Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space

Brown, K.A., Goodearl, K.R. and Yakimov, M. (2006) Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space. Advances in Mathematics, 206(2), pp. 567-629. (doi: 10.1016/j.aim.2005.10.004)

Abstract

The standard Poisson structure on the rectangular matrix variety Mm,n(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂ GLm+n(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of Mm,n(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GLm+n(C). Three different presentations of the T-orbits of symplectic leaves in Mm,n(C) are obtained – (a) as pullbacks of Bruhat cells in GLm+n(C) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GLm(C) and GLn(C). In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group Sm+n, such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed rowechelon form. Finally, decompositions of generalized double Bruhat cells in Mm,n(C) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.

• Item Type: Articles
• Keywords: Poisson varieties, poisson algebraic groups, poisson homogeneous spaces, symplectic leaves, flag varieties, Schubert cells, Bruhat cells
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Brown, Professor Ken
• Authors: Brown, K.A., Goodearl, K.R., and Yakimov, M.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Advances in Mathematics
• Publisher: Elsevier
• ISSN: 0001-8708
• ISSN (Online): 1090-2082
• Published Online: 22 November 2005
• Copyright Holders: Copyright © 2006 Elsevier
• First Published: First published in Advances in Mathematics 206(2):567-629
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher