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Proceedings of the London Mathematical Society Advance Access published online on April 22, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn018
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© 2008 London Mathematical Society

Rank Polynomials

Marco Brandt

Herrenberger Strasse 94/1
D-71069 Sindelfingen
Germany

Richard Dipper

Universität Stuttgart
Institut für Algebra und Zahlentheorie
Abteilung für Darstellungstheorie
Pfaffenwaldring 57
D-70569 Stuttgart
Germany
Richard.Dipper@mathematik.unistuttgart.de

Gordon James

Department of Mathematics
Imperial College London
180 Queen's Gate
London
SW7 2AZ
United Kingdom
g.james@imperial.ac.uk

Sinéad Lyle

School of Mathematics
University of East Anglia
Norwich
NR4 7TJ
United Kingdom

Received 24 November 2005. Revision received 5 September 2007.

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 how 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.

2000 Mathematics Subject Classification 20C20.

In the course of the work for this paper, Brandt, Dipper and James received support from DFG project No. DI 531/5-2 and Dipper and James received support from the Oberwolfach RIP-program.


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