Connected Components in the Intersection of Two Open Opposite Schubert Cells in SLn(ℝ)/B
Connected Components in the Intersection of Two Open Opposite Schubert Cells in SLn(ℝ)/B
复制标题
SLn(ℝ)/B 中两个开路相对舒伯特单元相交处的连通分量
DOI：
10.1155/s1073792897000329
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发表时间：
1997
影响因子：
1
通讯作者：
A. Vainshtein
中科院分区：
文献类型：
--
作者：
B. Shapiro;M. Shapiro;A. Vainshtein
一百多年来，完全和部分旗形的Schubert分解和Schubert演算一直是代数、几何、表示论和组合学的交汇点。本文研究了Schubert分解理论的进一步细化。我们远大而雄心勃勃的目标是描述两个任意Schubert胞格相交的拓扑。这个问题对于表示论来说似乎是非常重要的，特别是对于Kazhdan-Lusztig多项式的计算，这是一个众所周知的非常困难的问题。作为这一方向的一个较小的步骤，我们计算了实n维标志空间中两个开的相对的Schubert胞格的交集的连通分支的个数。这个问题在[A]中提出，后来在[SS]中关于完全积极的标准提出。Schubert胞格的两两交(不一定相反)的其他拓扑特征在我们以前的文章[SV]，[SSV1]，[SSV2]中也被考虑过。尽管在本文中我们只考虑实代数簇，但我们希望我们的方法也能给出一些关于复杂几何的信息。产生这种希望的原因如下。(A)计算方法是基于Berenstein、Fomin和Zlevinsky[BFZ]关于形成旗形流形中开放单元的幂等下三角矩阵的Lusztig参数化的美丽和非常深入的(“非常代数的”)结果。看起来可能会使用相同类型的参数化和像元分解
For more than one hundred years, Schubert decomposition of complete and partial flag manifolds and Schubert calculus have been an intertwining point of algebra, geometry, representation theory, and combinatorics. In this paper, we are concerned with the further refinement of Schubert decomposition theory. Our far and ambitious goal is to describe the topology of intersection of two arbitrary Schubert cells. This problem seems to be very important in connection with representation theory, in particular, with calculation of Kazhdan–Lusztig polynomials, which is known to be a very hard problem. As a minor step in this direction, we calculate in this paper the number of connected components in the intersection of two open opposite Schubert cells in the space of real n-dimensional flags. This question was raised in [A], and later, in connection with criteria of total positivity, in [SS]. Other topological characteristics of pairwise intersections of Schubert cells (not necessarily opposite) are considered in our previous papers [SV], [SSV1], [SSV2]. Despite the fact that in this paper we consider only real algebraic varieties, we hope that our approach can give some information about complex geometry as well. The reasons for this hope are as follows. (a) The calculation method is based on the beautiful and very deep (“very algebraic”) results by Berenstein, Fomin, and Zelevinsky [BFZ] on the Lusztig parametrization of the unipotent lower-triangular matrices,which form an open cell in the flag manifold. It looks like the same type of parametrization and cell decompositions might be used