  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.
