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- W3100780688 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-Dieudonn'e 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." @default.
- W3100780688 created "2020-11-23" @default.
- W3100780688 creator A5049974896 @default.
- W3100780688 date "2016-01-01" @default.
- W3100780688 modified "2023-09-25" @default.
- W3100780688 title "A 3D Spinorial View of 4D Exceptional Phenomena" @default.
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- W3100780688 doi "https://doi.org/10.1007/978-3-319-30451-9_4" @default.
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