Affine subgroups of the affine Coxeter group with the same Coxeter number

TL;DR

This paper constructs affine subgroups of affine Coxeter groups with the same Coxeter number using graph folding, revealing connections to quasicrystallography and lattice structures.

Contribution

It introduces a general method for constructing affine subgroups and root systems, including non-orthogonal vector sets with practical applications.

Findings

Affine subgroups are constructed via graph folding techniques.

Non-crystallographic groups like W(H3) relate to icosahedral quasicrystals.

New root system formulations aid in lattice and cell construction.

Abstract

Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-1) respectively. The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding. The latter two are the non-crystallographic groups where W(H3) plays a special role in the quasicrystallographic structures with icosahedral symmetry. A general construction of the affine dihedral subgroups is introduced, some of which, describe the existing planar quasicrystallography. In the construction of the root systems, sets of orthonormal vectors are used but a special non-orthogonal set of vectors in the formulation of the root system of W(An) is also introduced which has practical applications in the construction of the lattices An and An* and their Delone and Voronoi cells.