Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
By K. Brown, K. Goodearl, M. Yakimov. 2005
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Abstract The standard Poisson structure on the rectangular matrix variety M m , n ( C ) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂ GL m + n ( C ) . These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M m , n ( C ) , isomorphic to intersections of dual Schubert cells in the full flag variety of GL m + n ( C ) . Three different presentations of the T-orbits of symplectic leaves in M m , n ( C ) are obtained: (a) as pullbacks of Bruhat cells in GL m + 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 GL m ( C ) and GL n ( C ) . In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group S m + 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 M m , n ( C ) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d2d07d9df2cbcef2cc9aacb2a579416c881673fe
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
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Citations in the corpus
- Invariant prime ideals in quantizations of nilpotent Lie algebras (2009, background): “The torus orbits of symplectic leaves of πw were described in [3, 16].”