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

Alexander J. Hahn
1990 Journal of the Australian Mathematical Society  
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. Broue and M. Enguehard constructed, in every
more » ... ucted, in every dimension 2" , an even lattice (unimodular if n is odd) whose automorphism group is related to the simple Chevalley group of type D n . 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 2" which include those of Broue 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. 1980 Mathematics subject classification {Amer. Math. Soc.) (1985 Revision): 11 E 12, 11 E 25, 11 H 06, 11 H 31, 11 H 55. I wish to add a personal salute to Tim Wall on the occasion of his retirement. With the very elevated level of his research and with the distinguished quality of his character he has made a profound and lasting contribution to Australian mathematics.
doi:10.1017/s1446788700032420 fatcat:xnvsxisvpndz3b3zcmok7huene