Triple
T10641406
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Harish-Chandra isomorphism |
E250729
|
entity |
| Predicate | involves |
P1256
|
FINISHED |
| Object |
Harish-Chandra projection
The Harish-Chandra projection is a linear map from a universal enveloping algebra onto the symmetric algebra of a Cartan subalgebra that plays a central role in describing the center via the Harish-Chandra isomorphism.
|
E876153
|
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: Harish-Chandra projection | Statement: [Harish-Chandra isomorphism, involves, Harish-Chandra projection]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Harish-Chandra projection Context triple: [Harish-Chandra isomorphism, involves, Harish-Chandra projection]
-
A.
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.
-
B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
C.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
-
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.
Borel–Weil theorem
The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
- 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: Harish-Chandra projection Triple: [Harish-Chandra isomorphism, involves, Harish-Chandra projection]
Generated description
The Harish-Chandra projection is a linear map from a universal enveloping algebra onto the symmetric algebra of a Cartan subalgebra that plays a central role in describing the center via the Harish-Chandra isomorphism.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Harish-Chandra projection Target entity description: The Harish-Chandra projection is a linear map from a universal enveloping algebra onto the symmetric algebra of a Cartan subalgebra that plays a central role in describing the center via the Harish-Chandra isomorphism.
-
A.
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.
-
B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
C.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
-
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.
Borel–Weil theorem
The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
- 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_69d6aa5a4c4881908f39be6efe5981e5 |
completed | April 8, 2026, 7:19 p.m. |
| NER | Named-entity recognition | batch_69d6dfcd19648190882380d2c90be486 |
completed | April 8, 2026, 11:07 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69d96bcd8c0c8190a0fad6a85b5604bb |
completed | April 10, 2026, 9:29 p.m. |
| NEDg | Description generation | batch_69d9701de92881908c0b8f05eae97e35 |
completed | April 10, 2026, 9:48 p.m. |
| NED2 | Entity disambiguation (via description) | batch_69d970f609108190a499b542f0b10ee9 |
completed | April 10, 2026, 9:51 p.m. |
Created at: April 8, 2026, 9:05 p.m.