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

TL;DR

This paper systematically enumerates little groups of irreducible representations for O(3), SO(3), and infinite axial groups up to rank 9, using a new chain criterion, aiding symmetry analysis in materials science.

Contribution

It introduces a novel chain criterion for enumerating little groups of irreps, applicable to a wide range of groups and ranks, with verification by inspection.

Findings

Enumerated little groups for O(3) and SO(3) up to rank 9

Extended enumeration to infinite axial groups like C_infinity and D_infinity

Validated results through inspection and verification

Abstract

Little groups are enumerated for the irreps and their components in any basis of O(3) and SO(3) up to rank 9, and for all irreps of C$_{\infty}$, C$_{\infty h}$, C$_{\infty v}$, D$_{\infty}$ and D$_{\infty h}$. The results are obtained by a new chain criterion, which distinguishes massive (rotationally inequivalent) irrep basis functions and allows for multiple branching paths, and are verified by inspection. 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.