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Space of isospectral periodic tridiagonal matrices

Anton Ayzenberg

Algebraic & Geometric Topology 20 (2020) 2957–2994
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,λ Xn,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.

Keywords
isospectral space, matrix spectrum, Toda flow, periodic tridiagonal matrix, discrete Schrödinger operator, permutohedral tiling, simplicial poset, face ring, equivariant cohomology, torus action, crystallization
Mathematical Subject Classification 2010
Primary: 34L40, 52B70, 52C22, 55N91, 57R91
Secondary: 05E45, 13F55, 14H70, 15A18, 37C80, 37K10, 51M20, 55R80, 55T10
References
Publication
Received: 14 December 2018
Revised: 27 July 2019
Accepted: 28 November 2019
Published: 8 December 2020
Authors
Anton Ayzenberg
Faculty of Computer Science
National Research Institute Higher School of Economics
Moscow
Russia