Lifting polynomial representations of $\mathrm{SL}_2(p^r)$ from $\mathbb{F}_p$ to $\mathbb{Z}/p^s\mathbb{Z}$
Chris Parker, Martin van Beek
TL;DR
We address the lifting problem for modular representations of $G=SL_2(p^r)$ from $\,\mathbb{F}_p$ to $\mathbb{Z}/p^s\mathbb{Z}$ with $s>1$, focusing on the basic polynomial modules $V_n(p^r)$ and the dual module $\Lambda(p^r)$. Using Green's cohomological criterion and a concrete $p$-adic, block-matrix approach, the authors classify liftability: lifting to $\mathbb{Z}/p^2\mathbb{Z}$ occurs only in very restricted cases (namely $p^r=2$ or $r=1$ with odd $p$ and $n\in\{p-2,p-1\}$), and whenever a lift exists it lifts to $\mathbb{Q}_p$. They also show that certain related indecomposable $\mathbb{F}_pG$-modules do not lift, and they establish that $\Lambda(p^r)$ lifts only in the $p^r=2$ case. The results provide a complete lift classification for these basic representations and connect to broader questions about irreducible subgroups of $\mathrm{GL}_n(\mathbb{Z}/p^s\mathbb{Z})$ via cohomological obstructions and detailed $p$-adic analysis. This work informs the understanding of how modular representations of $SL_2(p^r)$ behave under $p$-adic lifting and has potential applications to the study of saturated fusion systems and related algebraic structures.
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 $s>1$.