Exceptional sequences in semidistributive lattices and the poset topology of wide subcategories
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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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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 | 0.000 |
| 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 it