Ideals Generated by Principal Minors.

Principal Minors;Determinantal Ideals;Generic Matrices;Matroid and Positroid Varieties;Commutative Algebra;Algebraic Geometry;Mathematics;Science;Mathematics

Wheeler, Ashley K.Derksen, Harm ;

University of Michigan

关键词: Principal Minors;    Determinantal Ideals;    Generic Matrices;    Matroid and Positroid Varieties;    Commutative Algebra;    Algebraic Geometry;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/108987/wheeles_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.