On the inverse problem for deformations of finite group representations

TL;DR

The paper demonstrates that certain deformation rings of finite group representations can be explicitly realized as quotients of universal deformation rings, using representations of SL(2, p^2) to illustrate multiple liftings.

Contribution

It provides the first explicit example of a finite group representation with multiple lifts to the ring of Witt vectors, revealing new insights into the inverse deformation problem.

Findings

Explicit construction of deformation ring quotients

Multiple lifts of a finite group representation over Witt vectors

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