Jordan degree type for codimension three Gorenstein algebras of small Sperner number
Bibliographic record
Abstract
The Jordan type P A , ℓ of a linear form ℓ acting on a graded Artinian algebra A over a field k is the partition describing the Jordan block decomposition of the multiplication map m ℓ , which is nilpotent. The Jordan degree type S A , ℓ is a finer invariant, describing also the initial degrees of the simple submodules of A in a decomposition of A as a direct sum of k [ ℓ ] -modules. The set of Jordan types of A or Jordan degree types (JDT) of A as ℓ varies, is an invariant of the algebra. This invariant has been studied for codimension two graded algebras. We here extend the previous results to certain codimension three graded Artinian Gorenstein (AG) algebras - those of small Sperner number. Given a Gorenstein sequence T - one possible for the Hilbert function of a codimension three graded AG algebra - the irreducible variety Gor ( T ) parametrizes all Gorenstein algebras of Hilbert function T . We here completely determine the JDT possible for all pairs ( A , ℓ ) , A ∈ Gor ( T ) , for Gorenstein sequences T of the form T = ( 1 , 3 , s k , 3 , 1 ) for Sperner number s = 3 , 4 , 5 and arbitrary multiplicity k . For s = 6 we delimit the prospective JDT, without verifying that each occurs.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".