The Global Dimension of Schur Algebras for GL_2 and GL_3

Details The Global Dimension of Schur Algebras for GL_2 and GL_3 The Global Dimension of Schur Algebras for GL_2 and GL_3

2005-07-26

arxiv.org/abs/math/0507521v1

J. Algebra, 241 (2001), 340-378

Mathematics

Representation Theory

uses xypic v 3.7 and amsart v. 2.08. 1 figure, 32 pages

Scientific paper

We first define the notion of good filtration dimension and Weyl filtration dimension in a quasi-hereditary algebra. We calculate these dimensions explicitly for all irreducible modules in SL_2 and SL_3. We use these to show that the global dimension of a Schur algebra for GL_2 and GL_3 is twice the good filtration dimension. To do this for SL_3, we give an explicit filtration of the modules \nabla(\lambda) by modules of the form \nabla(\mu)^F \otimes L(\nu) where \mu is a dominant weight and \nu is p-restricted.

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