Manifolds of isospectral arrow matrices

An arrow matrix is a matrix with zeros outside the main diagonal, the first row and the first column. We consider the space $M_{\operatorname{St}_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove that this space is a smooth $2n$-manifold with a locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{\operatorname{St}_n,\lambda}/T^n$ is not a polytope, hence $M_{\operatorname{St}_n,\lambda}$ is not a quasitoric manifold. However, there is an action of a semidirect product $T^n\rtimes\Sigma_n$ on $M_{\operatorname{St}_n,\lambda}$, and the orbit space of this action is a certain simple polytope $\mathscr{B}^n$ obtained from the cube by cutting off codimension-2 faces. In the case $n=3$, the space $M_{\operatorname{St}_3,\lambda}/T^3$ 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 $M_{\operatorname{St}_3,\lambda}$ and another manifold, its twin. Bibliography: 32 titles.

sparse matrix, group action, moment map, fundamental domain, codimension-2 face cuts