On graded presentations of Hecke algebras and their generalizations
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Bibliographic record
Abstract
In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the “Hecke family”; on the other, we find generalizations of KLR algebras in finite and affine type A, the “KLR family.” We show that these algebras have compatible isomorphisms generalizing those between Hecke and KLR algebras given by Brundan and Kleshchev. This allows us to organize a long list of algebras and categories into a single system, including (affine/cyclotomic) Hecke algebras, (affine/cyclotomic) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -Schur algebras, (weighted) KLR algebras, category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒪</mml:mi> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔤𝔩</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> and for the Cherednik algebras for the groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℤ</mml:mi> <mml:mo>/</mml:mo> <mml:mi>e</mml:mi> <mml:mi>ℤ</mml:mi> <mml:mo>≀</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , and give graded presentations of all of these objects.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it