The second cohomology of simple SL_3-modules

TL;DR

This paper computes the second cohomology groups of irreducible modules for the algebraic group SL_3 and its finite Chevalley groups over fields of positive characteristic, advancing understanding of their module structures.

Contribution

It determines the second cohomology groups H^2 for all irreducible modules of SL_3 and SL(3,q) in characteristic p>0, including new results for finite groups when p>7.

Findings

H^2(G,V) computed for all irreducible G-modules

H^2(G(q),V) determined for finite groups when p>7

Results extend knowledge of module cohomology in positive characteristic

Abstract

Let G be the simple, simply connected algebraic group SL_3 defined over an algebraically closed field K of characteristic p>0. In this paper, we find H^2(G,V) for any irreducible G-module V. When p>7 we also find H^2(G(q),V) for any irreducible G(q)-module V for the finite Chevalley groups G(q)=SL(3,q) where q is a power of p.