Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T16:37:34.055Z Has data issue: false hasContentIssue false
  • English
  • Français

The Coset lattices of E. S. Barnes and G. E. Wall

Part of: Geometry of numbers Forms and linear algebraic groups

Published online by Cambridge University Press:  09 April 2009

Alexander J. Hahn
Alexander J. Hahn
Affiliation:
University of Notre DameNotre Dame, Indiana 46556, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

John Conway's analysis in 1968 of the automorphism group of the Leech lattice and his discovery of three sporadic simple groups led to the immediate speculation that other Z-lattices might have interesting automorphism groups which give rise to (possibly new) finite simple groups. (The classification theorem for the finite simple groups has since told us that no new finite simple groups can arise in this or any other way.) For example in 1973, M. Broué and M. Enguehard constructed, in every dimension 2n, an even lattice (unimodular if n is odd) whose automorphism group is related to the simple Chevalley group of type Dn. This family of integral lattices received attention and acclaim in the subsequent literature. What escaped the attention of this literature, however, was the fact that these lattices had been discovered years earlier. Indeed in 1959, E. S. Barnes and G. E. Wall gave a uniform construction for a large class of positive definite Z-lattices in dimensions 2n which include those of Broué and Enguehard as special cases. The present article introduces an abstracted and generalized version of the construction of Barnes and Wall. In addition, there are some new observations about Barnes-Wall lattices. In particular, it is shown how to associate to each such lattice a continuous, piecewise linear graph in the plane from which all the important properties of the lattice, for example, its minimum, whether it is integral, unimodular, even, or perfect can be read off directly.

MSC classification

Secondary: 11E12: Quadratic forms over global rings and fields 11E25: Sums of squares and representations by other particular quadratic forms 11H06: Lattices and convex bodies 11H31: Lattice packing and covering 11H55: Quadratic forms (reduction theory, extreme forms, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 3 , December 1990 , pp. 418 - 433
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Barnes, E. S. and Wall, G. E., ‘Some extreme forms defined in terms of Abelian groups’, J. Austral. Math. Soc. 1 (1959), 47–63.CrossRefGoogle Scholar
[2]Broué, M. and Enguehard, M., ‘Une famille infinie de formes quadratiques entieres; leurs groupes d'automorphismes’, Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 17–53.CrossRefGoogle Scholar
[3]Chernyakov, A. G., ‘Examples of a 32-dimensional even unimodular lattice’, J. Soviet Math. 17 (1981), 2068–2075.CrossRefGoogle Scholar
[4]Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, Grundlehren der Math. Wissenschaften 290, (Springer-Verlag, Berlin, Heidelberg, New York, 1988).CrossRefGoogle Scholar
[5]Forney, G. D. Jr, ‘Coset codes. Part I: Introduction and geometrical classification’, IEEE Trans. Information Theory 34 (1988), 1123–1151.CrossRefGoogle Scholar
[6]Forney, G. D. Jr, ‘Coset codes. Part II: Binary lattices and related codes’, IEEE Trans. Information Theory 34 (1988), 1152–1187.CrossRefGoogle Scholar
[7]Koch, H., ‘Unimodular lattices and self-dual codes’, Proc. Internat. Congr. Math., (Berkeley, Calif., 1986, Vol. 1, Amer. Math. Soc., Providence, R. I., 1987).Google Scholar
[8]Milnor, J. and Husemoller, D., Symmetric Bilinear Forms, (Springer, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[9]Niemeier, H.-V., ‘Definite quadratische Formen der Dimension 24 und Diskriminante 1’, J. Number Theory 5 (1973), 142–178.CrossRefGoogle Scholar
[10]O'Meara, O. T., Introduction to Quadratic Forms, Grundlehren der Math. Wissenschaften 117, 2nd ed., (Springer, Berlin, Heidelberg, New York, 1971).Google Scholar
[11]O'Meara, O. T., ‘On indecomposable quadratic forms’, J. Reine Angew. Math. 317 (1980), 120–156.Google Scholar
[12]Steinhausen, G., ‘Definite, gerade Bilinearformen der Diskriminante I’, Bonner Mathematische Schriften 76 (Bonn, 1974).Google Scholar
[13]Wall, G. E., ‘On the Clifford collineations, transform and similarity groups (IV), an application to quadratic forms’, Nagoya Math. 21–22 (1962–1963), 199–222.CrossRefGoogle Scholar