Abstract
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.
| Original language | English (US) |
|---|---|
| Article number | #47 |
| Journal | Seminaire Lotharingien de Combinatoire |
| Issue number | 86 |
| State | Published - 2022 |
| Externally published | Yes |
Bibliographical note
Funding Information:∗oliver.pechenik@uwaterloo.ca. Partially supported by a Postdoctoral Fellowship (#1703696) from the NSF, as well as by a Discovery Grant (RGPIN-2021-02391) and Launch Supplement (DGECR-2021-00010) from NSERC. †speyer@umich.edu. Supported in part by National Science Foundation grants DMS-1600223, DMS-1855135 and DMS-1854225. ‡weigandt@mit.edu. Partially supported by Bill Fulton’s Oscar Zariski Distinguished Professor Chair funds.
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