# The symmetries of octupolar tensors

## TL;DR

This paper classifies all inequivalent symmetries of octupolar tensors in three dimensions using the octupolar potential, providing a systematic mathematical framework that extends previous less comprehensive studies.

## Contribution

It offers a complete classification of octupolar tensor symmetries in 3D, advancing understanding of their geometric and algebraic properties.

## Key findings

- Classification of all inequivalent octupolar symmetries in 3D
- Use of octupolar potential to analyze tensor symmetries
- Integration of previous partial results into a systematic framework

## Abstract

Octupolar tensors are third order, completely symmetric and traceless tensors. Whereas in 2D an octupolar tensor has the same symmetries as an equilateral triangle and can ultimately be identified with a vector in the plane, the symmetries that it enjoys in 3D are quite different, and only exceptionally reduce to those of a regular tetrahedron. By use of the octupolar potential that is, the cubic form associated on the unit sphere with an octupolar tensor, we shall classify all inequivalent octupolar symmetries. This is a mathematical study which also reviews and incorporates some previous, less systematic attempts.