`Extremal rays and vertices in eigenvalue problems'

Abstract

The Hermitian eigenvalue problem asks for the possible
eigenvalues of a sum of Hermitian matrices, given the eigenvalues of the
summands. The regular faces (i.e., not contained in  Weyl chambers) of the
cone controlling this problem have been characterized in terms of Schubert
calculus by the work of several authors.

We relate the extremal rays of the cones above (which are never regular
faces) 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.

A similar description also holds for the vertices in the multiplicative
eigenvalue problem (where one wants to characterise the possible
eigenvalues of a product of unitary matrices, given the eigenvalues of the
terms).