The Horn problem is a Linear Algebra question asking to determine the range of eigenvalues of the sum (a+b) of two Hermitian matrices with given spectra. The solution was conjectured by Horn, and it is given by a set of linear inequalities on eigenvalues. The proof of the conjecture is due to Klyachko and Knutson-Tao. It is interesting that exactly the same set of inequalities describes singular values of matrix products, maximal multipaths in concatenation of planar networks, and non-vanishing of Littlewood-Richardson coefficients for representations of GL(N).

In this talk, we consider the multiple Horn problem which is asking to determine the range of eigenvalues of (a+b), (b+c) and (a+b+c) for a, b and c with given spectra. Now the four different problems described above no longer have the same solution. We will present some results for the maximal multipaths problem, and for the singular value problem. It turns out that under some further assumptions the maximal multipaths is related to the octahedron recurrence from the theory of crystals.

The talk is based on a joint work in progress with A. Berenstein, A. Gurenkova, and Y. Li.