Activities per year
Abstract
E8 is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eightdimensional (8D) space very different from the space we inhabit; for instance, the Lie group E8 features heavily in 10D superstring theory. Contrary to that point of view, here we show that the E8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3D geometry. The 240 roots of E8 arise in the 8D Clifford algebra of 3D space as a double cover of the 120 elements of the icosahedral group, generated by the root system H3. As a byproduct, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.
Original language  English 

Journal  Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 
Early online date  13 Jan 2016 
DOIs  
Publication status  Published  2016 
Bibliographical note
© Author 2015. This content is made available by the publisher under a Creative Commons CC BY LicenceKeywords
 E8
 Icosahedral symmetry
 Clifford algebras
 exceptional Lie algebras
 viruses
 spinors
Profiles

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

Chern Institute: Nankai Symposium on Physics, Geometry and Number Theory
PierrePhilippe Dechant (Keynote/plenary speaker)
30 Jul 2017 → 5 Aug 2017Activity: Talk or presentation › Symposium