Ideals Generated by Principal Minors

TL;DR

This paper investigates the algebraic properties of ideals generated by principal minors of generic matrices, revealing structural isomorphisms and component decompositions, especially focusing on cases with fixed rank and specific matrix sizes.

Contribution

It introduces new isomorphisms between principal minor ideals in localized rings and analyzes the geometric structure of their algebraic sets, including component decomposition and linkage for specific cases.

Findings

For rank n, ideals of principal minors are isomorphic after localization.

The algebraic set defined by principal minors has a codimension n component.

In the case n=4, the components are linked, with implications for algebraic geometry.

Abstract

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$ denote the ideal generated by the size $t$ principal minors of $X$. When $t=2$ the resulting quotient ring $K[X]/\mathfrak P_2$ is a normal complete intersection domain. When $t>2$ we break the problem into cases depending on a fixed rank, $r$, of $X$. We show when $r=n$ for any $t$, the respective images of $\mathfrak P_t$ and $\mathfrak P_{n-t}$ in the localized polynomial ring, where we invert $\det X$, are isomorphic. From that we show the algebraic set given by $\mathfrak P_{n-1}$ has a codimension $n$ component, plus a codimension 4 component defined by the determinantal ideal (which is given by all the submaximal minors of $X$). When $n=4$ the two components are linked, and we prove some consequences.