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A 3D Spinorial View of 4D Exceptional Phenomena

  • Pierre-Philippe DechantEmail author
Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 159)

Abstract

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via ‘sandwiching’. This extends to a description of orthogonal transformations in general by means of ‘sandwiching’ with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflections by the Cartan-Dieudonné theorem. We begin by viewing the largest non-crystallographic reflection/Coxeter group \(H_4\) as a group of rotations in two different ways—firstly via a folding from the largest exceptional group \(E_8\), and secondly by induction from the icosahedral group \(H_3\) via Clifford spinors. We then generalise the second way by presenting a construction of a 4D root system from any given 3D one. This affords a new, spinorial, perspective on 4D phenomena, in particular as the induced root systems are precisely the exceptional ones in 4D, and their unusual automorphism groups are easily explained in the spinorial picture; we discuss the wider context of Platonic solids, Arnold’s trinities and the McKay correspondence. The multivector groups can be used to perform concrete group-theoretic calculations, e.g. those for \(H_3\) and \(E_8\), and we discuss how various representations can also be constructed in this Clifford framework; in particular, representations of quaternionic type arise very naturally.

Keywords

Root System Simple Root Clifford Algebra Dynkin Diagram Coxeter Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgments

I would like to thank Reidun Twarock, Anne Taormina, David Hestenes, Anthony Lasenby, John Stillwell, Jozef Siran, Robert Wilson and Ben Fairbairn.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK

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