Matrix Schubert varieties, binomial ideals, and reduced Gr\"obner bases

We prove a sharp lower bound on the number of terms in an element of the reduced Gr\"obner basis of a Schubert determinantal ideal $I_w$ under the term order of [Knutson-Miller '05]. 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-M\'esz\'aros '16] 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 '23] and [Almousa-Dochtermann-Smith '22] 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 Gr\"obner basis for $I_w$ given by minimal generators [Gao-Yong '22] is reduced if and only if the defining permutation $w$ is vexillary.