Multigraded generic initial ideals of determinantal ideals

TL;DR

This paper characterizes the multigraded generic initial ideals of determinantal ideals of minors of linear matrices, providing generators, prime decompositions, and Hilbert series formulas.

Contribution

It extends previous work by describing generators and prime decompositions of these ideals in terms of linear dependence data.

Findings

Generators and prime decompositions of gin(I) are explicitly described.

A closed formula for the multigraded Hilbert series of 2-minors is provided.

gin(I) is radical and largely independent of term order.

Abstract

Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.