Consider Hermitian matrices A, B : H → H on an n-dimensional Hilbert space and C = A + B. Denote the eigenvalues, counting multiplicity, of A, B, and C by
α = {α1 ≥ · · · ≥ αn }, β = {β1 ≥ · · · ≥ βn }, and γ = {γ1 ≥ · · · ≥ γn } respectively, arranged in decreasing order.
In 1962, A. Horn conjectured that the relations of (α, β, γ) can be characterized by a set of inequalities defined inductively. This conjecture was proved true by Klyachko and Knutson-Tao in the late 1990s, using highly sophisticated machinery from algebraic geometry and intricate combinatorics, including honeycombs and hives. In this talk I will give a brief overview of the history of Horn’s conjecture, some infinite dimensional generalizations, and some fascinating questions suggested by the combinatorics, in particular, the Danilov-Koshevoy conjecture.
