The fine structure of SU(2) intertwiners from U(N) representations

In this work, we study the Hilbert space space of N-valent SU(2) intertwiners with fixed total spin, which can be identified, at the classical level, with a space of convex polyhedra with N faces and fixed total boundary area. We show that this Hilbert space provides, quite remarkably, an irreducible representation of the U(N) group. This gives us therefore a precise identification of U(N) as a group of area-preserving diffeomorphisms of polyhedral spheres. We use this result to get new closed formulas for the black hole entropy in loop quantum gravity.