Space of isospectral periodic tridiagonal matrices

Ayzenberg, Anton

doi:10.2140/agt.2020.20.2957

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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 $X_{n, λ}$ of Hermitian periodic tridiagonal $n \times n$ matrices with a fixed simple spectrum $λ$ . Using the discretized Schrödinger operator we describe all spectra $λ$ for which $X_{n, λ}$ is a topological manifold. The space $X_{n, λ}$ carries a natural effective action of a compact $(n - 1)$ –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 $S^{4} \times T^{n - 3}$ . There is a classical dynamical system: the flow of the periodic Toda lattice, acting on $X_{n, λ}$ . Except for the degenerate locus $X_{n, λ}^{0}$ , the Toda lattice exhibits Liouville–Arnold behavior, so that the space $X_{n, λ} ∖ X_{n, λ}^{0}$ 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 $X_{n, λ}$ .

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

Volume 20, issue 6 (2020)