Thesis Abstract

Title

On The Global Dimension of Schur Algebras and Related Algebras

Abstract

We wish to determine the global dimension of all Schur Algebras, $S(n,r)$. We also consider quasi-hereditary algebras of which the Schur algebra is an example. The global dimension is known to be finite - since all quasi-hereditary algebras have finite global dimension.

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 $\GL_2$ and $\GL_3$ and hence for the corresponding Schur Algebras. We then 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 $\GL_3$, we give an explicit filtration of the induced modules $\nabla(\lambda)$ by modules of the form $\nabla(\mu)\frob \otimes L(\nu)$ where $\mu$ is a dominant weight, $L(\nu)$ is the irreducible $G$-module of highest weight $\nu$ and $\nu$ is $p$-restricted.

We then consider a more general situation and calculate the Weyl filtration dimension of all $\nabla(\lambda)$ for $\lambda$ a weight lying inside an alcove and $G$ an arbitary split, connected and reductive algebraic group whose root system is irreducible. We then deduce the global dimension of Schur algebras with $p >n$.

We also consider the general case and present a conjectured value for the global dimension of the remaining Schur algebras.