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Working paper

Manifolds of isospectral arrow matrices

1803.10449. 1803.10449. Arxiv Cornell University Library, 2018. No. 10449.
Ayzenberg A., Бухштабер В. М.
An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space MStn,λ of Hermitian arrow (n+1)×(n+1)-matrices with fixed simple spectrum λ. 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 MStn,λ/Tn is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on MStn,λ which induces the combined action of a semidirect product Tn⋊Σn. 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 MSt3,λ/T3 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 MSt3,λ using the general theory developed by the first author. This theory is also applied to a certain 6-dimensional manifold called the twin of MSt3,λ. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.