Bounds on the Dimension of Ext for Finite Groups of Lie Type

TL;DR

This paper extends bounds on the dimension of Ext^1 for modules over finite groups of Lie type in non-defining characteristic, using modular Harish-Chandra theory and illustrating results with examples.

Contribution

It generalizes previous bounds on Ext^1 dimensions for irreducible modules of finite groups of Lie type in non-defining characteristic, employing modular Harish-Chandra theory.

Findings

Derived new bounds on Ext^1 dimensions for certain irreducible modules.

Applied Dipper and Du's algorithms to illustrate the bounds.

Extended previous work on cohomological bounds 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 $r \neq p$ (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 $H^1(G,V)$ 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 $\mathrm{Ext^1}$ between certain irreducible $kG$-modules. We then use Dipper and Du's algorithms to illustrate our bounds in a series of examples.