On abstract representations of the groups of rational points of algebraic groups in positive characteristic

TL;DR

This paper investigates the structure of algebraic rings over algebraically closed fields of positive characteristic using Greenberg's functor, and applies these findings to rigidity problems in Chevalley group representations, including a known rigidity theorem.

Contribution

It introduces a novel analysis of algebraic rings in positive characteristic and applies it to solve rigidity problems for Chevalley group representations.

Findings

Structural insights into algebraic rings in positive characteristic

Application of Greenberg's functor to rigidity problems

Recovery of Seitz's rigidity theorem

Abstract

We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups, recovering, in particular, a rigidity theorem of Seitz.