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.
![prod](/content/JG74474328335J73/xxlarge8719.gif)
![prod](/content/JG74474328335J73/xxlarge8719.gif)
![Lambda](/content/JG74474328335J73/xxlarge923.gif)
![Lambda](/content/JG74474328335J73/xxlarge923.gif)
![sgr](/content/JG74474328335J73/xxlarge963.gif)
![sgr](/content/JG74474328335J73/xxlarge963.gif)
![Lambda](/content/JG74474328335J73/xxlarge923.gif)
![sgr](/content/JG74474328335J73/xxlarge963.gif)
![sgr](/content/JG74474328335J73/xxlarge963.gif)
![rarr](/content/JG74474328335J73/xxlarge8594.gif)
![prod](/content/JG74474328335J73/xxlarge8719.gif)
![le](/content/JG74474328335J73/xxlarge8804.gif)
![ldquo](/content/JG74474328335J73/xxlarge8220.gif)
![rdquo](/content/JG74474328335J73/xxlarge8221.gif)
Pages: 111–121
Keywords: matching field; Newton polytope; maximal minor
Full Text: PDF
References
1. B. Sturmfels and A. Zelevinsky, "Maximal minors and their leading terms," Advances in Math, 98 (1993), 65-112.