Manifolds of isospectral arrow matrices

TL;DR

This paper studies the geometric and topological properties of manifolds formed by Hermitian arrow matrices with fixed spectra, revealing their smooth structure, symmetries, and cohomology, especially for the case when n=3.

Contribution

It establishes that these manifolds are smooth, independent of spectrum, and describes their orbit spaces and symmetries, including the structure of associated polytopes and cohomology.

Findings

The space of Hermitian arrow matrices is a smooth 2n-manifold.

The orbit space under torus action is not a polytope for n≥3.

For n=3, the orbit space is a solid torus with a hexagon subdivision.

Abstract

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space $M_{St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove that this space is a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{St_n,\lambda}/T^n$ is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on $M_{St_n,\lambda}$ which induces the combined action of a semidirect product $T^n\rtimes\Sigma_n$. The orbit space of this large action is a simple polytope. The structure of this polytope is described in the paper. In case $n=3$, the space $M_{St_3,\lambda}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{St_3,\lambda}$ using the general theory developed by the first author. This theory is also applied to a certain $6$-dimensional manifold called the twin of $M_{St_3,\lambda}$. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.