Intersection Theory for Embeddings of Matroids into Uniform Geometries

Abstract

A uniform combinatorial geometry G is a finite geometric lattice which has the same number, WG(i,j), of flats of corank j in every upper interval of rank i. Examples of these geometries include affine and projective geometries, boolean algebras, and partition lattices as well as any homogeneous geometry (those in which flats of the same rank have equal cardinalities). Matrix equations relate the matroid structure M of a set S embedded into G to the intersection numbers of the embedding. In particular, if and PM(m,n) equals the number of m‐elements subsets of S with corank n, then T·PM·WG=IM→G, where IM→G(p,q) counts the flats in G of corank q which contain p points of S. The intersection matrix of an embedding of M or its dual into another uniform geometry G′ can be computed from IM→G, while PG−M may be computed from PM, where M is a sub geometry of a homogeneous geometry G, and G − M is its set complement. Applications for linearly representable matroids include matrix generalizations to the critical exponent of Crapo and Rota and to the codeweight polynomial of a linear code.