Hexagons of Christaller and Lösch: Using Equivariant Branching Lemma

Abstract

Following the framework of group-theoretic bifurcation analysis for the hexagonal lattice with \(\mathrm{D}_{6} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})\)-symmetry in Chaps. 5– 7, we highlight the equivariant branching lemma as a pertinent and sufficient means to test the existence of hexagonal bifurcating patterns on the hexagonal lattice. By the application of this lemma to the irreducible representations of the group \(\mathrm{D}_{6} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})\), all hexagonal distributions of Christaller and Losch (Chaps. 1 and 4) are shown to appear at critical points of multiplicity 2, 3, 6, or 12 for appropriate lattice sizes n. A complete classification of these hexagonal distributions is presented. As a main technical contribution of this book, a complete analysis of bifurcating solutions for hexagonal distributions from critical points of multiplicity 12 is conducted. In particular, hexagons of different types are shown to emerge simultaneously at bifurcation points of multiplicity 12 of certain types.