The rational Schur algebra

Details The rational Schur algebra The rational Schur algebra

2005-11-27

arxiv.org/abs/math/0511663v2

Mathematics

Representation Theory

30 Pages; revised Nov 2007; this is the final version accepted to appear in Representation Theory (Electronic)

Scientific paper

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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