Matroids on the Bases of Simple Matroids

Abstract

Let M be a simple matroid (= combinatorial geometry). On the bases of M we consider two matroids S ( M, F ) and H ( M, F ), which depend on a field F. S ( M, F ) is the simplicial matroid with coefficients in F on the bases of M considered as simplices. H ( M, F ) has been studied by Björner in [1]. It is defined in terms of the order homology of the associated geometric lattice L ( M ). We prove that H ( M, F ) is a minor contraction of the full simplicial matroid on all subsets of elements of size r = r ( M ). Dually this is equivalent to an isomorphism H ( M, F )* ≃ S ( M *, F ), where M * denotes the dual of M . It can be deduced that H ( M, F ) need not be unimodular, a problem in [1], which inspired this study.