Complex and integral laminated lattices

Conway, J. H.; Sloane, N. J. A.

doi:10.1090/S0002-9947-1983-0716832-0

Complex and integral laminated lattices
HTML articles powered by AMS MathViewer

by J. H. Conway and N. J. A. Sloane

Trans. Amer. Math. Soc. 280 (1983), 463-490

DOI: https://doi.org/10.1090/S0002-9947-1983-0716832-0

PDF | Request permission

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

A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI 10.1112/S0025579300002588
H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1935), no. 1, 1–15. MR 1545485, DOI 10.1007/BF01201341
Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379–436. MR 422448
J. H. Conway, Three lectures on exceptional groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 215–247. MR 0338152
Michael P. J. Curran (ed.), Topics in group theory and computation, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. MR 0460427
S. Norton, A bound for the covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1779, 259–260. MR 660414, DOI 10.1098/rspa.1982.0041
J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15 (1982), no. 1, 83–94. MR 666350, DOI 10.1016/0022-314X(82)90084-1
J. H. Conway and N. J. A. Sloane, Laminated lattices, Ann. of Math. (2) 116 (1982), no. 3, 593–620. MR 678483, DOI 10.2307/2007025
J. H. Conway and N. J. A. Sloane, The unimodular lattices of dimension up to $23$ and the Minkowski-Siegel mass constants, European J. Combin. 3 (1982), no. 3, 219–231. MR 679207, DOI 10.1016/S0195-6698(82)80034-6
J. H. Conway and N. J. A. Sloane, The Coxeter-Todd lattice, the Mitchell group, and related sphere packings, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 421–440. MR 698347, DOI 10.1017/S0305004100060746
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 2180, DOI 10.2307/2371466

Extreme forms

3

Regular complex polytopes

H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913
H. S. M. Coxeter and J. A. Todd, An extreme duodenary form, Canad. J. Math. 5 (1953), 384–392. MR 55381, DOI 10.4153/cjm-1953-043-4
R. T. Curtis, On subgroups of $^{\ast } O$. I. Lattice stabilizers, J. Algebra 27 (1973), 549–573. MR 340404, DOI 10.1016/0021-8693(73)90064-1
R. T. Curtis, A new combinatorial approach to $M_{24}$, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 25–42. MR 399247, DOI 10.1017/S0305004100052075
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909

Uber die Zahlentheorie der Quaternionen

Martin Kneser, Klassenzahlen definiter quadratischer Formen, Arch. Math. 8 (1957), 241–250 (German). MR 90606, DOI 10.1007/BF01898782
John Leech and N. J. A. Sloane, Sphere packings and error-correcting codes, Canadian J. Math. 23 (1971), 718–745. MR 285994, DOI 10.4153/CJM-1971-081-3
J. H. Lindsey II, A correlation between $\textrm {PSU}_{4}\,(3)$, the Suzuki group, and the Conway group, Trans. Amer. Math. Soc. 157 (1971), 189–204. MR 283097, DOI 10.1090/S0002-9947-1971-0283097-8
J. H. Lindsey II, On the Suzuki and Conway groups, Representation theory of finite groups and related topics (Proc. Sympos. Pure Math., Vol. XXI, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 107–109. MR 0316552
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215

The theory of error-correcting codes

John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372
Howard H. Mitchell, Determination of All Primitive Collineation Groups in More than Four Variables which Contain Homologies, Amer. J. Math. 36 (1914), no. 1, 1–12. MR 1506202, DOI 10.2307/2370513
Hans-Volker Niemeier, Definite quadratische Formen der Dimension $24$ und Diskriminante $1$, J. Number Theory 5 (1973), 142–178 (German, with English summary). MR 316384, DOI 10.1016/0022-314X(73)90068-1
S. Norton, A bound for the covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1779, 259–260. MR 660414, DOI 10.1098/rspa.1982.0041
O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Band 117, Springer-Verlag, New York-Heidelberg, 1971. Second printing, corrected. MR 0347768
W. Plesken and M. Pohst, Constructing integral lattices with prescribed minimum. I, Math. Comp. 45 (1985), no. 171, 209–221, S5–S16. MR 790654, DOI 10.1090/S0025-5718-1985-0790654-2

On integral lattice constructions

3

G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
N. J. A. Sloane, Self-dual codes and lattices, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978) Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 273–308. MR 525331
H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050
N. M. Vetčinkin, Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for $6\leq n\leq 8$, Trudy Mat. Inst. Steklov. 152 (1980), 34–86, 237 (Russian). Geometry of positive quadratic forms. MR 603814
G. L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Math. (Rozprawy Mat.) 84 (1971), 42. MR 318061
Robert A. Wilson, The maximal subgroups of Conway’s group $\bfcdot 2$, J. Algebra 84 (1983), no. 1, 107–114. MR 716772, DOI 10.1016/0021-8693(83)90069-8