First cohomology groups for finite groups of Lie type in defining characteristic

TL;DR

This paper establishes explicit bounds for the first cohomology groups of finite groups of Lie type over fields of positive characteristic, linking algebraic module structure to cohomological growth rates.

Contribution

It introduces new bounds for the dimension of first cohomology groups using composition factors of Weyl modules, applicable across all ranks and independent of characteristic p.

Findings

Bounded the number of composition factors of Weyl modules.

Derived growth rate estimates for first cohomology dimensions.

Established bounds that depend on the rank and conjectural assumptions.

Abstract

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0. We find explicit bounds for the dimension of the first cohomology group for G with coefficients in a simple kG-module. We proceed by bounding the number of composition factors of Weyl modules for simple algebraic groups independently of p and using this to deduce bounds for the 1-cohomology of simple algebraic groups. Finally, we use this to obtain estimations for the growth rate of the maximum dimension {\gamma_l} of these 1-cohomology groups over all groups of Lie type of rank l. We find that log \gamma_l is O(l^3 log l) (or if the Lusztig conjecture holds, O(l^2 log l)).