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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.
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Abstract

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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

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