Injective and non-injective realizations with symmetry

In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be embedded injectively in the space. In particular, we introduce a symmetry-adapted notion of ‘generic’ frameworks with respect to this classification and show that ‘almost all’ realizations in a given symmetry class are generic and all generic realizations in this class share the same infinitesimal rigidity properties. Within this classification we also clarify under what conditions techniques from group representation theory can be applied to further analyze the rigidity properties of a (not necessarily injective) symmetric realization.