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

Shalotenko, Veronica, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Parshall, Brian, Department of Mathematics, University of Virginia

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).

PHD (Doctor of Philosophy)
Ext between irreducibles for finite groups of Lie type, Ext groups in non-defining characteristic
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