Matroids on the Bases of Simple Matroids

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