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The magic square of reflections and rotations

Ragnar-Olaf Buchweitz, Eleonore Faber, Colin Ingalls

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.

The magic square of reflections and rotations

Abstract

We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in . 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.

Paper Structure

This paper contains 9 sections, 26 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The groups
  3. The Magic Square of Rotations and Reflections
  4. Quaternions
  5. The Clifford Algebra of Euclidean $3$--Space
  6. The Pin Groups for Euclidean $3$--Space
  7. Reflections
  8. General Clifford algebras
  9. Acknowledgements

Key Result

Theorem 3.4

The inclusion of groups $j:\mathop{\mathrm{SO}}\nolimits(3)\to \mathop{\mathrm{\mathsf {O}}}\nolimits(3)$ sets up, via $j^{-1}(\mathsf G)= \mathsf G\cap \mathop{\mathrm{SO}}\nolimits(3)$, a bijection between the conjugacy classes of finite reflection groups $\mathsf G\leqslant \mathop{\mathrm{\maths

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 3.1
  • Example 3.2
  • Theorem 3.4: H.S.M. Coxeter, Cox1
  • Theorem 3.7: C. Jordan Jordan, F. Klein KleinIcosaeder
  • Remark 3.8
  • Theorem 3.9
  • Theorem 3.10
  • Remark 3.11
  • Remark 3.13
  • ...and 61 more