论文标题
在两个真实bruhat细胞的相交的同喻类型上
On the homotopy type of intersections of two real Bruhat cells
论文作者
论文摘要
真实的bruhat单元对$ gl_ {n+1} $，$ flag_ {n+1} = sl_ {n+1}/b $，$ so_ so_ {n+1} $和$ spin_ {n+1} $等空间提供了重要且研究的分层。我们研究了顶维细胞与另一个细胞的相交（另一个基础）。自然而然地用下nilpotent组的子集$ lo_ {n+1}^{1} $自然识别了这种交叉点。我们对此类交叉路口的同质类型特别感兴趣。在本文中，我们定义了此类交叉点的分层。结果，我们获得了有限的CW复合物，该复合物在同质上等同于交叉点。 我们计算同型类型的几个示例。事实证明，对于$ n \ le 4 $，$ lo_ {n+1}^1 $的此类子集的所有连接组件都是合同的：我们通过明确构建相应的CW复合物来证明这一点。相反，对于$ n \ ge 5 $和最高排列，总是有一个连接的组件，甚至具有Euler特征，因此不可签约。这是从相应CW复合物的每个维度的细胞数量的公式开始的。例如，对于顶部排列$ s_6 $，存在一个连接的组件，其特性等于$ 2 $。我们还举了一个$ s_6 $中的排列示例，其中存在一个连接的组件，该组件在同型上等同于圆圈$ s^1 $。
Real Bruhat cells give an important and well studied stratification of such spaces as $GL_{n+1}$, $Flag_{n+1} = SL_{n+1}/B$, $SO_{n+1}$ and $Spin_{n+1}$. We study the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of the lower nilpotent group $Lo_{n+1}^{1}$. We are particularly interested in the homotopy type of such intersections. In this paper we define a stratification of such intersections. As a consequence, we obtain a finite CW complex which is homotopically equivalent to the intersection. We compute the homotopy type for several examples. It turns out that for $n \le 4$ all connected components of such subsets of $Lo_{n+1}^1$ are contractible: we prove this by explicitly constructing the corresponding CW complexes. Conversely, for $n \ge 5$ and the top permutation, there is always a connected component with even Euler characteristic, and therefore not contractible. This follows from formulas for the number of cells per dimension of the corresponding CW complex. For instance, for the top permutation $S_6$, there exists a connected component with Euler characteristic equal to $2$. We also give an example of a permutation in $S_6$ for which there exists a connected component which is homotopically equivalent to the circle $S^1$.
