对称刚性框架的商图
Quotient graphs of symmetrically rigid frameworks
AI总结：
本文将周期框架商图提升理论推广到任意维对称框架，刻画多类点群下商图提升为对称刚性框架的条件，并给出有限群标记概率结果。
摘要：
组合刚性理论中的一个自然问题涉及确定在 $\mathbb{R}^d$ 中具有某种非平凡对称性的杆-关节框架的刚性或柔性。当 $d=2$ 时，关于这一主题已有大量文献。特别地，典型做法是按群对对称图取商，并利用适当的群标记商图的组合结构来分析对称但在其他方面通用的框架的刚性。然而，与通用刚性的情形类似，当 $d\geq 3$ 时，从组合角度所知甚少。尽管如此，在周期情形下，Borcea 和 Streinu 的一项关键结果刻画了商图何时可被提升为 $\mathbb{R}^d$ 中的刚性周期框架。我们为 $\mathbb{R}^d$ 中的对称框架发展了一套类似理论。所得结果适用于所有有限和无限的二维点群；随后在任意维数中，它们涉及广泛的无限点群、足够大的有限群以及包含平移和旋转的群。对于有限群情形，我们还推导了关于为商图分配群标记以使所得提升在 $\mathbb{R}^d$ 中对称刚性的概率的结果。
英文摘要：
A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in $\mathbb{R}^d$ that admit some non-trivial symmetry. When $d=2$ there is a large literature on this topic. In particular, it is typical to quotient the symmetric graph by the group and analyse the rigidity of symmetric, but otherwise generic frameworks, using the combinatorial structure of the appropriate group-labelled quotient graph. However, mirroring the situation for generic rigidity, little is known combinatorially when $d\geq 3$. Nevertheless in the periodic case, a key result of Borcea and Streinu characterises when a quotient graph can be lifted to a rigid periodic framework in $\mathbb{R}^d$. We develop an analogous theory for symmetric frameworks in $\mathbb{R}^d$. The results obtained apply to all finite and infinite 2-dimensional point groups, and then in arbitrary dimension they concern a wide range of infinite point groups, sufficiently large finite groups and groups containing translations and rotations. For the case of finite groups we also derive results concerning the probability of assigning group labels to a quotient graph so that the resulting lift is symmetrically rigid in $\mathbb{R}^d$.
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