Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Complex and integral laminated lattices


Authors: J. H. Conway and N. J. A. Sloane
Journal: Trans. Amer. Math. Soc. 280 (1983), 463-490
MSC: Primary 11H99; Secondary 52A43
DOI: https://doi.org/10.1090/S0002-9947-1983-0716832-0
MathSciNet review: 716832
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In an earlier paper we studied real laminated lattices (or $ {\mathbf{Z}}$-modules) $ {\Lambda_n}$, where $ {\Lambda_1}$ is the lattice of even integers, and $ {\Lambda_n}$ is obtained by stacking layers of a suitable $ (n - 1)$-dimensional lattice $ {\Lambda_{n - 1}}$ as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing $ {\mathbf{Z}}$-module by $ J$-module, where $ J$ may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which $ {\Lambda_n}$ is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the $ 6$-dimensional integral laminated lattice over $ {\mathbf{Z}}[ \omega ]$ of minimal norm $ 2$. The paper includes tables of the best real integral lattices in up to $ 24$ dimensions.


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

  • [1] A. Baker, Linear forms in the logarithms of algebraic numbers. I, Mathematika 13 (1966), 204-216. MR 0258756 (41:3402)
  • [2] H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1934), 1-15. MR 1545485
  • [3] A. M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. 9 (1976), 379-436. MR 0422448 (54:10437)
  • [4] I. H. Conway, Three lectures on exceptional groups in Finite Simple Groups (M. B. Powell and G. Higman, eds.), Academic Press, New York, 1971, pp. 215-247. MR 0338152 (49:2918)
  • [5] -, The miracle octad generator in Topics in Group Theory and Computation (M. P. J. Curran, ed.), Academic Press, New York, 1977, pp. 215-247. MR 0460427 (57:421)
  • [6] J. H. Conway, R. A. Parker and N. J. A. Sloane, The covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982), 261-290. MR 660415 (84m:10022b)
  • [7] J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15 (1982), 83-94. MR 666350 (84b:10047)
  • [8] -, Laminated lattices, Ann. of Math. 116 (1982), 593-620. MR 678483 (84c:52015)
  • [9] -, The unimodular lattices of dimension up to $ 23$ and the Minkowski-Siegel mass constants, European J. Combin. 3 (1982), 219-231. MR 679207 (84b:10045)
  • [10] -, The Coxeter-Todd lattice, the Mitchell group, and related sphere packings, Math. Proc. Cambridge Philos. Soc. 93 (1983), 421-440. MR 698347 (84i:10032)
  • [11] H. S. M. Coxeter, The polytope $ {2_{21}}$, whose twenty-seven vertices correspond to the lines on the general cubic surface, Amer. J. Math. 62 (1940), 457-486. MR 0002180 (2:10a)
  • [12] -, Extreme forms, Canad. J. Math. 3 (1951), 391-441.
  • [13] -, Regular complex polytopes, Cambridge Univ. Press, Cambridge, 1974.
  • [14] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Springer-Verlag, New York, 1980. MR 562913 (81a:20001)
  • [15] H. S. M. Coxeter and J. A. Todd, An extreme duodenary form, Canad. J. Math. 5 (1953), 384-392. MR 0055381 (14:1066a)
  • [16] R. T. Curtis, On subgroups of $ \cdot 0.1.$ Lattice stabilizers, J. Algebra 27 (1973), 549-573. MR 0340404 (49:5159)
  • [17] -, A new combinatorial approach to $ {M_{24}}$, Math. Proc. Cambridge Philos. Soc. 79 (1976), 25-42. MR 0399247 (53:3098)
  • [18] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Oxford Univ. Press, Oxford, 1979. MR 568909 (81i:10002)
  • [19] A. Hurwitz, Uber die Zahlentheorie der Quaternionen, Nachr. Gesellschaft Wiss. Göttingen Math.Phys. K1. (1896), 313-340. Reprinted in Math. Werke. Vol. II, Birkhäuser, Basel, 1933, pp. 303-330.
  • [20] M. Kneser, Klassenzahlen definiter quadratischer Formen, Arch. Math. 8 (1957), 241-250. MR 0090606 (19:838c)
  • [21] J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Canad. J. Math. 23 (1971), 718-745. MR 0285994 (44:3211)
  • [22] J. H. Lindsey II, A correlation between $ PSU_4(3)$, the Suzuki group, and the Conway group, Trans. Amer. Math. Soc. 157 (1971), 189-204. MR 0283097 (44:330)
  • [23] -, On the Suzuki and Conway groups in Representation Theory of Finite Groups and Related Topics, Proc. Sympos. Pure Math., vol. 21, Amer. Math. Soc., Providence, R.I., 1971, pp. 107-109. MR 0316552 (47:5099)
  • [24] S. Mac Lane, Homology, Springer-Verlag, New York, 1963. MR 1344215 (96d:18001)
  • [25] F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, 1981.
  • [26] J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. MR 0506372 (58:22129)
  • [27] H. H. Mitchell, Determination of all primitive collineation groups in more than four variables, Amer. J. Math. 36 (1914), 1-12. MR 1506202
  • [28] H. V. Niemeier, Definite quadratische Formen der Dimension $ 24$ und Diskriminante $ 1$, J. Number Theory 5 (1973), 142-178. MR 0316384 (47:4931)
  • [29] S. Norton, A bound for the covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982), 259-260. MR 660414 (84m:10022a)
  • [30] O. T. O'Meara, Introduction to quadratic forms, Springer-Verlag, New York, 1971. MR 0347768 (50:269)
  • [31] W. Plesken and M. Pohst, Constructing integral lattices with prescribed minimum. I, preprint. MR 790654 (87e:11077)
  • [32] M. Pohst, On integral lattice constructions. Abstracts Amer. Math. Soc. 3 (1982), 152; Abstract #793-12-14.
  • [33] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304. MR 0059914 (15:600b)
  • [34] N. J. A. Sloane, Self-dual codes and lattices in Relations Between Combinatorics and Other Parts of Mathematics, Proc. Sympos. Pure Math., vol. 34, Amer. Math. Soc., Providence, R.I., 1979, pp. 273-308. MR 525331 (80f:10037)
  • [35] H. Stark, A complete determination of the complex fields of class-number one, Michigan Math. J. 14 (1967), 1-27. MR 0222050 (36:5102)
  • [36] J. G. Thompson, private communication.
  • [37] N. M. Vetchinkin, Uniqueness of classes of positive quadratic forms on which values of Hermite constants are attained for $ 6 \leqslant n \leqslant 8$, Trudy Mat. Inst. Steklov 152 (1980), 34-86. (Russian). English translation in Proc. Steklov Inst. Math. 1982, issue 3, pp. 37-95. MR 603814 (82f:10040)
  • [38] G. L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Math. 84 (1971), 42 pages. MR 0318061 (47:6610)
  • [39] R. A. Wilson, The complex Leech lattice and maximal subgroups of the Suzuki group, J. Algebra (to appear). MR 716777 (86e:20034b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11H99, 52A43

Retrieve articles in all journals with MSC: 11H99, 52A43


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0716832-0
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society