The object of study in this paper is the space \(X_{n,\lambda}\) of \(n \times n\) Hermitian matrices \([b_{ij}]\), having fixed simple spectrum \(\lambda=(\lambda_1<\cdots <\lambda_n)\), for which \(b_{ij}=0\) unless \(ij\) is an edge of the \(n\)-cycle. The real \(n\)-torus \(U(1)^n\) acts on \(X_{n,\lambda}\) by conjugation; scalar matrices act trivially and the quotient \((n-1)\)-torus \(\mathcal{T}\) acts effectively. The author uses methods from dynamical systems, mathematical physics, toric topology, and algebraic combinatorics to give an explicit description of \(X_{n,\lambda}\) and its decomposition into orbits.
The topology of \(X_{n,\lambda}\) is determined by the smallest positive and largest negative critical values, \(M\) and \(-m\), of the characteristic polynomial \(F(x)=\prod (x-\lambda_i)\); \(X_{n,\lambda}\) is a smooth manifold if and only if each of \(M\) and \(-m\) is achieved only once. The parameter \(B=b_{n,1}\prod_{i=1}^{n-1} b_{i,i+1}\) is constant on \(\mathcal{T}\)-orbits, taking values in a region \(\mathbb B\) bounded by two parabolic arcs, with common focus at 0, determined by \(M\) and \(m\). The action is free over the interior of \(\mathbb B\), away from zero. Over the boundary of \(\mathbb B\) there is nontrivial isotropy, constant on each parabolic arc, with further collapsing at the intersection points, all determined by the number of roots \(n_+\) and \(n_-\) of \(F(x)=M\) and \(F(x)=-m\), respectively.
The space of \(n \times n\) Hermitian tridiagonal matrices with spectrum \(\lambda\) is known to have orbit space equal to the permutohedron. The central fiber \(B=0\) is the union of \(n\) copies of this space, intersecting over lower-dimensional permutohedra in a cyclic manner. The space of orbits with \(B=0\) is then identified with a cyclic tiling of \(\mathcal{T}\) by \(n\) permutohedra. This allows computations in ordinary and equivariant cohomology using methods from algebraic combinatorics, showing in particular that \(X_{n,\lambda}\) is not equivariantly formal for \(n \geq 4\). The fundamental group of \(X_{n,\lambda}\) is free abelian of rank determined by \(n_+\) and \(n_-\), and is trivial if and only if \(-m\) and \(M\) are the only critical values of \(F(x)\), the most degenerate case.