Universal deformation rings and fusion

TL;DR

This paper investigates the structure of universal deformation rings of modules over finite groups, especially focusing on how these rings reflect the fusion of subgroups within the group, with explicit calculations for certain group extensions.

Contribution

It determines the cohomology groups and universal deformation rings for specific group extensions, revealing how these rings encode fusion information.

Findings

Cohomology groups are explicitly computed for the group extensions.

Universal deformation rings are characterized for various modules.

The extent to which deformation rings detect fusion is analyzed.

Abstract

Let p be a prime, M be a finite group, F be the field with p elements, and V be an absolutely irreducible FM-module. Then V has a universal deformation ring R(M,V) whose structure is closely related to the first and second cohomology groups of M with coefficients in Hom_F(V,V). We consider the case when M is an extension of a dihedral group G whose order is relatively prime to p by an elementary abelian p-group N of rank 2. We determine the cohomology groups and also R(M,V) for various V and show to what extent R(M,V) sees the fusion of N in M.