Algebra Seminar

Let G be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible. We denote the induced G-modules, \nabla(\lambda) and the simple G-modules L(\lambda), for \lambda a dominant weight. A G-module has a Weyl filtration if it is filtered by Weyl modules (which are dual to the induced modules). Following Friedlander and Parshall, we can measure how far a module is from having a Weyl filtration by considering its Weyl filtration dimension (just as the projective dimension measures in some sense how far a module is from being projective). Since all projectives have a Weyl filtration these two concepts are related. In fact we show how to calculate some projective (and injective) dimensions for modules for the Schur algebra using knowledge of the Weyl filtration dimension. The Weyl filtration dimension has another advantage in that it is finite for a finite dimensional G-module.

This talk will show how to calculate explicitly the Weyl filtration dimension for the modules L(\lambda) and \nabla(\lambda) for lambda a regular dominant weight. We then show how to deduce the projective and injective dimensions for these modules considered as modules for associated generalised Schur algebras (which includes the usual Schur algebras for GL_n). We also deduce the global dimension of the Schur algebras for GL_n, S(n,r), when p>n and for S(mp,p) with m an integer.