2018/03/31 by Anton Ayzenberg · 5 citations
Materials Science · Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Cohomology #Equivariant cohomology #Equivariant map #Invariant (physics) #Isospectral #Quasicrystal Structures and Properties #Topological space #Topology (electrical circuits) #Tridiagonal matrix #math.AT #math.CO #math.DS #math.KT #msc:05E45 #msc:13F55 #msc:14H70 #msc:15A18 #msc:34L40 #msc:37C80 #msc:37K10 #msc:51M20 #msc:52B70 #msc:52C22 #msc:55N91 #msc:55R80 #msc:55T10 #msc:57R91
paper · pdf · doi:10.2140/agt.2020.20.2957
published in Algebraic & Geometric Topology 20(6), 2957-2994 (Mathematical Sciences Publishers) · 31 pages, 7 figures, 2 tables. Certain inaccuracy was corrected in the second version, concerning the naming of the lattices. The third version contains new section, which contains the computation of Betti numbers and the fundamental group. 4-th version contains several improvements in the style of exposition, proposed by the anonymous referee
openalex created_date 2018/04/06 · openalex publication_date 2020/12/08 · arxiv created 2021/02/18 · arxiv updated 2021/02/19 · openalex updated_date 2026/08/05
A periodic tridiagonal matrix is a tridiagonal matrix with an additional two entries at the corners. We study the space [math] of Hermitian periodic tridiagonal [math] matrices with a fixed simple spectrum [math] . Using the discretized Schrödinger operator we describe all spectra [math] for which [math] is a topological manifold. The space [math] carries a natural effective action of a compact [math] –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 [math] . There is a classical dynamical system: the flow of the periodic Toda lattice, acting on [math] . Except for the degenerate locus [math] , the Toda lattice exhibits Liouville–Arnold behavior, so that the space [math] 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 [math] .