Universal deformation rings for the symmetric group S_4

Details Universal deformation rings for the symmetric group S_4 Universal deformation rings for the symmetric group S_4

2008-12-12

arxiv.org/abs/0812.2505v1

Algebr. Represent. Theory 13 (2010), no. 3, 255-270.

Mathematics

Group Theory

12 pages, 2 figures

Scientific paper

10.1007/s10468-008-9120-7

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.

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