Products of Borel fixed ideals of maximal minors

TL;DR

This paper investigates products of Borel fixed ideals of maximal minors, computing their initial ideals, primary decompositions, and demonstrating they have linear free resolutions, with implications for their algebraic and homological properties.

Contribution

It extends straightening law techniques and introduces a surprising primary decomposition formula for these ideals, advancing understanding of their algebraic structure.

Findings

Initial ideals and primary decompositions computed

Proved these ideals have linear free resolutions

Associated multi-Rees algebra is Cohen-Macaulay and Koszul

Abstract

We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very surprising primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen-Macaulay, Koszul and defined by a Gr\"obner basis of quadrics.