Rank polynomials

A long-standing open problem in the representation theory of the finite general linear groups is to determine a ‘standard basis’ for the Specht modules. Such a basis would be analogous to the most commonly used basis for the Specht modules of the symmetric groups which is indexed by standard tableaux of a given shape. Here we show the existence of such a basis when the Specht module is indexed by a partition with two parts. In order to prove the result, we introduce a class of polynomials which we call rank polynomials; the combinatorics of these rank polynomials turns out to be intriguing in its own right.