Generating the Ideals Defining Unions of Schubert Varieties

TL;DR

This paper computes a Gr"obner basis for the ideal defining unions of Schubert varieties, including determinantal and matrix Schubert varieties, using northwest rank conditions on matrices.

Contribution

It provides a method to explicitly compute Gr"obner bases for unions of schemes defined by northwest rank conditions, extending understanding of these algebraic varieties.

Findings

Computed Gr"obner basis for unions of Schubert varieties

Unified approach for determinantal and matrix Schubert varieties

Enhanced tools for algebraic and geometric analysis of these schemes

Abstract

This note computes a Gr\"obner basis for the ideal defining a union of Schubert varieties. More precisely, it computes a Gr\"obner basis for unions of schemes given by northwest rank conditions on the space of all matrices of a fixed size. Schemes given by northwest rank conditions include classical determinantal varieties and matrix Schubert varieties--closures of Schubert varieties lifted from the flag manifold to the space of matrices.