Lifting polynomial representations of $\mathrm{SL}_2(p^r)$ from $\mathbb{F}_p$ to $\mathbb{Z}/p^s\mathbb{Z}$

TL;DR

This paper classifies which irreducible polynomial representations of $ ext{SL}_2(p^r)$ over $ ext{F}_p$ can be lifted to representations over $ ext{Z}/p^s ext{Z}$, revealing that such lifts are rare and certain indecomposables cannot be lifted.

Contribution

It provides a complete description of the irreducible polynomial representations of $ ext{SL}_2(p^r)$ that lift from $ ext{F}_p$ to $ ext{Z}/p^s ext{Z}$, and shows that most do not lift, including some indecomposables.

Findings

Most irreducible polynomial representations do not lift to $ ext{Z}/p^s ext{Z}$ for $s>1$.

Two specific indecomposable representations cannot be lifted to $ ext{Z}/p^s ext{Z}$.

The paper characterizes all irreducible lifts explicitly.

Abstract

We describe all of the irreducible polynomial $\mathbb{F}_p\mathrm{SL}_2(p^r)$ representations which lift to $(\mathbb{Z}/p^s\mathbb{Z})\mathrm{SL}_2(p^r)$ representations for $s>1$, observing that they almost never do. We also show that two related indecomposable $\mathbb{F}_p \mathrm{SL}_2(p^r)$ representations cannot be lifted to $\mathbb{Z}/p^s\mathbb{Z}$ representations for $s>1$.