A 3D spinorial view of 4D exceptional phenomena

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Publication details

Title of host publicationSymmetries in Graphs, Maps, and Polytopes
DateAccepted/In press - 2015
DatePublished (current) - Apr 2016
Pages81-95
PublisherSpringer-verlag
Original languageEnglish
ISBN (Electronic)978-3-319-30451-9
ISBN (Print)978-3-319-30449-6

Publication series

NameSpringer Proceedings in Mathematics & Statistics
PublisherSpringer
Volume159
ISSN (Print)2194-1009

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 Cartan-Dieudonn\'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 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 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.

Bibliographical note

Dechant, P.P., 2014. A 3D spinorial view of 4D exceptional phenomena. In Symmetries in Graphs, Maps, and Polytopes (pp. 81-95). Springer International Publishing.

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