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.
