正交Grassmann图的自同构和测地性质
Automorphisms and some geodesic properties of ortho-Grassmann graphs
摘要：
设$H$是复希尔伯特空间.考虑正交Grassmann图$Γ^{\perp}_{k}(H)$，其顶点是$H$的$k$维子空间（秩为$k$的投影），如果两个子空间相容且相邻（对应的秩为$k$的投影可交换且它们的差是秩为$2$的算子），则这两个子空间通过边相连.主要结果是：若$\dim H\ne 2k$，则$Γ^{\perp}_{k}(H)$的每一个自同构都是由酉或反酉算子诱导的;若$\dim H=2k\ge 6$，则$Γ^{\perp}_{k}(H)$的每一个自同构都是由酉或反酉算子诱导的或者是这样的自同构与正交补映射的合成.对于$\dim H=2k=4$，语句失败。为了证明这一点，我们比较了长度为2的正交格拉斯曼图的测地线和正交格拉斯曼图和格拉斯曼图的测地线的相容性（交换性）。最后，我们将这一结果推广到有限秩自伴算子共轭类的广义正交Grassmann图上。
英文摘要：
Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $Γ^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $Γ^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $Γ^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
