Universal deformation rings for extensions of finite subgoups of $Gl_2(\mathbb{C})$

TL;DR

This paper investigates how universal deformation rings of irreducible modules over certain finite groups can reveal fusion patterns, focusing on groups extending subgroups of $Gl_2(\mathbb{C})$ by elementary abelian $p$-groups.

Contribution

It extends previous work by analyzing the detection of fusion in specific group extensions using universal deformation rings.

Findings

Universal deformation rings can encode fusion information.

Extensions of subgroups of $Gl_2(\mathbb{C})$ are studied.

Results enhance understanding of deformation theory in group extensions.

Abstract

In this paper we expand on previous results, studying the extent to which one can detect fusion in certain finite groups $\Gamma$, from information about the universal deformation rings of absolutely irreducible $\mathbb{F}_p\Gamma$-modules. We consider groups which are extensions of finite irreducible subgroups of $Gl_2({\mathbb{C}})$ by elementary abelian $p$-groups of rank $2$.