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

TL;DR

This paper computes the filtration dimensions and Ext groups of Weyl and simple modules for a linear algebraic group over an algebraically closed field, providing insights into their projective, injective, and global dimensions.

Contribution

It introduces explicit calculations of Weyl filtration dimensions and Ext groups for modules of algebraic groups, advancing understanding of their homological properties.

Findings

Calculated Weyl filtration dimensions for induced and simple modules.

Determined projective and injective dimensions for modules at regular weights.

Established the global dimension of Schur algebras in specific cases.

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.