Ideals Generated by Principal Minors

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 $t\le n$, 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.