We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph.

中文翻译：

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平面中的增益稀疏和对称强制刚性

我们考虑具有离散点群对称性的平面杆关节框架，其中关节位置在受对称约束的情况下尽可能通用。我们为此类结构的对称强制刚度提供组合表征，其具有奇数 k 的 2k 阶旋转对称或二面对称，统一并扩展了先前在该主题上的工作。我们还探索了我们结果的拟阵背景，并表明对称框架的轨道矩阵的行独立性引起的拟阵是同构的，以获得在框架商图上定义的稀疏拟阵，其边由对应的对称群。证明基于与所讨论拟阵的基相对应的增益图的新 Henneberg 型归纳构造，