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POISSON STRUCTURES ON AFFINE SPACES AND FLAG VARIETIES. I. MATRIX AFFINE POISSON SPACE
 

Summary: POISSON STRUCTURES ON AFFINE SPACES AND
FLAG VARIETIES. I. MATRIX AFFINE POISSON SPACE
K. A. Brown, K. R. Goodearl, and M. Yakimov
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 row-
echelon 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.
Introduction
0.1. We investigate the geometry of the affine variety Mm,n = Mm,n(C) of complex mn

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics