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 Mat. Sb., 2021, Volume 212, Number 5, Pages 3–36 (Mi msb9381)

Manifolds of isospectral arrow matrices

A. A. Ayzenberga*, V. M. Buchstaberb

a Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: 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.

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

 Funding Agency Grant Number HSE Basic Research Program Ministry of Education and Science of the Russian Federation 5-100 This research was carried out within the framework of the Basic Research Program at HSE University and funded by the Russian Academic Excellence Project “5-100”.

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm9381

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English version:
Sbornik: Mathematics, 2021, 212:5, 605–635

Bibliographic databases:

UDC: 515.146
MSC: Primary 52B11, 15A42, 57R19, 57R91; Secondary 05E45, 52B70, 15B57, 52C45, 55N91, 57S25, 20Bxx, 53D20

Citation: A. A. Ayzenberg, V. M. Buchstaber, “Manifolds of isospectral arrow matrices”, Mat. Sb., 212:5 (2021), 3–36; Sb. Math., 212:5 (2021), 605–635

Citation in format AMSBIB
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