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群与格引论 有限群与正定有理格 英文PDF|Epub|txt|kindle电子书版本网盘下载
- （美）Robert L. Griess 著
- 出版社： 北京：高等教育出版社
- ISBN：9787040292053
- 出版时间：2010
- 标注页数：251页
- 文件大小：11MB
- 文件页数：260页
- 主题词：有限群－英文；格－英文
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图书目录
1 Introduction1
1.1 Outline of the book2
1.2 Suggestions for further reading3
1.3 Notations,background,conventions5
2 Bilinear Forms,Quadratic Forms and Their Isometry Groups7
2.1 Standard results on quadratic forms and reflections,Ⅰ9
2.1.1 Principal ideal domains(PIDs)10
2.2 Linear algebra11
2.2.1 Interpretation of nonsingularity11
2.2.2 Extension of scalars13
2.2.3 Cyclicity of the values of a rational bilinear form13
2.2.4 Gram matrix14
2.3 Discriminant group16
2.4 Relations between a lattice and sublattices18
2.5 Involutions on quadratic spaces19
2.6 Standard results on quadratic forms and reflections,Ⅱ20
2.6.1 Involutions on lattices20
2.7 Scaled isometries:norm doublers and triplers23
3 General Results on Finite Groups and Invariant Lattices25
3.1 Discreteness of rational lattices25
3.2 Finiteness of the isometry group25
3.3 Construction of a G-invariant bilinear form26
3.4 Semidirect products and wreath products27
3.5 Orthogonal decomposition of lattices28
4 Root Lattices of Types A,D,E31
4.1 Background from Lie theory31
4.2 Root lattices,their duals and their isometry groups32
4.2.1 Definition of the An lattices33
4.2.2 Definition of the Dn lattices34
4.2.3 Definition of the En lattices34
4.2.4 Analysis of the An root lattices34
4.2.5 Analysis of the Dn root lattices37
4.2.6 More on the isometry groups of type Dn39
4.2.7 Analysis of the En root lattices41
5 Hermite and Minkowski Functions49
5.1 Small ranks and small determinants51
5.1.1 Table for the Minkowski and Hermite functions52
5.1.2 Classifications of small rank,small determinant lattices53
5.2 Uniqueness of the lattices E6,E7 and E854
5.3 More small ranks and small determinants57
6 Constructions of Lattices by Use of Codes61
6.1 Definitions and basic results61
6.1.1 A construction of the E8-lattice with the binary[8,4,4]code62
6.1.2 A construction of the E8-lattice with the ternary[4,2,3]code64
6.2 The proofs64
6.2.1 About power sets,boolean sums and quadratic forms64
6.2.2 Uniqueness of the binary[8,4,4]code65
6.2.3 Reed-Muller codes66
6.2.4 Uniqueness of the tetracode67
6.2.5 The automorphism group of the tetracode67
6.2.6 Another characterization of[8,4,4]269
6.2.7 Uniqueness of the E8-lattice implies uniqueness of the binary[8,4,4]code69
6.3 Codes over F7 and a(mod 7)-construction of E870
6.3.1 The A6-lattice71
7 Group Theory and Representations73
7.1 Finite groups73
7.2 Extraspecial p-groups75
7.2.1 Extraspecial groups and central products75
7.2.2 A normal form in an extraspecial group77
7.2.3 A classification of extraspecial groups77
7.2.4 An application to automorphism groups of extraspecial groups79
7.3 Group representations79
7.3.1 Representations of extraspecial p-groups80
7.3.2 Construction of the BRW groups82
7.3.3 Tensor products85
7.4 Representation of the BRW group G86
7.4.1 BRW groups as group extensions88
8 Overview of the Barnes-Wall Lattices91
8.1 Some properties of the series91
8.2 Commutator density93
8.2.1 Equivalence of 2/4-,3/4-generation and commutator density for Dih893
8.2.2 Extraspecial groups and commutator density96
9 Construction and Properties of the Barnes-Wall Lattices99
9.1 The Barnes-Wall series and their minimal vectors99
9.2 Uniqueness for the BW lattices101
9.3 Properties of the BRW groups102
9.4 Applications to coding theory103
9.5 More about minimum vectors104
10 Even unimodular lattices in small dimensions107
10.1 Classifications of even unimodular lattices107
10.2 Constructions of some Niemeier lattices108
10.2.1 Construction of a Leech lattice109
10.3 Basic theory of the Golay code111
10.3.1 Characterization of certain Reed-Muller codes111
10.3.2 About the Golay code112
10.3.3 The octad Triangle and dodecads113
10.3.4 A uniqueness theorem for the Golay code116
10.4 Minimal vectors in the Leech lattice116
10.5 First proof of uniqueness of the Leech lattice117
10.6 Initial results about the Leech lattice118
10.6.1 An automorphism which moves the standard frame118
10.7 Turyn-style construction of a Leech lattice119
10.8 Equivariant unimodularizations of even lattices121
11 Pieces of Eight125
11.1 Leech trios and overlattices125
11.2 The order of the group O(∧)128
11.3 The simplicity of M24130
11.4 Sublattices of Leech and subgroups of the isometry group132
11.5 Involutions on the Leech lattice134
References137
Index143
Appendix A The Finite Simple Groups149
Appendix B Reprints of Selected Articles153
B.1 Pieces of Eight:Semiselfdual Lattices and a New Foundation for the Theory of Conway and Mathieu Groups155
B.2 Pieces of 2d:Existence and Uniqueness for Barnes-Wall and Ypsilanti Lattices181
B.3 Involutions on the Barnes-Wall Lattices and Their Fixed Point Sublattices,Ⅰ223