Space of isospectral periodic tridiagonal matrices

Ayzenberg, Anton¹

¹Faculty of Computer Science, National Research Institute Higher School of Economics, Moscow, Russia

Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 2957-2994

Résumé

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 (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 S4 × Tn−3. 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.

Détail

Classification : 34L40, 52B70, 52C22, 55N91, 57R91, 05E45, 13F55, 14H70, 15A18, 37C80, 37K10, 51M20, 55R80, 55T10
Keywords: isospectral space, matrix spectrum, Toda flow, periodic tridiagonal matrix, discrete Schrödinger operator, permutohedral tiling, simplicial poset, face ring, equivariant cohomology, torus action, crystallization

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