The representation ring of SL2(Fp) and stable modular plethysms of its natural module in characteristic p

Let p be an odd prime and let k be a field of characteristic p. We provide a practical algebraic description of the representation ring of kSL2(Fp) modulo projectives. We then investigate a family of modular plethysms of the natural kSL2(Fp)-module E of the form for a partition ν of size less than p and 0≤l≤p−2. Within this family we classify both the modular plethysms of E which are projective and the modular plethysms of E which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of E are replaced by kSL2(Fp)-modules of the form ∇νV, where V is a non-projective indecomposable kSL2(Fp)-module and |ν|<p.