Résumé

Local unitary invariant polynomials of tensor variables play an  important role in the study of multipartite entanglement.  They also  characterize tensor distributions that possess this invariance, and are  in bijection with certain triangulations in dimension three or more.  After reviewing this, I will describe how due to these different facets,  some recent results with Collins and Gurau concerning local spherical  integrals that generalize the HCIZ integral have repercussions in  entanglement detection, random tensors, and random geometry.  

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