Generic Symmetry-Forced Infinitesimal Rigidity: Translations and Rotations

We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the underlying symmetry group consists of rotations and translations. Along the way, we use tropical geometry to show how a construction of Edmonds and Rota that associates a matroid to a submodular function can be used to give a description of the algebraic matroid underlying a Hadamard product of two linear spaces in terms of the matroids underlying each linear space. This leads to new, short, proofs of Laman's theorem, and a theorem of Jord{a}n, Kaszanitzky, and Tanigawa characterizing the minimally generically symmetry-forced rigid graphs in the plane when the symmetry group contains only rotations.