Schubert Decomposition for Milnor Fibers of the Varieties of Singular Matrices

Journal of Singularities
volume 18 (2018), 358-396

Received: 9 December 2017. Accepted: 29 May 2018.

DOI: 10.5427/jsing.2018.18s

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Abstract:

We consider the varieties of singular m x m complex matrices which may be either general, symmetric or skew-symmetric (with m even). For these varieties we have shown in another paper that they had compact "model submanifolds", for the homotopy types of the Milnor fibers which are classical symmetric spaces in the sense of Cartan. In this paper we use these models, combined with results due to a number of authors concerning the Schubert decomposition of Lie groups and symmetric spaces via the Cartan model, together with Iwasawa decomposition, to give cell decompositions of the global Milnor fibers.

The Schubert decomposition is in terms of "unique ordered factorizations", of matrices in the Milnor fibers as products of "pseudo-rotations". In the case of symmetric or skew-symmetric matrices, this factorization has the form of iterated "Cartan conjugacies", by pseudo-rotations. The decomposition respects the towers of Milnor fibers and symmetric spaces ordered by inclusions. Furthermore, the "Schubert cycles", which are the closures of the Schubert cells, are images of products of suspensions of projective spaces (complex, real, or quaternionic as appropriate). In the cases of general or skew-symmetric matrices the Schubert cycles have fundamental classes, and for symmetric matrices mod 2 classes, which give a basis for the homology. They are also shown to correspond to the cohomology generators for the symmetric spaces. For general matrices the duals of the Schubert cycles are represented as explicit monomials in the generators of the cohomology exterior algebra; and for symmetric matrices they are related to Stiefel-Whitney classes of an associated real vector bundle.

Furthermore, for a matrix singularity of any of these types. the pull-backs of these cohomology classes generate a characteristic subalgebra of the cohomology of its Milnor fiber.

We also indicate how these results extend to exceptional orbit hypersurfaces, complements and links, including a characteristic subalgebra of the cohomology of the complement of a matrix singularity.

Keywords:

varieties of singular matrices, global Milnor fibration, classical symmetric spaces, Cartan Model, Cartan conjugacy, pseudo-rotations, ordered symmetric and skew-symmetric factorizations, Schubert decomposition, Schubert cycles, Iwasawa decomposition, characteristic subalgebra

Mathematical Subject Classification:

Primary: 11S90, 32S25, 55R80; Secondary: 57T15, 14M12, 20G05

Author(s) information:

James Damon
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599-3250, USA