Bounds on the dimension of Ext for finite groups of Lie type

Abstract

Let G be a finite group of Lie type defined in characteristic p, and let k be an algebraically closed field of characteristic r>0. We will assume that r≠p (so, we are in the non-defining characteristic case). Let V be a finite-dimensional irreducible left kG-module. In 2011, Guralnick and Tiep found bounds on the dimension of H1(G,V) in non-defining characteristic, which are independent of V. The aim of this paper is to generalize the work of Gurlanick and Tiep. We assume that G is split and use methods of modular Harish-Chandra theory to find bounds on the dimension of ExtkG1(Y,V), where Y and V are irreducible kG-modules. We then use Dipper and Du's algorithms to illustrate our bounds in a series of examples.