Representation of Crystallographic Subperiodic Groups by Geometric Algebra

TL;DR

This paper extends the geometric algebra representation from 3D crystallographic space groups to the 162 subperiodic groups, introducing a new compact symbolic notation for their generators.

Contribution

It introduces a novel geometric algebra group representation symbol for subperiodic groups, enhancing clarity and computational utility.

Findings

New compact geometric algebra group symbols for subperiodic groups

Explicit generator sets for each group in the new notation

Enhanced understanding of subperiodic group structures

Abstract

We explain how following the representation of 3D crystallographic space groups in geometric algebra it is further possible to similarly represent the 162 socalled subperiodic groups of crystallography in geometric algebra. We construct a new compact geometric algebra group representation symbol, which allows to read off the complete set of geometric algebra generators. For clarity we moreover state explicitly what generators are chosen. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford group, Lipschitz elements).