Matrix Schubert varieties, binomial ideals, and reduced Gröbner bases

Abstract:

We prove a sharp lower bound on the number of terms in an element of the reduced Gröbner basis of a Schubert determinantal ideal $I_w$ under the term order of Knutson–Miller [Ann. of Math. (2) 161 (2005), pp. 1245–1318]. We give three applications. First, we give a pattern-avoidance characterization of the matrix Schubert varieties whose defining ideals are binomial. This complements a result of Escobar–Mészáros [Proc. Amer. Math. Soc. 144 (2016), pp. 5081–5096] on matrix Schubert varieties that are toric with respect to their natural torus action. Second, we give a combinatorial proof that the recent formulas of Rajchgot–Robichaux–Weigandt [J. Algebra 617 (2023), pp. 160–191] and Almousa–Dochtermann–Smith [Preprint, arXiv:2209.09851, 2022] computing the Castelnuovo–Mumford regularity of vexillary $I_w$ and toric edge ideals of bipartite graphs respectively agree for binomial $I_w$. Third, we demonstrate that the Gröbner basis for $I_w$ given by the minimal generators of Gao–Yong [J. Commut. Algebra 16 (2024), pp. 267–273] is reduced if and only if the defining permutation $w$ is vexillary.