Affine extensions of non-crystallographic Coxeter groups induced by projection

Abstract

In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset of the Kac-Moody-type extensions considered in Dechant et al. This class of extensions was motivated by physical applications in icosahedral systems in biology (viruses), physics (quasicrystals) and chemistry (fullerenes). By connecting these here to extensions of E_8, D_6 and A_4, we place them into the broader context of crystallographic lattices such as E_8, suggesting their potential for applications in high energy physics, integrable systems and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.

34 pages, 11 figures

References in corpus (4)

• Dense Packings of the Platonic and Archimedean Solids
• Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups
• Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups
• Platonic solids generate their four-dimensional analogues

Cited by in corpus (11)

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• Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
• The $E_8$ geometry from a Clifford perspective
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• Group Theoretical Analysis of Quasicrystallography from Projections of Higher Dimensional Lattices Bn
• The Birth of $E_8$ out of the Spinors of the Icosahedron
• Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction
• From the Trinity $(A_3,B_3,H_3)$ to an ADE correspondence
• A 3D spinorial view of 4D exceptional phenomena
• Machine Learning Clifford invariants of ADE Coxeter elements
• From Affine $A_4$ to Affine $H_2$: Group Theoretical Analysis of Five-fold Tilings