The fusion rules of some free wreath product quantum groups and applications

In this paper we find the fusion rules of the free wreath products $\widehat{\Gamma}\wr_*S_N^+$ for any (discrete) group $\Gamma$. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced $C^*$-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type $II_1$-factor and that the dual of $\widehat{\Gamma}\wr_*S_N^+$ has the Haagerup approximation property for all finite groups $\Gamma$.