Combinatorics of maximal minors

Bernstein, David ; Zelevinsky, Andrei

Journal of Algebraic Combinatorics, Tome 2 (1993) no. 2, pp. 111-121.

Résumé

Summary: We continue the study of the Newton polytope $prod_{ m, n }$ of the product of all maximal minors of an $m \times $n-matrix of indeterminates. The vertices of $prod_{ m, n }$ are encoded by coherent matching fields $Lambda = ( Lambda _$ sgr), where $sgr$ runs over all $m$-element subsets of columns, and each $Lambda _$ sgr is a bijection $sgrrarr$ [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 $prod_{ m, n }$ for $mlen$ by means of $ldquo$generalized permutations $rdquo$, which are bijections between $( n - m + 1)$-element subsets of columns and $( n - m + 1)$-element submultisets of rows.

Détail

Keywords: matching field, Newton polytope, maximal minor