Combinatorics of Fulton's Essential Set

Fulton introduced the essential set of a permutation, together with a rank function. In this paper we study some combinatorial aspects of the essential set: We present an algorithm that retrieves a permutation from its ranked essential set. Thereby we can characterize the class of ranked sets that arise as ranked essential sets of permutations, answering a question of Fulton. Several classes of permutations are characterized in terms of their essential set. Results in higher dimensions are discussed. 1. Introduction An n by n permutation matrix can be represented by an n by n array of squares with one dot in each row and column and all other squares empty. The diagram of a permutation matrix (deened in 1800 by Rothe) is obtained by shading every row from the dot and eastwards and shading every column from the dot and southwards. The essential set is the set of southeast corners of the connected components of the diagram. The essential set, together with a rank function, was introduced by Fulton 7] in a pioneering paper from 1992, in order to study the irreducible loci in spaces of matrices. Though in the present paper we are not going to pursue this geometric issue much but rather concentrate on the essential set per se, we need a short recollection of the background. Fulton studies varieties given by ideals of minors, subject to certain rank conditions , in a generic matrix. One concern is, given a prescribed rank function r(i; j), to determine if there exist matrices such that the rank of the upper left i by j subma-trices is r(i; j) for every position (i; j). He observes that if such matrices exist, then there is in particular some permutation matrix with this property, so it is enough to consider permutation matrices. More generally, he is interested in prescribing the ranks only for a few of the submatrices, such that all the other ranks follow from these. In other words, nd a subset of the upper left submatrices of a permutation matrix such that the permutation is uniquely determined by their ranks. Fulton