The representation ring of ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$ 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 $k{\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$ modulo projectives. We then investigate a family of modular plethysms of the natural $k{\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$-module $E$ of the form ${\mathrm{\nabla }}^{\nu }{\mathrm{S}\mathrm{y}\mathrm{m}}^{l}E$ for a partition $\nu$ of size less than $p$ and $0\le l\le p-2$. 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 $k{\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$-modules of the form ${\mathrm{\nabla }}^{\nu }V$, where $V$ is a non-projective indecomposable $k{\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$-module and $\mathrm{\mid }\nu \mathrm{\mid }<p$.