  In this paper we introduce a novel family of Markov chains on the simple
representations of $\mathrm{SL}_2(\mathbb{F}_p)$ in defining characteristic,
defined by tensoring with a fixed simple module and choosing an indecomposable
non-projective summand. We show these chains are reversible and find their
connected components and their stationary distributions. We draw connections
between the properties of the chain and the representation theory of
$\mathrm{SL}_2(\mathbb{F}_p)$, emphasising symmetries of the tensor product. We
also provide an elementary proof of the decomposition of tensor products of
simple $\mathrm{SL}_2(\mathbb{F}_p)$-representations.
