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The Global Dimension of Schur Algebras for GL_2 and GL_3

Alison E. Parker
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Abstract

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(λ) by modules of the form \nabla(μ)^F \otimes L(ν) where μis a dominant weight and νis p-restricted.

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Alison E. Parker. 2005-07-26. The Global Dimension of Schur Algebras for GL_2 and GL_3. https://arxiv.org/abs/math/0507521

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