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Sphere Packings, Lattices and Groups
John ConwayNeil J. A. Sloane
出版Springer Science & Business Media, 2013-06-29
主題Mathematics / AppliedMathematics / Algebra / AbstractTechnology & Engineering / Engineering (General)Science / Chemistry / GeneralMathematics / GeneralMathematics / Group TheoryComputers / Artificial Intelligence / General
ISBN14757656819781475765687
URLhttp://books.google.com.hk/books?id=5-UlBQAAQBAJ&hl=&source=gbs_api
EBookSAMPLE
註釋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.