Manifolds of isospectral arrow matrices

Abstract

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space $M_{S{t}_n,\lambda }$ 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 $n⩾3$, the orbit space $M_{S{t}_n,\lambda }/T^n$ is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on $M_{S{t}_n,\lambda }$ which induces the combined action of a semidirect product $T^n⋊{\Sigma }_n$. 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 $M_{S{t}_3,\lambda }/T^3$ 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 $M_{S{t}_3,\lambda }$ using the general theory developed by the first author. This theory is also applied to a certain $6$-dimensional manifold called the twin of $M_{S{t}_3,\lambda }$. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.

29 pages, 8 figures