In Search of Bounds on the Dimension of Ext betweenIrreducible Modules for Finite Groups of Lie Type

Abstract

Let q be a power of a prime p, and let G(q) be a finite group of Lie type defined over the finite field of q elements. Let k be an algebraically closed field of characteristic r. There are three distinct cases to consider in the representation theory of kG(q): r = 0 (the characteristic 0 case), r = p (the defining characteristic case), and r > 0, r distinct from p (the non-defining characteristic case). In defining and non-defining characteristic, we are interested in finding bounds on the dimension of cohomology groups H^i(G(q),V), where V is any irreducible kG(q)-module and i is fixed. In the defining characteristic, such bounds exist when the rank is fixed (this is due to Cline, Parshall, and Scott in the i = 1 case and Bendel, Nakano, Parshall, Pillen, Scott, and Stewart in the i>1 case). In 2011, Guralnick and Tiep showed that 1-cohomology groups are bounded in non-defining characteristic. In this dissertation, we use techniques of modular Harish-Chandra theory to find bounds on the dimension of Ext^1 between irreducible kG(q)-modules in non-defining characteristic when G has a split BN-pair. We also consider higher Ext groups in the case that G(q) is a finite general linear group. Generalizing work of Cline, Parshall, and Scott, we relate certain higher Ext calculations for the general linear group (in non-defining characteristic) to higher Ext calculations over a q-Schur algebra (which in turn, can translate to higher Ext calculations over an appropriate quantum group).