## The rational Schur algebra

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- by Richard Dipper and Stephen Doty PDF
- Represent. Theory
**12**(2008), 58-82 Request permission

## 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
- MR Author ID: 59395
- ORCID: 0000-0003-3927-3009
- Email: doty@math.luc.edu
- 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.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**12**(2008), 58-82 - MSC (2000): Primary 16G99; Secondary 20G05
- DOI: https://doi.org/10.1090/S1088-4165-08-00303-8
- MathSciNet review: 2375596