Universal deformation rings and generalized quaternion defect groups

TL;DR

This paper determines the universal deformation rings for certain mod 2 representations of finite groups with generalized quaternion defect groups, confirming a conjecture and analyzing their algebraic structure.

Contribution

It explicitly computes universal deformation rings for specific representations and confirms a conjecture relating these rings to the defect groups.

Findings

R(G,V) is a complete intersection for the studied cases.

The relation between R(G,V) and the defect group D is affirmed.

R(G/N,V) may not be a complete intersection for some normal subgroups N.

Abstract

We determine the universal deformation ring R(G,V) of certain mod 2 representations V of a finite group G which belong to a 2-modular block of G whose defect groups are isomorphic to a generalized quaternion group D. We show that for these V, a question raised by the author and Chinburg concerning the relation of R(G,V) to D has an affirmative answer. We also show that R(G,V) is a complete intersection even though R(G/N,V) need not be for certain normal subgroups N of G which act trivially on V.