The magic square of reflections and rotations

TL;DR

This paper explores the connections between complex and real reflection groups of rank two, using Coxeter's work and Clifford algebras, revealing a 'magic square' of symmetries and their interpretations in low dimensions.

Contribution

It establishes a bijection between complex rank-two reflection groups and real groups in three dimensions, and interprets these via Clifford algebras and spin groups.

Findings

Bijection between complex and real reflection groups of rank two.

Interpretation of reflection groups using Clifford algebras.

Analysis of these groups in small dimensions.

Abstract

We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in $O(3)$. We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions.