Matroids and Semimodular Lattices

Abstract

In this chapter we continue the process of explaining the intimate connection between matroids and the symmetric group by first switching to semimodular lattices and seeing how they are related to the symmetric group. We develop a viewpoint of a semimodular lattice as a chamber system with a kind of metric that gives it a structure only slightly weaker than that of a building over Sym n . This leads to a natural way to represent flag matroids in semimodular lattices, which in turn leads up to our representation of arbitrary Coxeter matroids in buildings in Chapter 7.