Tableaux in the Whitney Module of a Matroid

TL;DR

This paper explores the use of tableaux to describe the Whitney module of matroids, providing a basis for certain classes and analyzing symmetric group representations through combinatorial formulas.

Contribution

It introduces tableaux-based descriptions for the Whitney module and derives a formula for multiplicities in symmetric group representations related to matroids.

Findings

Basis description for Whitney modules of freedom matroids

Formula for hook shape multiplicities in symmetric group representations

Connection between no broken circuit sets and module multiplicities

Abstract

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.