Manifolds of isospectral arrow matrices

Abstract

An arrow matrix is a matrix with zeros outside the main diagonal, the first row and the first column. We consider the space of Hermitian arrow -matrices with fixed simple spectrum . We prove that this space is a smooth -manifold with a locally standard torus action: we describe the topology and combinatorics of its orbit space. If , the orbit space is not a polytope, hence is not a quasitoric manifold. However, there is an action of a semidirect product on , and the orbit space of this action is a certain simple polytope obtained from the cube by cutting off codimension-2 faces. In the case , the space is a solid torus with boundary subdivided into hexagons in a regular way. This description allows us to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold and another manifold, its twin. Bibliography: 32 titles.

Orbit spaces of torus actions on Hessenberg varieties

“…Note, however, that while H h are symplectic submanifolds of the full complex flag variety F l n , the restriction of the symplectic form from Herm(Λ) to M h is degenerate (see [AB1], [AB2]; it follows from the fact that their moment polytopes are not convex).…”