The Green correspondence for SL(2,p)

Denver-James Logan Marchment

Published 2025-03-10, updated 2025-05-14Version 2

Let ${p > 2}$ be an odd prime and ${G = SL_2(\mathbb{F}_p)}$. 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 ${\text{Res}^G_B}$.