We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches ofarbitrary Grassmannian Schubert varieties.
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This content will become publicly available on July 3, 2025
Castelnuovo-Mumford regularity of matrix Schubert varieties
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order.
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- NSF-PAR ID:
- 10521261
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Selecta Mathematica
- Volume:
- 30
- ISSN:
- 1022-1824
- Page Range / eLocation ID:
- 66
- Subject(s) / Keyword(s):
- Castelnuovo–Mumford regularity Matrix Schubert variety Grothendieck polynomial Major index Weak Bruhat order Rajchgot index
- Format(s):
- Medium: X Other: pdf
- Sponsoring Org:
- National Science Foundation
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