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 di- mension of the induced G-modules,∇(�) and the simple G-modules L(�), fora regular weight. We use this to calculate some Ext groups of the form Ext � ∇(�),�(µ) � , ExtL(�),L(µ) � , and Ext � ∇(�),∇(µ) � , where �,µ are regular and �(µ) is the Weyl module of highest weight µ. We then deduce the projective dimensions and injective dimensions for L(�), ∇(�) and �(�) for � a regular weight in associated generalised Schur algebras. We also deduce the global dimension of the Schur algebras for GLn, S(n,r), when p > n and for S(mp,p) with m an integer.