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The lattices of six-dimensional Euclidean space


Authors: W. Plesken and W. Hanrath
Journal: Math. Comp. 43 (1984), 573-587
MSC: Primary 11H06; Secondary 20H15, 52A43
DOI: https://doi.org/10.1090/S0025-5718-1984-0758205-5
MathSciNet review: 758205
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Abstract: The lattices of full rank of the six-dimensional Euclidean space are classified according to their automorphism groups (Bravais classification). We find 826 types of such lattices.


References [Enhancements On Off] (What's this?)

  • [1] H. Brown, R. Bülow, J. Neubüser, H. Wondratschek & H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space, Wiley-Interscience, New York, 1978.
  • [2] H. Brown, J. Neubüser & H. Zassenhaus, "On integral groups II; The irreducible case," Numer. Math., v. 20, 1972, pp. 22-31. MR 0318335 (47:6882)
  • [3] J. Cannon, A General Purpose Group Theory Program, Lecture Notes in Math., Vol. 372, Springer-Verlag, Berlin and New York, 1973. MR 0354823 (50:7300)
  • [4] J. D. Jarrat, The Decomposition of Crystal Families, Report Series No. 151, Dept. of Math., Univ. of Auckland, New Zealand, 1979. MR 578269 (81i:20057)
  • [5] J. Neubüser, Computing Moderately Large Groups: Some Methods and Applications, SIAM-AMS-Proc., Vol. 4, Amer. Math. Soc., Providence, R. I., 1971. MR 0379633 (52:538)
  • [6] J. Neubüser, W. Plesken & H. Wondratschek, An Emendatory Discursion on Defining Crystal Systems, Proc. Conf. on Crystallographic Groups (Univ. Bielefeld, Bielefeld, 1979), Match No. 10, 1981, pp. 77-96. MR 620802 (82i:20059)
  • [7] W. Plesken, Beiträge zur Bestimmung der endlichen irreduziblen Untergruppen von $ GL(n,{\mathbf{Z}})$, Dissertation, RWTH Aachen, 1974.
  • [8] W. Plesken, "The Bravais group and the normalizer of a reducible finite subgroup of $ GL(n,{\mathbf{Z}})$," Comm. Algebra, v. 5, 1977, pp. 375-396. MR 0439948 (55:12829)
  • [9] W. Plesken, "On reducible and decomposable representations of orders," J. Reine Angew. Math., v. 297, 1978, pp. 188-210. MR 0466193 (57:6073)
  • [10] W. Plesken, Bravais Groups in Low Dimensions, Proc. Conf. on Crystallographic Groups (Univ. Bielefeld, Bielefeld, 1979), Part II, Match No. 10, 1981, pp. 97-119. MR 620803 (82k:20077)
  • [11] W. Plesken, "Applications of the theory of orders to crystallographic groups," in Integral Representations and Applications (Proceedings, Oberwolfach, Germany 1980); edited by K. W. Roggenkamp, Lecture Notes in Math., Vol. 882, Springer-Verlag, Berlin and New York, 1981, pp. 37-92. MR 646093 (83i:82030)
  • [12] W. Plesken, "Counting with groups and rings," J. Reine Angew. Math., v. 334, 1982, pp. 40-68. MR 667449 (84a:20008)
  • [13] W. Plesken & M. Pohst, "On maximal finite irreducible subgroups of $ GL(n,{\mathbf{Z}})$. I. The five and seven dimensional case," Math. Comp., v. 31, 1977, pp. 536-551; "II. The six dimensional case," ibid., pp. 552-573. MR 0444789 (56:3137a)
  • [14] W. Plesken & M. Pohst, "On maximal finite irreducible subgroups of $ GL(n,{\mathbf{Z}})$. III. The nine dimensional case," Math. Comp., v. 34, 1980, pp. 245-258; "IV. Remarks on even dimensions with applications to $ n = 8$," ibid., pp. 259-275; "V. The eight dimensional case and a complete description of dimensions less than ten," ibid., pp. 277-301. MR 551303 (81b:20012a)
  • [15] S. S. Ryskow & S. D. Lomakina, "The proof of the theorems on the maximal finite groups of integral $ 5 \times 5$-matrices," Trudy Mat. Inst. Steklov., v. 152, 1980, pp. 204-215. (Russian) MR 603825 (82e:20058)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1984-0758205-5
Keywords: Bravais lattices, integral matrix groups
Article copyright: © Copyright 1984 American Mathematical Society

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