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Bifurcation Theory for Hexagonal Agglomeration in Economic Geography pp 175–202Cite as

Matrix Representation for Economy on Hexagonal Lattice

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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.

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Notes

  1. 1.

    The choice of this matrix Q will be explained in Sect. 7.5, and μ 1, μ 2, and μ 3 are irreducible representations denoted in Sect. 6.3 as \((1; +, +)\), (2; +), and (6; 1, 0, +), respectively.

  2. 2.

    This is a basic fact, sometimes called Gelfand’s lemma, in the context of Gelfand pairs (Diaconis, 1988 [2]; Macdonald, 1995 [3]; Ceccherini-Silberstein, Scarabotti, and Tolli, 2010 [1]). See Proposition 1.4.8 of [1], in particular.

  3. 3.

    The notation ⟨⋅ ⟩ here should not be confused with that for the generators of a group.

  4. 4.

    The theoretical background of this consideration will be given in Sect. 8.4.

  5. 5.

    The theoretical background of this consideration will be given in Sect. 8.5.

  6. 6.

    The theoretical background of this consideration will be given in Sect. 8.7.

References

  1. Ceccherini-Silberstein T, Scarabotti F, Tolli F (2010) Representation theory of the symmetric groups: the Okounkow–Vershik approach, character formulas, and partition algebras. Cambridge studies in advanced mathematics, vol 121. Cambridge University Press, Cambridge

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  2. Diaconis P (1988) Group representations in probability and statistics. Lecture notes-monograph series, vol 11. Institute of Mathematical Statistics, Hayward

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  3. Macdonald IG (1995) Symmetric functions and Hall polynomials, 2nd edn. Oxford mathematical monograph. Oxford University Press, New York

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Authors and Affiliations

  1. Civil and Environmental Engineering Graduate School of Engineering, Tohoku University, Sendai, Japan

    Kiyohiro Ikeda

  2. Information Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo, Japan

    Kazuo Murota

Authors
  1. Kiyohiro Ikeda
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  2. Kazuo Murota
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Ikeda, K., Murota, K. (2014). Matrix Representation for Economy on Hexagonal Lattice. In: Bifurcation Theory for Hexagonal Agglomeration in Economic Geography. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54258-2_7

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  • DOI: https://doi.org/10.1007/978-4-431-54258-2_7

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  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-54257-5

  • Online ISBN: 978-4-431-54258-2

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