Ideals Generated by Principal Minors.

Abstract

Let X be a square matrix of indeterminates. Let K[X] denote the polynomial ring in those indeterminates over an algebraically closed field, K. A minor is principal means it is defined by the same row and column indices. We prove various statements about ideals generated by principal minors of a fixed size t. When t=2 the resulting quotient ring is a normal complete intersection domain. When t>2 we break the problem into cases by intersecting with the locally closed variety of rank r matrices. We show when r=n for any t, there is a K-automorphism of that maps the ideal generated by size t principal minors to the ideal generated by size n-t principal minors, inducing an isomorphism on the respectively defined schemes. When t=r we factor a matrix in the algebraic set as the product of its row space matrix, an invertible size t matrix, and its column space matrix. We show that for the analysis of components it is enough to consider irreducible algebraic sets in the product of two Grassmannians, Grass(t,n). For t=r we also observe the connection between such decompositions and matroid theory.