In 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups.We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general.

We study Karoubian tensor categories, which interpolate representation categories of families of the so-called easy quantum groups in the same sense in which Deligne’s interpolation categories $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ interpolate the representation categories of the symmetric groups. As such categories can be described using a graphical calculus of partitions, we call them interpolating partition categories. They include $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ as a special case and can generally be viewed as subcategories of the latter. Focusing on semisimplicity and descriptions of the indecomposable objects, we prove uniform generalisations of results known for special cases, including $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ or Temperley–Lieb categories. In particular, we identify those values of the interpolation parameter, which correspond to semisimple and non-semisimple categories, respectively, for all the so-called group-theoretical easy quantum groups. A crucial ingredient is an abstract analysis of certain subobject lattices developed by Knop, which we adapt to categories of partitions. We go on to prove a parametrisation of the indecomposable objects in all interpolating partition categories for non-zero interpolation parameters via a system of finite groups, which we associate to any partition category, and which we also use to describe the associated graded rings of the Grothendieck rings of these interpolation categories.

We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing R-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in W *probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group U + n . Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in W * -probability spaces. On the other hand, if we drop the assumption of faithful states in W * -probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.

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We prove several de Finetti Theorems for the unitary dual group, also called the Brown algebra.Firstly, we provide a finite de Finetti Theorem characterizing R-diagonal elements with identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti Theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in W * -probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group U + n . Thirdly, the above de Finetti Theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti Theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in W * -probability spaces. On the other hand, if we drop the assumption of faithful states in W * -probability spaces, we obtain a non-trivial half a de Finetti Theorem similar to the case of the dual group action.