2005/07/27 by Alison E. Parker · 1 citation
Mathematics · #math.RT #math.GR #msc:16G99 #msc:20G05 #msc:20G10
published as J. reine angew. Math. 562 (2003), 5-21 · 17 pages, uses amsart v. 2.08
arxiv created 2005/07/27 · arxiv updated 2009/12/01
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, ∇(λ) and the simple G-modules L(λ), for λa regular weight. We use this to calculate some Ext groups of the form Ext^*(∇(λ),Δ(μ)), Ext^*(L(λ),L(μ)), and Ext^*(∇(λ), ∇(μ)), where λ, μare regular and Δ(μ) is the Weyl module of highest weight μ. We then deduce the projective dimensions and injective dimensions for L(λ), ∇(λ) and Δ(λ) for λa 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.