Rigidity of symmetric frameworks with non-free group actions on the vertices

TL;DR

This paper extends the combinatorial characterizations of symmetry-generic infinitesimal rigidity in plane frameworks to cases with non-free group actions, introducing phase-symmetric orbit rigidity matrices for various symmetries.

Contribution

It introduces generalized orbit rigidity matrices for non-free group actions and provides complete combinatorial characterizations for certain symmetry groups in the plane.

Findings

Established necessary conditions for infinitesimal rigidity with non-free group actions.

Provided complete combinatorial characterizations for reflection, half-turn, and three-fold rotational symmetries.

Developed refined sparsity counts using generalized group-labelled quotient graphs.

Abstract

For plane frameworks with reflection or rotational symmetries, where the group action is not necessarily free on the vertex set, we introduce a phase-symmetric orbit rigidity matrix for each irreducible representation of the group. We then use these generalised orbit rigidity matrices to provide necessary conditions for infinitesimal rigidity for frameworks that are symmetric with a cyclic group that acts freely or non-freely on the vertices. Moreover, for the reflection, the half-turn, and the three-fold rotational group in the plane, we establish complete combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks. This extends well-known characterisations for these groups to the case when the group action is not necessarily free on the vertices. The presence of vertices that are fixed by non-trivial group elements requires the introduction of generalised versions of group-labelled quotient graphs leads to more refined types of combinatorial sparsity counts for characterising symmetry-generic infinitesimal rigidity.