The irreducible unipotent modules of the finite general linear groups via tableaux

Abstract

We construct the irreducible unipotent modules of the finite general linear groups using tableaux. Our construction is analogous to that of James (1976) for the symmetric groups, answering an open question as to whether such a construction exists. Our modules are defined over any field containing a nontrivial $p^\text{th}$ root of unity (where $p$ is the defining characteristic of the group). We show that our modules are isomorphic to those constructed by James (1984), although the two constructions utilize quite different approaches. Finally we look closer at the complex irreducible unipotent modules, providing motivation for our construction.