Universal deformation rings for the symmetric group S_4

TL;DR

This paper explicitly determines the universal deformation rings for certain modules over the symmetric group S_4 in characteristic 2, showing they are always isomorphic to specific subquotients of group rings over Witt vectors.

Contribution

It provides a complete classification of universal deformation rings for modules with stable endomorphism ring k over S_4 in characteristic 2, confirming a conjecture about their structure.

Findings

Universal deformation rings are isomorphic to k, W[t]/(t^2,2t), or W[Z/2].

The classification confirms the conjecture for S_4 modules.

The results support the idea that deformation rings relate closely to defect groups.

Abstract

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.