Partition quantum spaces

We propose a definition of partition quantum spaces. They are given by universal $C^*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first $d$ columns of easy quantum groups. However, we define them as universal $C^*$-algebras rather than as $C^*$-subalgebras of easy quantum groups. We also investigate the minimal number $d$ needed to recover an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, $d$ takes the values one or two.