Structure and representations of bimodule categories

This thesis consists of four research papers in the field of representation theory. Three of the papers are concerned with bicategories of finite-dimensional bimodules over a family of radical square zero Nakayama algebras. In the first, we study the tensor combinatorics of these bimodules. This amounts to an explicit description of the tensor combinatorics in terms of so called left, right, and two-sided cells, which are inspired by Green's relations for semigroups. In the second and third paper we study the problem of classifying simple transitive birepresentations of these bicategories. This results in a complete classification for all but one possible value of an invariant called apex. The fourth paper is concerned with the cell structure within the bicategory of finite-dimensional bimodules over all algebras over a fixed field. We specifically study two-sided relations between the regular bimodules over different algebras, that is, the question of when the regular bimodule can appear as a direct summand in a tensor product of bimodules