Spaces of isospectral Hermitian matrices having zeroes at prescribed positions

The space of all Hermitian matrices with the given simple spectrum is diffeomorphic to the variety of complete complex flags. We consider and study its subspaces of matrices having zeroes at prescribed positions. All these subspaces carry the natural action of a compact torus.
The most well-known case is the manifold of isospectral tridiagonal matrices. This manifold is closely related to the toric variety of type A known in representation theory. This relation can be extended to the relation between manifolds of isospectral staircase matrices and semisimple regular Hessenberg varieties: they have homeomorphic orbit spaces and isomorphic equivariant cohomology rings.
We study two more examples: the manifold of arrow matrices, and the space of periodic tridiagonal matrices. The study of topology in these two examples had lead us to surprisingly interesting objects from combinatorial geometry: the maximal cubical subcomplex of a permutohedron and the regular tiling of Euclidean space by permutohedra.
The talk is partly based on joint works with Victor Buchstaber.