Domain walls in topological phases and the Brauer-Picard ring for $\mathrm{Vec}(\mathbb{Z}\mathrm{/}p\mathbb{Z})$

We show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer-Picard ring for the fusion category $\mathrm{Vec}(\mathbb{Z}\mathrm{/}p\mathbb{Z})$. Moreover, we provide a physical interpretation of all indecomposable bimodule categories in terms of domain walls in the associated topological phase. We show how this interpretation can be used to compute the Brauer-Picard ring from a physical perspective.