Tensor products of modular representations of $\operatorname{SL}_2(\mathbb{F}_p)$ and a random walk on their indecomposable summands
In this paper we give a novel, concise and elementary proof of the decomposition of tensor products of simple modular $\operatorname{SL}_2(\mathbb{F}_p)$-representations. This result is used to decompose tensor products involving their projective covers and to decompose symmetric squares. We define a Markov chain on the simple modular $\operatorname{SL}_2(\mathbb{F}_p)$-representations via tensoring with a fixed simple module and choosing an indecomposable summand according to a specified weighting; we show this chain is reversible and find its stationary distributions.