Cohomology and Ext for rank one finite groups of Lie type in cross characteristic

TL;DR

This paper calculates the dimensions of Ext and cohomology groups for irreducible modules in specific blocks of certain finite groups of Lie type in cross characteristic, providing explicit results for these algebraic structures.

Contribution

It provides explicit computations of Ext and cohomology dimensions for modules in cyclic defect blocks of specific finite groups of Lie type, extending understanding in modular representation theory.

Findings

Computed Ext group dimensions for modules in Sz(q), PSU_3(q), and 2G_2(q) groups.

Determined cohomology group dimensions for all modules in these blocks.

Extended results to blocks with Brauer trees as stars or lines.

Abstract

We compute the dimensions of $\operatorname{Ext}_G^n(V, W)$ for all irreducible $V$, $W$ lying in $r$-blocks of cyclic defect in the simple groups $\operatorname{Sz}(q)$, $\operatorname{PSU}_3(q)$ and $\operatorname{{}^2G}_2(q)$ in cross characteristic, obtaining in particular the dimensions of all cohomology groups for such modules. Along the way, we also obtain an analogous result for any $r$-block of cyclic defect whose Brauer tree is either a star or line (open polygon).