The p-Modular Green Correspondence for SL₂(Fₚ)

Let p be an odd prime. Let G = SL₂(Fₚ) G = S L 2 ( F p ) and let B denote the subgroup of upper triangular matrices of G . Finally, let F F be an algebraically closed field of characteristic p . The Green correspondence gives a bijection between the non-projective indecomposable F[G] F [ G ] modules and non-projective indecomposable F[B] F [ B ] modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the F[B] F [ B ] module decomposition of an F[G] F [ G ] module, and a complete description of IndBG Ind B G and ResGB Res B G .