  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.
