Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
Bibliographic record
Abstract
Let \(P\) be a graded poset of rank \(r\) and let \(\mathbf{c}\) be a \(c\)-element chain. A plane partition on \(P\) is an order ideal of \(P \times \mathbf{c}\). For an order ideal \(I\) of \(P \times \mathbf{c}\), its rowmotion \(\psi(I)\) is the smallest ideal containing the minimal elements of the complementary filter of \(I\). The map \(\psi\) defines invertible dynamics on the set of plane partitions. We say that \(P\) has NRP (`not relatively prime') rowmotion if no \(\psi\)-orbit has cardinality relatively prime to \(r+c+1\). In recent work, R. Patrias and the author (2020) proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product \(P = \mathbf{a} \times \mathbf{b}\) of two chains, the poset whose order ideals correspond to the Schubert varieties of a Grassmann variety \(\mathrm{Gr}_a(\mathbb{C}^{a+b})\) under containment. Here, we initiate the general study of posets with NRP rowmotion. Our first main result establishes NRP rowmotion for all minuscule posets \(P\), posets whose order ideals reflect the Schubert stratification of minuscule flag varieties. Our second main result is that NRP rowmotion depends only on the isomorphism class of the comparability graph of \(P\).Mathematics Subject Classifications: 05E18, 06A07, 06D99
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.011 | 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".