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

الأصل · EN

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.