Horn's problem is concerned with characterizing the eigenvalues (a,b,c) of Hermitian matrices (A,B,C) satisfying the constraint A+B=C and forming the edges of a triangle in the space of Hermitian matrices. It has deep connections to tensor product invariants, Littlewood-Richardson coefficients, geometric invariant theory and the intersection theory of Schubert varieties. This paper concerns the tetrahedral Horn problem which aims to characterize the tuples of eigenvalues (a,b,c,d,e,f) of Hermitian matrices (A,B,C,D,E,F) forming the edges of a tetrahedron, and thus satisfying the constraints A+B=C, B+D=F, D+C=E and A+F=E.
Here we derive new inequalities satisfied by the Schur-polynomials of such eigenvalues and, using eigenvalue estimation techniques from quantum information theory, prove their satisfaction up to degree k implies the existence of approximate solutions with error O(\ln k / k). Moreover, the existence of these tetrahedra is related to the semiclassical asymptotics of the 6j-symbols for the unitary group U(n), which are maps between multiplicity spaces that encode the associativity relation for tensor products of irreducible representations. Using our techniques, we prove the asymptotics of norms of these 6j-symbols are either inverse-polynomial or exponential depending on whether there exists such tetrahedra of Hermitian matrices.
