Sphere Packings, Lattices and Groups

John Conway, Neil J. A. Sloane

Springer Science & Business Media, Dec 7, 1998 - Mathematics - 706 pages

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.

Contents

• IV: lxxiii
• IX: 1
• X: 5
• XI: 6
• XII: 9
• XIII: 15
• XV: 18
• XVI: 20
• XVII: 21
• XVIII: 23
• XIX: 25
• XXII: 27
• XXIII: 30
• XXIV: 34
• XXV: 35
• XXVII: 36
• XXVIII: 38
• XXIX: 41
• XXX: 44
• XXXI: 46
• XXXII: 50
• XXXIV: 53
• XXXVI: 57
• XXXIX: 60
• XL: 63
• XLI: 65
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• XLV: 73
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• XLIX: 77
• L: 78
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• LIII: 81
• LVI: 82
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• LIX: 84
• LX: 86
• LXI: 88
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• LXIV: 93
• LXV: 95
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• LXIX: 104
• LXX: 106
• LXXI: 107
• LXXIII: 109
• LXXIV: 110
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• LXXVII: 113
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• LXXIX: 118
• LXXX: 119
• LXXXI: 121
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• XC: 132
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• CI: 139
• CIII: 140
• CV: 141
• CVI: 142
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• CX: 144
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• CXXXIX: 196
• CXL: 199
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• CXLIX: 209
• CL: 215
• CLI: 216
• CLII: 218
• CLV: 221
• CLVII: 223
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• CLX: 227
• CLXI: 229
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• CLXIV: 239
• CLXVII: 243
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• CLXIX: 246
• CLXX: 247
• CLXXI: 250
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• CLXXXII: 263
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• CXC: 270
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• CXCVI: 284
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• CXCIX: 288
• CCI: 289
• CCII: 290
• CCIII: 293
• CCVI: 294
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• CCIX: 297
• CCX: 299
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• CCLI: 362
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• CCLXIX: 380
• CCLXX: 382
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• CCLXXV: 385
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• CCLXXVIII: 390
• CCLXXX: 393
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• CCLXXXIX: 413
• CCXCIII: 421
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• CCXCV: 425
• CCXCVI: 428
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• CCCII: 435
• CCCIII: 437
• CCCVI: 438
• CCCVII: 440
• CCCVIII: 441
• CCCIX: 442
• CCCXI: 443
• CCCXV: 445
• CCCXVII: 446
• CCCXIX: 447
• CCCXXI: 448
• CCCXXII: 449
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• CCCXXVI: 453
• CCCXXVII: 456
• CCCXXIX: 457
• CCCXXX: 466
• CCCXXXI: 468
• CCCXXXII: 470
• CCCXXXV: 472
• CCCXXXVIII: 474
• CCCXXXIX: 478
• CCCXL: 489
• CCCXLI: 496
• CCCXLII: 500
• CCCXLIV: 504
• CCCXLV: 507
• CCCXLVI: 508
• CCCXLVII: 513
• CCCXLVIII: 516
• CCCL: 517
• CCCLI: 521
• CCCLV: 522
• CCCLVI: 526
• CCCLX: 535
• CCCLXI: 541
• CCCLXIII: 544
• CCCLXIV: 548
• CCCLXVIII: 550
• CCCLXXI: 551
• CCCLXXIII: 552
• CCCLXXIV: 553
• CCCLXXVI: 554
• CCCLXXVII: 555
• CCCLXXIX: 556
• CCCLXXX: 557
• CCCLXXXII: 558
• CCCLXXXIII: 560
• CCCLXXXIV: 561
• CCCLXXXV: 562
• CCCLXXXVIII: 566
• CCCLXXXIX: 634
• CCCXC: 673

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Bibliographic information

• Title: Sphere Packings, Lattices and Groups; Volume 290 of Grundlehren der mathematischen Wissenschaften
• Authors: John Conway, Neil J. A. Sloane
• Edition: 3, illustrated
• Publisher: Springer Science & Business Media, 1998
• ISBN: 0387985859, 9780387985855
• Length: 706 pages
• Subjects: Computers / Intelligence (AI) & Semantics; Mathematics / Algebra / Abstract; Mathematics / Applied; Mathematics / General; Mathematics / Group Theory; Science / Chemistry / General