Space of isospectral periodic tridiagonal matrices

Abstract

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized Shr\"{o}dinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ 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 $X_{n,\lambda}$. Except for the degenerate locus $X_{n,\lambda}^0$, the Toda lattice exhibits Liouville--Arnold behavior, so that the space $X_{n,\lambda}\setminus X_{n,\lambda}^0$ 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 $X_{n,\lambda}$.