The Lattice property of biclosed sets for infinite root systems
Bibliographic record
Abstract
A root system is a finite collection of vectors in Euclidean space satisfying certain symmetry conditions. Work by Killing and Cartan in the late 19th century showed that these come in seven families, and uses this fact to classify the semisimple Lie algebras. Root systems play an important role in many fields, including the representation theory of algebras, algebraic geometry, and particle physics. Relaxing the finiteness assumption in the definition of root system leads to the more general notion of a Kac–Moody root system. In the finite setting, root systems can be studied via their inversion sets, which are objects carrying geometric information. This does not carry over to Kac-Moody root systems. However, the inversion sets are completely characterised by two closure properties, making them so-called biclosed sets. Following work of Barkley and Speyer, we investigate biclosed sets as a generalization of inversion sets to infinite root systems, especially a family called the hyperbolic root systems. Their result that these sets form a lattice in affine type is found not to hold when the sets are allowed to contain imaginary roots, and the related “2-out-of-3 rule” also fails without this restriction. We explored distinctions between two different notions of closure used to define a biclosed set. Several interesting questions arose from these areas.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.008 |
| Scholarly communication | 0.005 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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".