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

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 $s>1$.