Shi arrangements and low elements in Coxeter groups
Bibliographic record
Abstract
Abstract Given an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".