Extremal rays in the Hermitian eigenvalue problem

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $n\times n$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $\Gamma_n(s)$ controlling this problem have been characterized in terms of classical Schubert calculus by the work of several authors. We determine extremal rays of $\Gamma_n(s)$ (which are never regular faces) by relating them to the geometry of flag varieties: The extremal rays either arise from "modular intersection loci", or by "induction" from extremal rays of smaller groups. Explicit formulas are given for both the extremal rays coming from such intersection loci, and for the induction maps.