turkmath.org

Given three $n \times n$ hermitian matrices $A, B, C$ with $A + B = C$, one can write down several inequalities between the eigenvalues of $A, B$ and $C$. The first inequalities were discovered by Weyl and the list was expanded by several mathematicians till 1958 when Alfred Horn conjectured a complete set of (linear) inequalities. Horn additionally conjectured that these inequalities formed a complete set, that is, three lists of real numbers that satisfy all these inequalities can be realised as the eigenvalues of $A, B, C$ as above. That all the inequalities hold was proved by Dooley, Repka and Wildberger in 1991 and that the inequalities was sufficient was proved by Knutson and Tao, with critical contributions by Klyachko. The proof uses some beautiful ideas from both representation theory and algebraic geometry. An analogous problem can be stated in von Neumann algebras and partial answers to this have been given by Bercovici, Li, Collins and Dykema. I'll give an overview of this problem and end with recent results on mine that leave open some tantalising questions.