Tableaux in the Whitney Module of a Matroid

The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom matroids (also known as Schubert, or shifted matroids). The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a formula for the multiplicity of hook shapes in this representation in terms of no broken circuit sets.