Gr\"obner geometry for skew-symmetric matrix Schubert varieties

TL;DR

This paper extends the study of matrix Schubert varieties to skew-symmetric cases, providing Gr"obner bases, primary decompositions, and shellable simplicial complexes, with applications to symplectic Grothendieck polynomials.

Contribution

It introduces skew-symmetric matrix Schubert varieties, describes their prime ideals, computes Gr"obner bases, and links initial ideals to shellable complexes, expanding geometric and algebraic understanding.

Findings

Prime ideals generated by natural sets for skew-symmetric varieties

Computed Gr"obner bases for these ideals

Identified initial ideals as Stanley-Reisner ideals of shellable complexes

Abstract

Matrix Schubert varieties are the closures of the orbits of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gr\"obner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gr\"obner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from Knutson and Miller's and can be used to give new proofs of some of their results, as we explain at the end of this article.