A Gr\"obner Basis for Schubert Patch Ideals

TL;DR

This paper proves that the essential minors of Schubert patch ideals form a Gr"obner basis, confirming a conjecture, and shows these ideals are glicci, advancing understanding of their algebraic structure.

Contribution

It adapts linkage-theoretic methods to prove the Gr"obner basis property for Schubert patch ideals and related ideals, confirming conjectures and extending known results.

Findings

Essential minors of Schubert patch ideals form a Gr"obner basis.

Homogeneous Schubert patch and Kazhdan-Lusztig ideals are glicci.

The approach recovers and extends previous results on determinantal ideals.

Abstract

Schubert patch ideals are a class of generalized determinantal ideals. They are prime defining ideals of open patches of Schubert varieties in the type $A$ flag variety. In this paper, we adapt the linkage-theoretic approach of E. Gorla, J. Migliore, and U. Nagel to prove a conjecture of A. Yong, namely, that the essential minors of every Schubert patch ideal form a Gr\"{o}bner basis. Using the same approach, we recover the result of A. Woo and A. Yong that the essential minors of a Kazhdan-Lusztig ideal form a Gr\"{o}bner basis. With respect to the standard grading of assigning degree 1 to each variable, we also show that homogeneous Schubert patch ideals and homogeneous Kazhdan-Lusztig ideals (and hence, Schubert determinantal ideals) are glicci.