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# From the Trinity $(A_3, B_3, H_3)$ to an $ADE$ correspondence

Dechant, Pierre-Philippe (2017) From the Trinity $(A_3, B_3, H_3)$ to an $ADE$ correspondence. Acta Mathematica. (In Press)

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## Abstract

In this paper we present a novel $ADE$ correspondence. In previous work we have constructed the Trinity of exceptional 4D root systems $(D_4, F_4, H_4)$from the Trinity of symmetries of the Platonic solids $(A_3, B_3, H_3)$ using a simple Clifford algebraic construction. Arnold's original indirect link between these two Trinities was very cumbersome and involved exponents of the 4D groups showing up in the Springer cone Weyl chamber decomposition of the Platonic root systems.The above Clifford construction in fact included additional cases beyond the Trinity part since its general construction yielded a 4D root system for each 3D root system, thus extending the correspondence to $(A_1\times I_2(n), A_3, B_3, H_3)$ and $(I_2(n)\times I_2(n), D_4, F_4, H_4)$. We show that Arnold's original link also extends to these extra cases, establishing it more firmly as a novel correspondence. Since the 4D root systems are closely related to the discrete subgroups of $SU(2)$, which in turn are in an $ADE$ correspondence with the Lie algebras$(A_n, D_n, E_6, E_7, E_8)$ via the McKay correspondence, this makes an indirect link between these 3D root systems and the $ADE$ cases. However, since the latter includes two countably infininite families we argue that the 2D root systems $I_2(n)$ should be included in the correspondence, enlarging it to $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$. Our original observation that the Trinity $(12, 18, 30)$ is simultaneously the number of roots of $(A_3, B_3, H_3)$, the sum of the dimensions of irreducible representations of the binary polyhedral groups and the Coxeter numbers of $(E_6, E_7, E_8)$ in fact extends to the full correspondence: $(2n, 2n+2, 12, 18, 30)$. We finish by extending another correspondence between Trinities to a full $ADE$ correspondence between root systems. There is a connection between $(A_3, B_3, H_3)$ and $(E_6, E_7, E_8)$ based on the triples $(233, 234, 235)$ of orders of rotations and lengths of legs in the Dynkin diagrams, respectively. Extending this construction to the full newly proposed set $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$ yield precisely the $(A_n, D_{n+2}, E_6, E_7, E_8)$ Coxeter-Dynkin diagrams.

Item Type: Article In Press Q Science > QA Mathematics STEM http://ray.yorksj.ac.uk/id/eprint/3374

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