Castelnuovo-Mumford regularity of matrix Schubert varieties

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.

Minimal equations for matrix Schubert varieties

“…Since [2], there has been interest in matrix Schubert varieties and the Schubert determinantal ideals; see, e.g., [7,8,4,1,3,5,6,10,9] and references therein.…”

Vexillary Grothendieck Polynomials via Bumpless Pipe Dreams

“….. We conjecture the following generalization of Theorem 1.1 for Grothendieck polynomials corresponding to all permutations. Note that this would yield an alternative proof of Pechenik, Speyer, and Weigandt's result for the degree of the Grothendieck polynomial [8].…”

“…Note that it is always true that r i ≥ c i . Pechenik, Speyer, and Weigandt proved the following result [8].…”

On the support of Grothendieck polynomials

“…The main objective of this paper is to shed light on the combinatorial structure of the support of Grothendieck polynomials. While there have been recent breakthroughs in the degree of Grothendieck polynomials [24,28,29], much less is known about the structure of the support. The support has previously been conjecturally connected to generalized permutahedra via flow polytopes [25,Conjecture 5.1], and via the Lorentzian property [15,Conjecture 22].…”

“…[28, Definition 3.5]). A permutation w ∈ S n is called fireworks if the initial elements of its decreasing runs occur in increasing order.…”