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Bibliographic record
Abstract
Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> , there is a limit crystal, usually denoted by B (−∞), which contains all the other crystals. When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B (−∞). This polytope sits in the dual space of a Cartan subalgebra of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> , and its edges are parallel to the roots of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> . In this paper, we generalize this construction to the case where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ . We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B (−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> . The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mtext>-</mml:mtext> <mml:mi>mod</mml:mi> </mml:mrow> </mml:math> . The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ .
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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