Permutation representations defined by \(G\)-clusters with application to quasicrystals (Q1283206)

It is well-known that practically all quasicrystal models of physical significance can be described in the cut-and-project setting with simple embedding lattices (e.g., root lattices). This article takes a complementary point of view and starts from orbits of \(\mathbb{R}\)-irreps of a finite group \(G\). An embedding into a suitable higher-dimensional space (of usually non-minimal dimension) links it to the cut-and-project setting, but with the extra benefit of having originated from certain symmetric clusters that can be chosen according to what is observed in physics. The self-similarities of such sets are investigated and two examples with icosahedral symmetry are discussed in detail.