Geometric vertex decomposition, Gröbner bases, and Frobenius splittings for regular nilpotent Hessenberg varieties
Bibliographic record
Abstract
We initiate a study of the Gröbner geometry of local defining ideals of Hessenberg varieties by studying the special case of regular nilpotent Hessenberg varieties in Lie type A, and focusing on the affine coordinate chart on $\mathrm{Flags}(\mathbb{C}^n) \cong GL_n(\mathbb{C})/B$ corresponding to the longest element $w_0$ of the Weyl group $S_n$ of $GL_n(\mathbb{C})$. Our main results are as follows. Let $h$ be an indecomposable Hessenberg function. We prove that the local defining ideal $I_{w_0,h}$ in the $w_0$-chart of the regular nilpotent Hessenberg variety $\mathrm{Hess}(\mathsf{N},h)$ associated to $h$ has a Gröbner basis with respect to a suitably chosen monomial order. Our Gröbner basis consists of a collection $\{f^{w_0}_{k,\ell}\}$ of generators of $I_{w_0,h}$ obtained by Abe, DeDieu, Galetto, and the second author. We also prove that $I_{w_0,h}$ is geometrically vertex decomposable in the sense of Klein and Rajchgot (building on work of Knutson, Miller, and Yong). We give two distinct proofs of the above results. We make this unconventional choice of exposition because our first proof introduces and utilizes a notion of a triangular complete intersection which is of independent interest, while our second proof using liaison theory is more likely to be generalizable to the general $w$-charts for $w \neq w_0$. Finally, using our Gröbner analysis of the $f^{w_0}_{k,\ell}$ above and for $p>0$ any prime, we construct an explicit Frobenius splitting of the $w_0$-chart of $\mathrm{Flags}(\mathbb{C}^n)$ which simultaneously compatibly splits all the local defining ideals of $I_{w_0,h}$, as $h$ ranges over the set of indecomposable Hessenberg functions. This last result is a local Hessenberg analogue of a classical result known for $\mathrm{Flags}(\mathbb{C}^n)$ and the collection of Schubert and opposite Schubert varieties in $\mathrm{Flags}(\mathbb{C}^n)$.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".