Compactified isospectral sets of complex tridiagonal Hessenberg matrices

Pages: 788 - 797, Issue Special, July 2003

 Abstract        Full Text (193.3K)              

Barbara A. Shipman - Department of Mathematics Box 19408, The University of Texas at Arlington, Arlington, TX 76019-0408, United States (email)

Abstract: The completed flows of the complex Toda lattice hierarchy are used to compactify an arbitrary isospectral set $J$^ of complex tridiagonal Hessenberg matrices. When the eigenvalues are distinct, this compactification, as found by other authors, is a toric variety; it is the closure of a generic orbit of a complex maximal torus inside a flag manifold. This torus becomes a direct product, A, of a nonmaximal diagonal subgroup and a unipotent group when eigenvalues coincide. We describe the compactification of $J$^ in this case as the closure of a generic orbit of A. We are interested mainly in the structure of its boundary, which is a union of nonmaximal orbits of A. There is a one-to-one correspondence between the connected components of the intersections of $J$^ with the lower-dimensional symplectic leaves and the faces of the moment polytope where at least one vertex is a minimal coset representative of a certain quotient of the Weyl group.

Keywords:  Isospectral Set, Toda Lattice, Hamiltonian Flows.
Mathematics Subject Classification:  Primary: 37J35; Secondary: 70H06.

Received: August 2002;      Revised: March 2003;      Published: April 2003.