### Abstract

In this work, we show that the asymptotic limit of the recoupling coefficients of the symmetric group is characterized by the existence of quantum states of three particles with given eigenvalues for their reduced density matrices. This parallels Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2)---fundamental to the quantum theory of angular momentum---is governed by the existence of Euclidean tetrahedra. We explain how to deduce solely from symmetry properties of the recoupling coefficients the strong subadditivity of the von Neumann entropy, first proved by Lieb and Ruskai, and discuss possible generalizations of our result.