Horn's conjecture with positivity of Littlewood-Richardson coefficients

Abstract

We discuss the problem given by Hermann Weyl about relations of real sequences $\alpha$, $\beta$ and $\gamma$ that are eigenvalues of $n \times n$ Hermitian matrices $A$, $B$ and $A+B$ respectively. In the conjecture of Alfred Horn, triples $(\alpha, \beta, \gamma)$ form a convex set defined by some inequalities. This corresponds to another conjecture about positivity of Littlewood-Richardson coefficients called the saturation conjecture. After several proofs of the saturation conjecture appeared, the Horn's conjecture turned out to be true. We focus on the duality between hive models and honeycomb models which are main ideas of two distinct proofs of the saturation conjecture.