Universal deformation rings for the symmetric group \(S_5\) and one of its double covers. (Q616330)

Let \(k\) be an algebraically closed field of characteristic 2, let \(G\) be the symmetric group of degree 5 or its covering group which has generalized quaternion Sylow 2-subgroups, and let \(V\) be an indecomposable \(kG\)-module in the principal block of \(kG\) whose stable endomorphism ring has dimension 1 over \(k\). Moreover, suppose that \(Z(G)\) acts trivially on \(V\). The authors compute the universal deformation ring \(R(G,V)\) and show that \(R(G,V)\) is isomorphic to a subquotient of the group ring of a Sylow 2-subgroup of \(G\) over the ring of infinite Witt vectors over \(k\). This confirms a conjecture by Bleher and Chinburg in a special case.