# "Tetrahedral sums of Hermitian matrices and related problems" (Anton Alekseev, Unige)

15.08.2024 10:00 – 11:00

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 additive, multiplicative, and maximal multipaths problems. It turns out that under some further assumptions the maximal multipaths problem is related to the octahedron recurrence from the theory of crystals.

Based on joint works in progress with A. Berenstein, M. Christandl, T. Fraser, A. Gurenkova and Y. Li.

### Lieu

Bâtiment: Conseil Général 7-9

Room 1-05

### Organisé par

Section de mathématiques### Intervenant-e-s

Anton Alekseev, UNIGEentrée libre