Effective computation of $\mathrm{SO}(3)$ and $\mathrm{O}(3)$ linear representations symmetry classes

TL;DR

This paper introduces a comprehensive algorithm for determining all symmetry classes of any linear representation of SO(3) or O(3), utilizing a novel binary operation called clips on conjugacy classes.

Contribution

It presents a new general method based on the clips operation to compute symmetry classes for SO(3) and O(3) representations, providing explicit tables for practical use.

Findings

Developed a general algorithm for symmetry class computation.

Introduced the clips operation for conjugacy classes.

Provided explicit tables for the operation.

Abstract

We propose a general algorithm to compute all the symmetry classes of any $\mathrm{SO}(3)$ or $\mathrm{O}(3)$ linear representation. This method relies on the introduction of a binary operator between sets of conjugacy classes of closed subgroups, called the clips. We compute explicit tables for this operation which allows to solve definitively the problem.