A Bruhat type decomposition of the set of conditioned invariant subspaces

Jochen Trumpf

In 1998 J. Ferrer, F. Puerta and X. Puerta derived a description of conditioned invariant subspaces in terms of images of block Toeplitz type matrices. They used this description to construct a stratification of the set of conditioned invariant subspaces of fixed dimension into smooth manifolds. These so called Brunovsky strata consist of all the subspaces with fixed restriction indices. A similar result was derived by P.A. Fuhrmann and U. Helmke (published in 2001) with the help of polynomial models. Their work built on earlier work by D. Hinrichsen, H.F. Münzner, and D. Prätzel-Wolters (1981), who associated to each conditioned invariant subspace a module of Laurent series and parametrized these modules using their so-called Kronecker-Hermite bases. In 2002 F. Puerta, X. Puerta and I. Zaballa constructed a cell decomposition of the Brunovsky strata into so called Kronecker cells. In this talk I will show that in the tight case this cell decomposition is induced by a Bruhat decomposition of a generalized flag manifold. I identify the adherence order of the cell decomposition as being induced by the reverse Bruhat order. This result is derived using a retraction of the Brunovsky strata onto generalized flag manifolds which was derived by X. Puerta and U. Helmke in 2000. I will indicate how this result generalises to the non tight case.