Activities per year
Abstract
In this paper, we discuss a Clifford algebra framework for discrete symmetries  e.g. reflection, Coxeter, conformal, modular groups  that also leads to a surprising number of new results in itself. Clifford algebra affords a particularly simple description for performing reflections (via `sandwiching' with vectors in the Clifford algebra), and since via the CartanDieudonn\'e theorem all orthogonal transformations can be written as products of reflections, all such operations can be performed via `sandwiching' with Clifford algebra multivectors. We begin by viewing the largest noncrystallographic 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 this latter observation and present a procedure by which starting with any 3D root system one constructs a corresponding 4D root system. 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.
Original language  English 

Title of host publication  Symmetries in Graphs, Maps, and Polytopes 
Publisher  Springer 
Pages  8195 
ISBN (Electronic)  9783319304519 
ISBN (Print)  9783319304496 
DOIs  
Publication status  Published  Apr 2016 
Event  5th SIGMAP Workshop  West Malvern, United Kingdom Duration: 7 Jul 2014 → 11 Jul 2014 
Publication series
Name  Springer Proceedings in Mathematics & Statistics 

Publisher  Springer 
Volume  159 
ISSN (Print)  21941009 
Conference
Conference  5th SIGMAP Workshop 

Country/Territory  United Kingdom 
City  West Malvern 
Period  7/07/14 → 11/07/14 
Bibliographical note
Dechant, P.P., 2014. A 3D spinorial view of 4D exceptional phenomena. In Symmetries in Graphs, Maps, and Polytopes (pp. 8195). Springer International Publishing.Activities
 1 Workshop

Yau Institute: Tsinghua Summer Workshop in Geometry and Physics 2017
PierrePhilippe Dechant (Keynote/plenary speaker)
6 Aug 2017 → 12 Aug 2017Activity: Talk or presentation › Workshop