THE RENNER MONOIDS AND CELL DECOMPOSITIONS OF THE SYMPLECTIC ALGEBRAIC MONOIDS
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Bibliographic record
Abstract
In this paper we explicitly determine the Renner monoid ℛ and the cross section lattice Λ of the symplectic algebraic monoid MSp n in terms of the Weyl group and the concept of admissible sets; it turns out that ℛ is a submonoid of ℛ n , the Renner monoid of the whole matrix monoid M n , and that Λ is a sublattice of Λ n , the cross section lattice of M n . Cell decompositions in algebraic geometry are usually obtained by the method of [1]. We give a more direct definition of cells for MSp n in terms of the B × B-orbits, where B is a Borel subgroup of the unit group G of MSp n . Each cell turns out to be the intersection of MSp n with a cell of M n . We also show how to obtain these cells using a carefully chosen one parameter subgroup.
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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