The Horn inequalities, which can be used to characterize the eigenvalues of sums of self-adjoint matrices, are associated with certain combinatorial objects called balanced measures. These measures play a central role in our understanding of the Horn inequalities and the extension of Horn inequalities to the type II$_1$ von Neumann factors. In this talk I will discuss some of the recent developments of our continued investigation of the algebraic relations of these measures and also

their geometric implications. This is a joint work with H. Bercovici.