Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
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DOI：
10.1016/j.aim.2005.10.004
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发表时间：
2005-01
影响因子：
1.7
通讯作者：
Kenneth A. Brown;K. Goodearl;M. Yakimov
中科院分区：
文献类型：
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作者：
Kenneth A. Brown;K. Goodearl;M. Yakimov
利用辛叶在极大环面T⊂GLm+n(C)作用下的轨道，研究了矩形矩阵簇mm，n(C)上的标准泊松结构.证明了这些有限个轨道是mm，n(C)的光滑不可约局部闭子簇，同构于全旗簇GLm+n(C)中的对偶Schubert胞格的交集.得到了mm，n(C)中辛叶的T轨道的三种不同的表示形式：(A)GLm+n(C)中Bruhat胞格在特定映射下的拉回；(B)矩形子矩阵上的秩条件；(C)GLm(C)和Gln(C)中类似于双Bruhat胞格的集合的矩阵乘积.在表示(A)中，叶的轨道被Weyl群Sm+n的子集参数化，使得Zariski闭包的包含对应于Bruhat序。列报(B)允许显式计算轨道。从列报(C)可以看出，直到Zariski闭合，每个树叶轨道都是一个具有固定列梯形的轨道和一个具有固定行梯形的轨道的矩阵乘积。最后，得到了Mm，n(C)(关于部分置换矩阵对)中的广义双Bruhat胞胞分解为辛叶的T轨道的并.
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.