Triple

T10550238
Position Surface form Disambiguated ID Type / Status
Subject Robert Langlands E248927 entity
Predicate notableIdea P4 FINISHED
Object Langlands dual group
The Langlands dual group is an algebraic group constructed from a given reductive group by interchanging its root and coroot data, playing a central role in the Langlands program’s connections between number theory and representation theory.
E870220 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: Langlands dual group | Statement: [Robert Langlands, notableIdea, Langlands dual group]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Langlands dual group
Context triple: [Robert Langlands, notableIdea, Langlands dual group]
  • A. Weyl group
    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.
  • B. 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.
  • C. Weil–Deligne group
    The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
  • D. Weil representation
    The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
  • E. 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.
  • 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: Langlands dual group
Triple: [Robert Langlands, notableIdea, Langlands dual group]
Generated description
The Langlands dual group is an algebraic group constructed from a given reductive group by interchanging its root and coroot data, playing a central role in the Langlands program’s connections between number theory and representation theory.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Langlands dual group
Target entity description: The Langlands dual group is an algebraic group constructed from a given reductive group by interchanging its root and coroot data, playing a central role in the Langlands program’s connections between number theory and representation theory.
  • A. Weyl group
    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.
  • B. 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.
  • C. Weil–Deligne group
    The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
  • D. Weil representation
    The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
  • E. 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.
  • 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_69d381c733c08190ab1dd6239f5f34ae completed April 6, 2026, 9:49 a.m.
NER Named-entity recognition batch_69d526d3e45c819099b360f9cfd3dd50 completed April 7, 2026, 3:46 p.m.
NED1 Entity disambiguation (via context triple) batch_69d934639b3481908204db41101132c3 completed April 10, 2026, 5:33 p.m.
NEDg Description generation batch_69d938c8b25c8190bb048053d8668e5c completed April 10, 2026, 5:52 p.m.
NED2 Entity disambiguation (via description) batch_69d939b1844881908c8fbcb9488863f6 completed April 10, 2026, 5:56 p.m.
Created at: April 6, 2026, 12:34 p.m.