  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 face 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 diffeomorphism of
polyhedral spheres. We use this results to get new closed formulae for the
black hole entropy in loop quantum gravity.
