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In this paper, we pursue our study of asymptotic properties of families ofrandom matrices that have a tensor structure. In previous work, the first- andsecond-named authors provided conditions under which tensor products of unitaryrandom matrices are asymptotically free with respect to the normalized trace.Here, we extend this result by proving that asymptotic freeness of tensorproducts of Haar unitary matrices holds with respect to a significantly largerclass of states. Our result relies on invariance under the symmetric group, andtherefore on traffic probability.  As a byproduct, we explore two additional generalisations: (i) we stateresults of freeness in a context of general sequences of representations of theunitary group -- the fundamental representation being a particular case thatcorresponds to the classical asymptotic freeness result for Haar unitarymatrices, and (ii) we consider actions of the symmetric group and the freegroup simultaneously and obtain a result of asymptotic freeness in this contextas well.

Asymptotic Freeness of Unitary Matrices in Tensor Product Spaces for Invariant States