The Global Dimension of Schur Algebras for GL2 and GL3

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 SL2 and SL3. We use these to show that the global dimension of a Schur algebra for GL2 and GL3 is twice the good filtration dimension. To do this for SL3, we give an explicit filtration of the modules (λ) by modules of the form (μ)F L(ν) where μ is a dominant weight and ν is p-restricted.