Combinatorics of Maximal Minors
David Bernstein
and Andrei Zelevinsky
DOI: 10.1023/A:1022492222930
Abstract
We continue the study of the Newton polytope 
m, n of the product of all maximal minors of an m \times n-matrix of indeterminates. The vertices of m, n are encoded by coherent matching fields 
= ( 
), where runs over all m-element subsets of columns, and each is a bijection 
[m]. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid theory. Their analysis implies that maximal minors form a universal Gröbner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of m, n for m 
n by means of 
generalized permutations 
, which are bijections between ( n - m + 1)-element subsets of columns and ( n - m + 1)-element submultisets of rows.
m, n of the product of all maximal minors of an m \times n-matrix of indeterminates. The vertices of m, n are encoded by coherent matching fields = ( ), where runs over all m-element subsets of columns, and each is a bijection [m]. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid theory. Their analysis implies that maximal minors form a universal Gröbner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of m, n for m n by means of generalized permutations , which are bijections between ( n - m + 1)-element subsets of columns and ( n - m + 1)-element submultisets of rows.Pages: 111–121
Keywords: matching field; Newton polytope; maximal minor
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References
1. B. Sturmfels and A. Zelevinsky, "Maximal minors and their leading terms," Advances in Math, 98 (1993), 65-112.