Bibliographic record
Abstract
Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra 𝔤 has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given 𝔤 , there is a limit crystal, usually denoted by B (−∞), which contains all the other crystals. When 𝔤 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 𝔤 , and its edges are parallel to the roots of 𝔤 . In this paper, we generalize this construction to the case where 𝔤 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 𝔤 . The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category Λ - mod . The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".