Little groups of irreps ofO(3),SO(3), and the infinite axial subgroups

A new chain criterion, for the determination of little groups, is presented. This result distinguishes massive (rotationally inequivalent) irrep basis functions and allows for multiple branching paths; consequently, it is suitable for application to Lie groups. Applied to the groups O (3), SO (3), D ∞ h , D ∞ , C ∞ v , C ∞ h , and C ∞ , the result enables the enumeration of all possible little groups, together with their associated basis functions. These results are relevant to the determination of the symmetry of a material from its linear and nonlinear optical properties and to the choices of order parameters for symmetry breaking in liquid crystals.