Tetrahedral Horn problem (Anton ALEKSEEV, UNIGE)

30.09.2025 15:00 – 17:00

The tetrahedral Horn problem is asking to determine the range of eigenvalues of Hermitian n by n matrices C=A+B, E=A+B+D, and F=B+D if the eigenvalues of A, B and D are fixed. Geometrically, this corresponds to constructing tetrahedra in the space of Hermitian matrices H_n, and for n=2 the problem reduces to constructing Euclidean tetrahedra with given side lengths in R^3.

We will present a necessary and sufficient condition for the tetrahedral Horn problem. This condition consists of several easy trace equalities, and of an infinite number of inequalities on Schur polynomials of (A, B, C, D, E, F). We show that by increasing the polynomial degree one can find finite systems of inequalities which approximate the solution with an arbitrarily small error. The tetrahedral Horn problem is closely related to Wigner 6j-symbols for the unitary group U(n). We will show estimates on 6j symbols which establish a new instance of correspondence between geometry of tetrahedra and representation theory of U(n). These estimates shed light on the nature of inequalities in our necessary and sufficient condition.

The talk is based on a joint work in progress with M. Christandl and T.C. Fraser.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Séminaire "Groupes de Lie et espaces de modules"

entrée libre