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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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.