Generic symmetry-forced infinitesimal rigidity: translations and rotations
Frameworks that arise in certain applications are constrained to have particular symmetries. In such cases, one is often interested in whether a given framework allows motions that preserve the symmetry. The combinatorial setup for symmetry-forced rigidity is analogous to that for classical rigidity, but instead of studying frameworks on graphs, one studies symmetric frameworks on gain graphs (i.e. directed multigraphs whose arcs are labeled by elements of some group). I will begin my talk with an introduction to the basics of symmetry-forced rigidity. Then, I will present the talk's main result, which is a characterization of the gain graphs whose generic frameworks are infinitesimally rigid, when the gain group is an orientation-preserving subgroup of the Euclidean group. I will also sketch the proof, drawing particular attention to the matroidal constructions that go into the proof.