A Positive Monk Formula in the S<sup>1</sup>-Equivariant Cohomology of Type A Peterson Varieties
Bibliographic record
Abstract
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type An−1, with respect to a natural S1 -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac-Moody flag varieties G/P.\nOur main results are as follows. First, we identify a computationally convenient basis of H∗S1(Y ), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley-Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both H∗S1(Y ) and H ∗ (Y ) are generated in degree 2. Finally, by using our Chevalley-Monk formula we give explicit descriptions (via generators and relations) of both the S1 -equivariant cohomology ring H∗S1(Y ) and the ordinary cohomology ring H ∗ (Y ) of the type An−1 Peterson variety. Our methods are both directly from and inspired by those of GKM (Goresky-Kottwitz-MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.
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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 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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.
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