The rational Schur algebra

Dipper, Richard; Doty, Stephen

doi:10.1090/S1088-4165-08-00303-8

The rational Schur algebra
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by Richard Dipper and Stephen Doty

Represent. Theory 12 (2008), 58-82

DOI: https://doi.org/10.1090/S1088-4165-08-00303-8

Published electronically: February 12, 2008

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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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