Abstract
The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.
Highlights
The discovery of quasicrystals in 1984 by Shechtman et al has spurred the mathematical and physical community to develop mathematical tools in order to study structures with noncrystallographic symmetry.Quasicrystals are alloys with five, eight, ten- and 12-fold symmetry in their atomic positions (Steurer, 2004), and they cannot be organized as lattices
In this work we explored the subgroup structure of the hyperoctahedral group in six dimensions
In particular we found the class of the crystallographic representations of the icosahedral group, whose size is 192
Summary
The discovery of quasicrystals in 1984 by Shechtman et al has spurred the mathematical and physical community to develop mathematical tools in order to study structures with noncrystallographic symmetry. The noncrystallographic symmetry leaves a lattice invariant in higher dimensions, providing an integral representation of G If such a representation is reducible and contains a two- or three-dimensional invariant subspace, it is referred to as a crystallographic representation, following terminology given by Levitov & Rhyner (1988). The rationale behind this approach is that the two corresponding lattice groups share a common subgroup These two approaches are shown to be related (Indelicato et al, 2012), the idea is that it is possible to study the transitions between icosahedral quasicrystals by considering two distinct crystallographic representations of I in B6 which share a common subgroup. In x5 we study their subgroup structure, introducing the concept of G-graph, where G is a subgroup of I
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