Two results on the rank partition of a matroid

Abstract

The rank partition of a matroid M is the maximum dominance ordered partition $\rho$ such that the ground set of M can be partitioned into independent sets of sizes ${\rho }_1$, ${\rho }_2$, …. We prove two structural results on this partition, both motivated by representation theory of the general linear group. The first result characterizes the rank partition in terms of standard Young tableaux with a certain matroidal property. The second result says that the rank partition interacts nicely with certain polytopal decompositions of the matroid polytope of M. We also describe the representation theoretical motivation of these results.