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2020 Space of isospectral periodic tridiagonal matrices
Anton Ayzenberg
Algebr. Geom. Topol. 20(6): 2957-2994 (2020). DOI: 10.2140/agt.2020.20.2957

Abstract

A periodic tridiagonal matrix is a tridiagonal matrix with an additional two entries at the corners. We study the space Xn,λ of Hermitian periodic tridiagonal n×n matrices with a fixed simple spectrum λ. Using the discretized Schrödinger operator we describe all spectra λ for which Xn,λ is a topological manifold. The space Xn,λ carries a natural effective action of a compact (n1)–torus. We describe the topology of its orbit space and, in particular, show that whenever the isospectral space is a manifold, its orbit space is homeomorphic to S4×Tn3. There is a classical dynamical system: the flow of the periodic Toda lattice, acting on Xn,λ. Except for the degenerate locus Xn,λ0, the Toda lattice exhibits Liouville–Arnold behavior, so that the space Xn,λXn,λ0 is fibered into tori. The degenerate locus of the Toda system is described in terms of combinatorial geometry: its structure is encoded in the special cell subdivision of a torus, which is obtained from the regular tiling of the euclidean space by permutohedra. We apply methods of commutative algebra and toric topology to describe the cohomology and equivariant cohomology modules of Xn,λ.

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Anton Ayzenberg. "Space of isospectral periodic tridiagonal matrices." Algebr. Geom. Topol. 20 (6) 2957 - 2994, 2020. https://doi.org/10.2140/agt.2020.20.2957

Information

Received: 14 December 2018; Revised: 27 July 2019; Accepted: 28 November 2019; Published: 2020
First available in Project Euclid: 16 December 2020

MathSciNet: MR4185932
Digital Object Identifier: 10.2140/agt.2020.20.2957

Subjects:
Primary: 34L40 , 52B70 , 52C22 , 55N91 , 57R91
Secondary: 05E45 , 13F55 , 14H70 , 15A18‎ , 37C80 , 37K10 , 51M20 , 55R80 , 55T10

Keywords: crystallization , discrete Schrödinger operator , equivariant cohomology , face ring , isospectral space , matrix spectrum , periodic tridiagonal matrix , permutohedral tiling , simplicial poset , Toda flow , torus action

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.20 • No. 6 • 2020
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