Universal deformation rings for the symmetric group S_5 and one of its double covers

TL;DR

This paper explicitly determines the universal deformation rings for certain mod 2 representations of the symmetric group S_5 and its double cover, confirming a conjecture relating these rings to the Sylow 2-subgroups.

Contribution

It provides the first explicit calculations of universal deformation rings for these groups' representations, confirming a conjecture about their relation to Sylow 2-subgroups.

Findings

Universal deformation rings are explicitly computed for selected representations.

The conjecture relating deformation rings to Sylow 2-subgroups is affirmed.

Results apply to representations in the principal 2-modular block of S_5.

Abstract

Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ denote a non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalized quaternion. We determine the universal deformation rings $R(S_5,V)$ and $R(\hat{S}_5,V)$ for each mod 2 representation $V$ of $S_5$ that belongs to the principal 2-modular block of $S_5$ and whose stable endomorphism ring is given by scalars when it is inflated to $\hat{S}_5$. We show that for these $V$, a question raised by the first author and Chinburg concerning the relation of the universal deformation ring of $V$ to the Sylow 2-subgroups of $S_5$ and $\hat{S}_5$, respectively, has an affirmative answer.