Sphere Packings, Lattices and Groups

Couverture
Springer Science & Business Media, 7 déc. 1998 - 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.
 

Table des matières

IV
lxxiii
VI
1
VII
5
VIII
6
IX
8
X
15
XII
18
XIII
20
CLXXXVIII
280
CLXXXIX
281
CXC
284
CXCI
286
CXCII
287
CXCIII
288
CXCV
289
CXCVI
290

XIV
21
XV
23
XVI
25
XVIII
27
XIX
30
XX
34
XXI
35
XXIII
36
XXIV
38
XXV
41
XXVI
44
XXVII
46
XXVIII
50
XXIX
53
XXXI
57
XXXV
60
XXXVI
63
XXXVII
65
XXXVIII
69
XL
71
XLI
73
XLIII
75
XLIV
76
XLV
77
XLVI
78
XLVII
79
XLVIII
80
XLIX
81
LII
82
LIV
83
LV
84
LVI
86
LVII
88
LIX
89
LX
93
LXI
95
LXII
96
LXIII
100
LXIV
102
LXV
104
LXVI
106
LXVII
107
LXIX
109
LXX
110
LXXI
111
LXXII
112
LXXIII
113
LXXIV
114
LXXV
118
LXXVI
119
LXXVII
121
LXXVIII
123
LXXIX
125
LXXX
130
LXXXI
131
LXXXV
132
LXXXVIII
133
LXXXIX
134
XC
135
XCII
136
XCV
138
XCVII
139
XCVIII
140
C
141
CI
142
CIII
143
CV
144
CVIII
145
CX
146
CXI
147
CXII
149
CXIII
151
CXVI
157
CXVII
162
CXVIII
164
CXIX
168
CXX
170
CXXI
171
CXXII
173
CXXIII
175
CXXVII
176
CXXVIII
179
CXXIX
183
CXXX
185
CXXXII
187
CXXXIII
191
CXXXIV
196
CXXXV
199
CXXXVI
200
CXXXIX
201
CXLI
204
CXLII
205
CXLIII
209
CXLIV
215
CXLV
216
CXLVI
218
CXLIX
221
CLI
223
CLII
226
CLIV
227
CLV
229
CLVI
230
CLVII
232
CLVIII
239
CLX
243
CLXI
244
CLXII
246
CLXIII
247
CLXIV
250
CLXV
251
CLXVII
252
CLXVIII
254
CLXIX
257
CLXX
259
CLXXI
261
CLXXVI
263
CLXXIX
265
CLXXX
267
CLXXXII
268
CLXXXIV
270
CLXXXV
272
CLXXXVI
273
CLXXXVII
277
CXCVII
293
CC
294
CCI
296
CCIII
297
CCIV
299
CCV
301
CCVI
302
CCVII
303
CCVIII
305
CCIX
308
CCX
310
CCXI
312
CCXII
313
CCXIII
314
CCXIV
317
CCXV
321
CCXVI
325
CCXIX
331
CCXXII
332
CCXXIII
334
CCXXVII
336
CCXXVIII
338
CCXXIX
339
CCXXX
343
CCXXXI
346
CCXXXV
348
CCXXXVII
349
CCXXXVIII
350
CCXL
351
CCXLI
353
CCXLII
358
CCXLIII
360
CCXLV
361
CCXLVII
362
CCXLVIII
363
CCXLIX
364
CCLI
366
CCLII
367
CCLIII
369
CCLIV
371
CCLV
372
CCLVI
373
CCLVII
374
CCLIX
375
CCLX
376
CCLXII
378
CCLXIII
379
CCLXV
380
CCLXVI
382
CCLXVIII
383
CCLXIX
384
CCLXXI
385
CCLXXII
386
CCLXXIII
387
CCLXXIV
390
CCLXXVI
393
CCLXXVIII
396
CCLXXIX
400
CCLXXXIII
402
CCLXXXIV
402
CCLXXXV
405
CCLXXXVI
413
CCXC
421
CCXCI
422
CCXCII
425
CCXCIII
428
CCXCIV
431
CCXCV
433
CCXCIX
435
CCC
437
CCCIV
438
CCCV
440
CCCVI
441
CCCVII
442
CCCIX
443
CCCXIII
445
CCCXV
446
CCCXVIII
447
CCCXX
448
CCCXXI
449
CCCXXIV
450
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
CCCXLVII
508
CCCXLVIII
513
CCCXLIX
516
CCCLI
517
CCCLII
521
CCCLV
522
CCCLVI
526
CCCLX
535
CCCLXI
541
CCCLXIII
544
CCCLXIV
548
CCCLXVIII
550
CCCLXX
551
CCCLXXII
552
CCCLXXIII
553
CCCLXXV
554
CCCLXXVI
555
CCCLXXVIII
556
CCCLXXIX
557
CCCLXXXI
558
CCCLXXXII
560
CCCLXXXIII
561
CCCLXXXIV
562
CCCLXXXVII
566
CCCLXXXVIII
634
CCCLXXXIX
673
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