The Green correspondence for SL(2,p)

Let p > 2 be an odd prime and G = SL₂(Fₚ). Denote the subgroup of upper triangular matrices as B. Finally, let F be an algebraically closed field of characteristic p. The Green correspondence gives a bijection between the non-projective indecomposable F[G] modules and non-projective indecomposable 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 F[B] module decomposition of an F[G] module, and a complete description of IndBG and ResGB.