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 $\mathrm{SL}(2,{\mathbb{F}}_{p^2})$.

10 pages