On the Reduced Gr\"obner Bases of Blockwise Determinantal Ideals

TL;DR

This paper investigates the structure of reduced Gr"obner bases for blockwise determinantal ideals, providing criteria for minimality and explicit formulas, especially for Schubert determinantal ideals, enhancing understanding of their algebraic properties.

Contribution

It establishes criteria for reduced Gr"obner bases of blockwise determinantal ideals and derives explicit formulas for Schubert cases, advancing algebraic understanding of these ideals.

Findings

Reduced Gr"obner bases criteria for blockwise determinantal ideals.

Explicit formula for non-vexillary Schubert determinantal ideals.

Proven properties of normality and strength for associated characteristic sets.

Abstract

Blockwise determinantal ideals are those generated by the union of all the minors of specified sizes in certain blocks of a generic matrix, and they are the natural generalization of many existing determinantal ideals like the Schubert and ladder ones. In this paper we establish several criteria to verify whether the Gr\"obner bases of blockwise determinantal ideals with respect to (anti-)diagonal term orders are minimal or reduced. In particular, for Schubert determinantal ideals, while all the elusive minors form the reduced Gr\"obner bases when the defining permutations are vexillary, in the non-vexillary case we derive an explicit formula for computing the reduced Gr\"obner basis from elusive minors which avoids all algebraic operations. The fundamental properties of being normal and strong for W-characteristic sets and characteristic pairs, which are heavily connected to the reduced Gr\"obner bases, of Schubert determinantal ideals are also proven.