Colorings of lattices based on subgroup orbits

Abstract

In this paper, orbit colorings of a lattice Λ were considered, extending the idea of obtaining lattice colorings based on cosets of a sublattice. Given a subgroup S of finite index in the symmetry group G of the lattice Λ, if T(S) is the subgroup of translations in G and L is the sublattice of Λ determined by T(S), then a homomorphism f from N = N G (T(S)) to the subgroup S C of permutations of the cosets of L in Λ is defined. This homomorphism was used to obtain S-orbit colorings of Λ and the corresponding color group H and color fixing group K. Moreover, S ker f ≤ K ≤ H ≤ N.

Colorings of patterns fixed by an arbitrary finite-index subgroup of the symmetry group

“…These results have been expanded and used to study patterns without the restriction that the pattern is the orbit of a subset of the fundamental domain of the symmetry group (Felix & Gentuya, 2013;Felix & Eclarin, 2014;Loquias & Frettlo ¨h, 2017). Two particularly important research studies in relation to this paper are those of Say-awen et al (2015) and Walo & Felix (2011). The former deals with color fixing groups of colorings, which also serve as the symmetry groups of those colorings.…”