Abstract
Group-theoretic properties of the symmetry group \(\mathrm{D}_{6} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})\) of the hexagonal lattice, such as subgroups and irreducible representations, were derived in Chaps. 5 and 6. In this chapter, the matrix representation of this group for the economy on the hexagonal lattice is investigated in preparation for the group-theoretic bifurcation analysis in search of bifurcating hexagonal patterns in Chaps. 8 and 9. Irreducible decomposition of the matrix representation is obtained with the aid of characters to identify irreducible representations that are relevant to our analysis of the mathematical model of an economy on the hexagonal lattice. Formulas for the transformation matrix for block-diagonalization of the Jacobian matrix of the equilibrium equation of the economy on the hexagonal lattice are derived and put to use in numerical bifurcation analysis of hexagonal patterns.
Published Version
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