Cosets, characters and fusion for admissible-level $\mathrm{osp}\left(1\right|2)$ minimal models

We study the minimal models associated to $\mathrm{osp}\left(1\right|2)$, otherwise known as the fractional-level Wess–Zumino–Witten models of $\mathrm{osp}\left(1\right|2)$. Since these minimal models are extensions of the tensor product of certain Virasoro and ${\mathrm{sl}}_2$ minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of $\mathrm{osp}\left(1\right|2)$. In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles.