On the good filtration dimension of Weyl modules for a linear algebraic group

Abstract

Let G be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible. In this paper we calculate the Weyl filtration dimension of the induced G-modules, \nabla(\lambda) and the simple G-modules L(\lambda), for \lambda a regular weight. We use this to calculate some Ext groups of the form Ext^*(\nabla(\lambda),\Delta(\mu)), Ext^*(L(\lambda),L(\mu)), and Ext^*(\nabla(\lambda), \nabla(\mu)), where \lambda, \mu are regular and \Delta(\mu) is the Weyl module of highest weight \mu. We then deduce the projective dimensions and injective dimensions for L(\lambda), \nabla(\lambda) and \Delta(\lambda) for \lambda a regular weight in associated generalised Schur algebras. 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.