The Green correspondence for SL(2,p)

Abstract

Let $p>2$ be an odd prime and $G=S{L}_2({\mathbb{F}}_p)$. Denote the subgroup of upper triangular matrices as $B$. Finally, let $\mathbb{F}$ be an algebraically closed field of characteristic $p$. The Green correspondence gives a bijection between the non-projective indecomposable $\mathbb{F}[G]$ modules and non-projective indecomposable $\mathbb{F}[B]$ modules, realised by restriction and induction. In this paper, we start by recalling a suitable description of the non-projective indecomposable modules for these group algebras. Next, we explicitly describe the Green correspondence bijection by pinpointing the modules' position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of these descriptions: formulae for lifting the $\mathbb{F}[B]$ module decomposition of an $\mathbb{F}[G]$ module, and a complete description of ${\text{Ind}}_B^G$ and ${\text{Res}}_B^G$.