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Rank polynomials
Author(s) -
Brandt Marco,
Dipper Richard,
James Gordon,
Lyle Sinéad
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn018
Subject(s) - mathematics , basis (linear algebra) , rank (graph theory) , partition (number theory) , combinatorics , representation (politics) , order (exchange) , symmetric group , algebra over a field , pure mathematics , geometry , finance , politics , political science , law , economics
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.