Injective and non-injective realizations with symmetry

TL;DR

This paper develops a classification for symmetric bar and joint frameworks, extending rigidity theory to include frameworks with various symmetries and injectivity conditions, and establishes genericity and rigidity properties within this framework.

Contribution

It introduces a symmetry-based classification of frameworks, defines symmetry-adapted genericity, and clarifies when group representation theory applies to analyze their rigidity.

Findings

Most realizations in a symmetry class are generic.

All generic realizations in a class share the same infinitesimal rigidity.

Conditions identified for applying group representation theory to symmetric frameworks.

Abstract

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 group representation theory techniques can be applied to further analyze the rigidity properties of a (not necessarily injective) symmetric realization.