  We prove that the asymptotic behavior of the recoupling coefficients of the
symmetric group is characterized by a quantum marginal problem -- namely, by
the existence of quantum states of three particles with given eigenvalues for
their reduced density operators. This generalizes 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. As a corollary, we deduce solely from symmetry considerations the
strong subadditivity property of the von Neumann entropy. Lastly, we show that
the problem of characterizing the eigenvalues of partial sums of Hermitian
matrices arises as a special case of the quantum marginal problem. We establish
a corresponding relation between the recoupling coefficients of the unitary and
symmetric groups, generalizing a classical result of Littlewood and Murnaghan.
