Irreducible Representations of the Group for Hexagonal Lattice

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An n × n hexagonal lattice was introduced as a two-dimensional discretized uniform space for economic agglomeration, and the symmetry of this lattice was described in Chap. 5 by the group \(\mathrm{D}_{6} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})\), which is the semidirect product of D6 by \(\mathbb{Z}_{n} \times \mathbb{Z}_{n}\). In this chapter, the irreducible representations of this group are found according to a standard procedure in group representation theory known as the method of little groups, which exploits the semidirect product structure of the group. The group has a structure that admits irreducible representations of various kinds; the dimensions of the irreducible representations over \(\mathbb{R}\) are 1, 2, 3, 4, 6, or 12. The concrete forms of these irreducible representations are presented. These forms will be used in the group-theoretic bifurcation analysis in Chaps. 8 and 9 to prove the existence of the hexagonal distributions of Christaller and Losch.