We prove a variant of the well-known result that intertwiners for the bilateral shift on ${\ell }^2(\mathbb{Z})$ are unitarily equivalent to multiplication operators on $L^2(\mathbb{T})$. This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

中文翻译：

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对称框架的符号函数

我们证明了互为因果的双边转移的著名结果的一种变体 ${\ell }^2（\mathbb{ž}）$ 分别等价于 ${\mathrm{大号}}^2（\mathbb{Ť}）$。这使我们能够统一和扩展具有阿贝尔对称性组的钢筋连接框架的刚度理论的基本方面。特别是，我们为各种框架制定了符号函数，并展示了如何在各种新环境中构造广义刚性单元模式。