Combinatorics of Maximal Minors

We continue the study of the Newton polytope Pm,n of the product of all maximal minors of an m x n-matrix of indeterminates. The vertices of Pm,n are encoded by coherent matching fields A = (A z) , where z runs over all m-element subsets of columns, and each Az is a bijection z —> [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 Grobner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of Pm,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.