On the representation theory of Schur algebras in type B
Bibliographic record
Abstract
We study the representation theory of the type B Schur algebra L n ( m ) with unequal parameters introduced by Lai and Luo. For generic values of ( Q , q ) , this algebra is semi-simple and Morita equivalent to the type B Hecke algebra, but for special values, its category of modules is more complicated. We study this representation theory by comparison with the cyclotomic q -Schur algebra of Dipper, James, and Mathas, and use this to construct a cellular algebra structure on L n ( m ) . This allows us to index the simple L n ( m ) -modules as a subset of the set of bipartitions of n . For m large, this will be all bipartitions of n if and only if L n ( m ) is quasi-hereditary. In this case, the algebra L n ( m ) is Morita equivalent to the cyclotomic q -Schur algebra. We prove a modified version of a conjecture of Lai, Nakano, and Xiang giving the values of ( Q , q ) where this holds: if m is large and odd, Q ≠ − q k for all k satisfying 4 − n 2 ≤ k < n ; if m is large and even, Q ≠ − q k for all k satisfying − n < k < n . We also prove two strengthenings of this result: an indexing of the simple modules when q is not a root of unity, and a characterization of the quasi-hereditary blocks of L n ( m ) .
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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 | 0.002 |
| 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".