Triple

T11098937
Position Surface form Disambiguated ID Type / Status
Subject Clebsch–Aronhold invariants E262451 entity
Predicate relatedTo P37 FINISHED
Object projective linear group PGL(2)
The projective linear group PGL(2) is the group of Möbius (fractional linear) transformations acting on the projective line, fundamental in projective geometry, complex analysis, and invariant theory.
E898490 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: projective linear group PGL(2) | Statement: [Clebsch–Aronhold invariants, relatedTo, projective linear group PGL(2)]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: projective linear group PGL(2)
Context triple: [Clebsch–Aronhold invariants, relatedTo, projective linear group PGL(2)]
  • A. PSL(2,7)
    PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
  • B. modular group PSL(2,Z)
    The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
  • C. PSL(2,ℤ/Nℤ)
    PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
  • D. PSL(2,\mathbb{C})
    PSL(2,ℂ) is the group of Möbius transformations acting as all biholomorphic automorphisms of the Riemann sphere.
  • E. general linear group GL(n,C)
    The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
  • 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: projective linear group PGL(2)
Triple: [Clebsch–Aronhold invariants, relatedTo, projective linear group PGL(2)]
Generated description
The projective linear group PGL(2) is the group of Möbius (fractional linear) transformations acting on the projective line, fundamental in projective geometry, complex analysis, and invariant theory.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: projective linear group PGL(2)
Target entity description: The projective linear group PGL(2) is the group of Möbius (fractional linear) transformations acting on the projective line, fundamental in projective geometry, complex analysis, and invariant theory.
  • A. PSL(2,7)
    PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
  • B. modular group PSL(2,Z)
    The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
  • C. PSL(2,ℤ/Nℤ)
    PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
  • D. PSL(2,\mathbb{C}) chosen
    PSL(2,ℂ) is the group of Möbius transformations acting as all biholomorphic automorphisms of the Riemann sphere.
  • E. general linear group GL(n,C)
    The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
  • F. None of above.

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_69d6aa9a40d88190a373e2c7e48285db completed April 8, 2026, 7:20 p.m.
NER Named-entity recognition batch_69d79a0c46308190889b94c23ebaca62 completed April 9, 2026, 12:22 p.m.
NED1 Entity disambiguation (via context triple) batch_69e3e7eca9bc8190b43bae081d97d804 completed April 18, 2026, 8:22 p.m.
NEDg Description generation batch_69e3f2cbb4708190a328cff473104d14 completed April 18, 2026, 9:08 p.m.
NED2 Entity disambiguation (via description) batch_69e3f497a01881909d1dae70a02e5f97 completed April 18, 2026, 9:16 p.m.
Created at: April 8, 2026, 9:27 p.m.