论文标题
统一空间框架复合体的同置特性
Homotopy properties of the complex of frames of a unitary space
论文作者
论文摘要
令$ v $是一个有限的维矢量空间，配备了非脱位式遗传学形式，上面是$ \ mathbb {k} $。令$ \ mathcal {g}（v）$为顶点的图形设置$ 1 $ - 二维非授权子空间$ v $和正交性给出的邻接关系。我们对$ \ MATHCAL {G}（V）$的何时完整说明是根据$ V $的尺寸和地面场$ \ Mathbb {k} $连接的。此外，我们证明，如果$ \ dim（v）> 4 $，则$ \ MATHCAL {g}（g}（v）$的集团复杂$ \ mathcal {f}（v）$简单地连接。对于有限字段$ \ mathbb {k} $，我们还计算了$ \ Mathcal {g}（v）$的邻接矩阵的特征值。然后，通过加兰的方法，我们得出结论，$ \ tilde {h} _m（\ mathcal {f}（v）（v）; \ \ mathbb {k}）= 0 $ 0 $ 0 \ leq m \ leq m \ leq m \ leq \ leq \ leq \ dim（v）-3 $ | \ Mathbb {K} | $。在这些假设下，我们推断出$ \ Mathcal {f}（v）$变形的Barycentric细分将$ \ Mathcal {f}（f}（v）$的一定等级选择的顺序缩回，这是Cohen-Macaulay，这是$ \ nathbb {k} $。 最后，我们将结果应用于有限组的基本Abelian $ p $ -subgroups的Quillen Poset，并研究了$ v $的非分类子空间和$ v $的正交分解的Poset的几何属性。
Let $V$ be a finite dimensional vector space equipped with a non-degenerate Hermitian form over a field $\mathbb{K}$. Let $\mathcal{G}(V)$ be the graph with vertex set the $1$-dimensional non-degenerate subspaces of $V$ and adjacency relation given by orthogonality. We give a complete description of when $\mathcal{G}(V)$ is connected in terms of the dimension of $V$ and the size of the ground field $\mathbb{K}$. Furthermore, we prove that if $\dim(V) > 4$ then the clique complex $\mathcal{F}(V)$ of $\mathcal{G}(V)$ is simply connected. For finite fields $\mathbb{K}$, we also compute the eigenvalues of the adjacency matrix of $\mathcal{G}(V)$. Then by Garland's method, we conclude that $\tilde{H}_m(\mathcal{F}(V);\mathbb{k}) = 0$ for all $0\leq m\leq \dim(V)-3$, where $\mathbb{k}$ is a field of characteristic $0$, provided that $\dim(V)^2 \leq |\mathbb{K}|$. Under these assumptions, we deduce that the barycentric subdivision of $\mathcal{F}(V)$ deformation retracts to the order complex of the certain rank selection of $\mathcal{F}(V)$ which is Cohen-Macaulay over $\mathbb{k}$. Finally, we apply our results to the Quillen poset of elementary abelian $p$-subgroups of a finite group and to the study of geometric properties of the poset of non-degenerate subspaces of $V$ and the poset of orthogonal decompositions of $V$.
