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2003, 2003(Special): 788-797. doi: 10.3934/proc.2003.2003.788

Compactified isospectral sets of complex tridiagonal Hessenberg matrices

Barbara A. Shipman 1,

1. 

Department of Mathematics Box 19408, The University of Texas at Arlington, Arlington, TX 76019-0408, United States

Received  August 2002 Revised  March 2003 Published  April 2003

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: 70H0.
Citation: Barbara A. Shipman. Compactified isospectral sets of complex tridiagonal Hessenberg matrices. Conference Publications, 2003, 2003 (Special) : 788-797. doi: 10.3934/proc.2003.2003.788
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