A periodic tridiagonal matrix is a tridiagonal matrix with
an additional two entries at the corners. We study the space
of Hermitian periodic
tridiagonal
matrices with
a fixed simple spectrum
.
Using the discretized Schrödinger operator we describe all spectra
for which
is a topological
manifold. The space
carries a natural effective action of a compact
–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
.
There is a classical dynamical system: the flow of the periodic Toda lattice, acting on
. Except for the
degenerate locus
,
the Toda lattice exhibits Liouville–Arnold behavior, so that the space
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 .
