Georgia Institute of Technology College of Sciences

Consider self-adjoint operators $A, B, C : \mathcal{H} \to \mathcal{H}$ on a finite-dimensional Hilbert space such that $A + B + C = 0$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$,$\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively. This problem was eventually solved by A. Klyachko and Knutson-Tao in the late 1990s. Recently together with H. Bercovici, Collins, Dykema, and Timotin, we are able to find a proof to show that the inequalities are valid for self-adjoint elements that satisfies the relation $A+B+C=0$,  and the proof can be applied to finite von Neumann algebra. The major difficulty in our argument is to show that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requiresa good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions.