Exceptional sequences in semidistributive lattices and the poset topology of wide subcategories
Bibliographic record
Abstract
Let [Formula: see text] be a finite-dimensional algebra over a field. We describe how Buan and Marsh’s [Formula: see text]-exceptional sequences can be used to give a “brick labeling” of a certain poset of wide subcategories of finitely generated [Formula: see text]-modules. When [Formula: see text] is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn toward the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with a bijection between their completely join-irreducible and completely meet-irreducible elements, known as rowmotion or the “[Formula: see text]-map.” Generalizing known results for finite semidistributive lattices, we show that the [Formula: see text]-map determines exactly when a set of completely join-irreducible elements forms a “canonical join representation.” A consequence is that the corresponding “canonical join complex” is a flag simplicial complex, as has been shown for finite semidistributive lattices and lattices of torsion classes. Finally, we demonstrate how Jasso’s [Formula: see text]-tilting reduction of finite-dimensional algebras can be encoded using the [Formula: see text]-map. We use this to define [Formula: see text]-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to [Formula: see text]-exceptional sequences in the algebra setting.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".