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.