  We propose new symmetry-adapted rigidity matrices to analyze the
infinitesimal rigidity of arbitrary-dimensional bar-joint frameworks with
Abelian point group symmetries. These matrices define new symmetry-adapted
rigidity matroids on group-labeled quotient graphs. Using these new tools, we
establish combinatorial characterizations of infinitesimally rigid
two-dimensional bar-joint frameworks whose joints are positioned as generic as
possible subject to the symmetry constraints imposed by a reflection, a
half-turn or a three-fold rotation in the plane. For bar-joint frameworks which
are generic with respect to any other cyclic point group in the plane, we
provide a number of necessary conditions for infinitesimal rigidity.
