Universal deformation rings for the symmetric group S₄

الأصل · EN

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S₄ denote the symmetric group on 4 letters. We determine the universal deformation ring R(S₄,V) for every kS₄-module V which has stable endomorphism ring k and show that R(S₄,V) is isomorphic to either k, or W[t]/(t²,2t), 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.