Eigenvalues of tetrahedral sums of Hermitian matrices (Thomas Coolican Fraser, Copenhagen)

04.03.2025 16:10 – 17:00

Horn's problem aims to determine set of possible eigenvalues (a,b,c) for a triple of n x n Hermitian matrices (A,B,C) satisfying A+B=C. This talk is based on joint work with A. Alekseev and M. Christandl concerning a generalization of Horn's problem called the tetrahedral Horn problem. The tetrahedral Horn problem aims to determine the set of possible eigenvalues (a,b,c,d,e,f) for a sextuplet of n x n Hermitian matrices (A,B,C,D,E,F) satisfying A+B=C, B+D=F, C+D=E, and A+F=E. We will derive the complete solution for the tetrahedral Horn problem for n=2 and see how, unlike Horn's original problem, the solution space is neither convex nor a polytope.

For general n, we describe various inequality conditions and symmetries of the tetrahedral Horn problem, as well as prove how the set of tetrahedral eigenvalues is characterized by the asymptotics of the Wigner 6j symbol for the unitary group U(n), generalizing the well-known correspondence between the Littlewood-Richardson coefficients and Horn's original problem.

Lieu

Bâtiment: Conseil Général 7-9

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

entrée libre