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Representation Theory

ISSN 1088-4165

 
 

 

The rational Schur algebra


Authors: Richard Dipper and Stephen Doty
Journal: Represent. Theory 12 (2008), 58-82
MSC (2000): Primary 16G99; Secondary 20G05
DOI: https://doi.org/10.1090/S1088-4165-08-00303-8
Published electronically: February 12, 2008
MathSciNet review: 2375596
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Abstract: We extend the family of classical Schur algebras in type $ A$, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general linear groups directly through finite dimensional algebras. We show that rational Schur algebras are quasihereditary over any field, and thus have finite global dimension.

We obtain explicit cellular bases of a rational Schur algebra by a descent from a certain ordinary Schur algebra. We also obtain a description, by generators and relations, of the rational Schur algebras in characteristic zero.


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Additional Information

Richard Dipper
Affiliation: Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany
Email: Richard.Dipper@mathematik.uni-stuttgart.de

Stephen Doty
Affiliation: Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60626
Email: doty@math.luc.edu

DOI: https://doi.org/10.1090/S1088-4165-08-00303-8
Keywords: Schur algebras, finite dimensional algebras, quasihereditary algebras, general linear groups
Received by editor(s): November 28, 2005
Received by editor(s) in revised form: October 23, 2007
Published electronically: February 12, 2008
Additional Notes: This work was partially supported by DFG project DI 531/5-2 and NSA grant DOD MDA904-03-1-00.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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