Sphere Packings, Lattices and Groups [electronic resource] /

We now apply the algorithm above to find the 121 orbits of norm -2 vectors from the (known) nann 0 vectors, and then apply it again to find the 665 orbits of nann -4 vectors from the vectors of nann 0 and -2. The neighbors of a strictly 24 dimensional odd unimodular lattice can be found as follows. If a norm -4 vector v E II . corresponds to the sum 25 1 of a strictly 24 dimensional odd unimodular lattice A and a !-dimensional lattice, then there are exactly two nonn-0 vectors of ll25,1 having inner product -2 with v, and these nann 0 vectors correspond to the two even neighbors of A. The enumeration of the odd 24-dimensional lattices. Figure 17.1 shows the neighborhood graph for the Niemeier lattices, which has a node for each Niemeier lattice. If A and B are neighboring Niemeier lattices, there are three integral lattices containing A n B, namely A, B, and an odd unimodular lattice C (cf. [Kne4]). An edge is drawn between nodes A and B in Fig. 17.1 for each strictly 24-dimensional unimodular lattice arising in this way. Thus there is a one-to-one correspondence between the strictly 24-dimensional odd unimodular lattices and the edges of our neighborhood graph. The 156 lattices are shown in Table 17 .I. Figure I 7. I also shows the corresponding graphs for dimensions 8 and 16.

Bibliographic Details

Main Authors:	Conway, J. H. author., Sloane, N. J. A. author., SpringerLink (Online service)
Format:	Texto biblioteca
Language:	eng
Published:	New York, NY : Springer New York : Imprint: Springer, 1999
Subjects:	Mathematics., Chemometrics., Group theory., Applied mathematics., Engineering mathematics., Computational intelligence., Applications of Mathematics., Group Theory and Generalizations., Computational Intelligence., Math. Applications in Chemistry.,
Online Access:	http://dx.doi.org/10.1007/978-1-4757-6568-7
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