Paper Abstract

Title

The Global Dimension of Schur Algebras for $\GL_2$ and $\GL_3$

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 its good filtration dimension. To do this for $SL_3$, we give an explicit filtration of the modules $\nabla(\lambda)$ by modules of the form $\nabla(\mu)^{\mathrm F} \otimes L(\nu)$ where $\mu$ is a dominant weight and $\nu$ is $p$-restricted.