On the inverse problem for deformations of finite group representations

Abstract

Let $s$ be even and $q=p^s$. We show that the ring $W(\mathbb{F}_{q})[\![X]\!]/(X^2-pX)$ is a quotient of the universal deformation ring of a representation of a finite group. This amounts to giving an example of a finite group and its $\mathbb{F}_q$-representation that lifts to $W(\mathbb{F}_q)$ in two different ways and satisfies certain subtle extra conditions. We achieve this by studying representations of $\mathrm{SL}(2,\mathbb{F}_{p^2})$.