The Green correspondence for SL(2,p)

2025/03/10 by Marchment, Denver-James Logan
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2503.07581

Abstract

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

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