyl group, usedIn, Kazhdan–Lusztig theory]
  • A. Deligne–Lusztig theory
    Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
  • B. 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.
  • C. Gelfand–Tsetlin basis
    The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
  • D. 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.
  • E. Representation Theory and Automorphic Functions
    "Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Kazhdan–Lusztig theory
Triple: [Weyl group, usedIn, Kazhdan–Lusztig theory]
Generated description
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.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Kazhdan–Lusztig theory
Target entity description: 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.
  • A. Deligne–Lusztig theory
    Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
  • B. 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.
  • C. Gelfand–Tsetlin basis
    The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
  • D. 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.
  • E. Representation Theory and Automorphic Functions
    "Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69bd4464ba3c8190bc16b2ebbe42ddb0 completed March 20, 2026, 12:58 p.m.
NER Named-entity recognition batch_69bd7a7166848190805152142e184529 completed March 20, 2026, 4:48 p.m.
NED1 Entity disambiguation (via context triple) batch_69beefd9e0b481908db7d6e2907b3b2b completed March 21, 2026, 7:22 p.m.
NEDg Description generation batch_69bef0b1fe9c8190bfc1be621c7c1c76 completed March 21, 2026, 7:25 p.m.
NED2 Entity disambiguation (via description) batch_69bef16739148190b9700228be7d07f9 completed March 21, 2026, 7:28 p.m.
Created at: March 20, 2026, 1:47 p.m.