# The Fine Structure of SU(2) Intertwiners from U(N) Representations

## TL;DR

This paper explores the structure of SU(2) intertwiners with fixed total spin, revealing their relation to U(N) representations and applying this to derive new formulas for black hole entropy in loop quantum gravity.

## Contribution

It establishes a novel connection between SU(2) intertwiners and U(N) representations, providing a geometric interpretation and new entropy formulas.

## Key findings

- SU(2) intertwiners form irreducible U(N) representations
- U(N) acts as area-preserving diffeomorphisms of polyhedral spheres
- New closed formulas for black hole entropy in loop quantum gravity

## Abstract

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.