群与格引论 有限群与正定有理格 英文PDF格式文档图书下载

1 Introduction 1

1.1 Outline of the book 2

1.2 Suggestions for further reading 3

1.3 Notations,background,conventions 5

2 Bilinear Forms,Quadratic Forms and Their Isometry Groups 7

2.1 Standard results on quadratic forms and reflections,Ⅰ 9

2.1.1 Principal ideal domains(PIDs) 10

2.2 Linear algebra 11

2.2.1 Interpretation of nonsingularity 11

2.2.2 Extension of scalars 13

2.2.3 Cyclicity of the values of a rational bilinear form 13

2.2.4 Gram matrix 14

2.3 Discriminant group 16

2.4 Relations between a lattice and sublattices 18

2.5 Involutions on quadratic spaces 19

2.6 Standard results on quadratic forms and reflections,Ⅱ 20

2.6.1 Involutions on lattices 20

2.7 Scaled isometries:norm doublers and triplers 23

3 General Results on Finite Groups and Invariant Lattices 25

3.1 Discreteness of rational lattices 25

3.2 Finiteness of the isometry group 25

3.3 Construction of a G-invariant bilinear form 26

3.4 Semidirect products and wreath products 27

3.5 Orthogonal decomposition of lattices 28

4 Root Lattices of Types A,D,E 31

4.1 Background from Lie theory 31

4.2 Root lattices,their duals and their isometry groups 32

4.2.1 Definition of the An lattices 33

4.2.2 Definition of the Dn lattices 34

4.2.3 Definition of the En lattices 34

4.2.4 Analysis of the An root lattices 34

4.2.5 Analysis of the Dn root lattices 37

4.2.6 More on the isometry groups of type Dn 39

4.2.7 Analysis of the En root lattices 41

5 Hermite and Minkowski Functions 49

5.1 Small ranks and small determinants 51

5.1.1 Table for the Minkowski and Hermite functions 52

5.1.2 Classifications of small rank,small determinant lattices 53

5.2 Uniqueness of the lattices E6,E7 and E8 54

5.3 More small ranks and small determinants 57

6 Constructions of Lattices by Use of Codes 61

6.1 Definitions and basic results 61

6.1.1 A construction of the E8-lattice with the binary[8,4,4]code 62

6.1.2 A construction of the E8-lattice with the ternary[4,2,3]code 64

6.2 The proofs 64

6.2.1 About power sets,boolean sums and quadratic forms 64

6.2.2 Uniqueness of the binary[8,4,4]code 65

6.2.3 Reed-Muller codes 66

6.2.4 Uniqueness of the tetracode 67

6.2.5 The automorphism group of the tetracode 67

6.2.6 Another characterization of[8,4,4]2 69

6.2.7 Uniqueness of the E8-lattice implies uniqueness of the binary[8,4,4]code 69

6.3 Codes over F7 and a(mod 7)-construction of E8 70

6.3.1 The A6-lattice 71

7 Group Theory and Representations 73

7.1 Finite groups 73

7.2 Extraspecial p-groups 75

7.2.1 Extraspecial groups and central products 75

7.2.2 A normal form in an extraspecial group 77

7.2.3 A classification of extraspecial groups 77

7.2.4 An application to automorphism groups of extraspecial groups 79

7.3 Group representations 79

7.3.1 Representations of extraspecial p-groups 80

7.3.2 Construction of the BRW groups 82

7.3.3 Tensor products 85

7.4 Representation of the BRW group G 86

7.4.1 BRW groups as group extensions 88

8 Overview of the Barnes-Wall Lattices 91

8.1 Some properties of the series 91

8.2 Commutator density 93

8.2.1 Equivalence of 2/4-,3/4-generation and commutator density for Dih8 93

8.2.2 Extraspecial groups and commutator density 96

9 Construction and Properties of the Barnes-Wall Lattices 99

9.1 The Barnes-Wall series and their minimal vectors 99

9.2 Uniqueness for the BW lattices 101

9.3 Properties of the BRW groups 102

9.4 Applications to coding theory 103

9.5 More about minimum vectors 104

10 Even unimodular lattices in small dimensions 107

10.1 Classifications of even unimodular lattices 107

10.2 Constructions of some Niemeier lattices 108

10.2.1 Construction of a Leech lattice 109

10.3 Basic theory of the Golay code 111

10.3.1 Characterization of certain Reed-Muller codes 111

10.3.2 About the Golay code 112

10.3.3 The octad Triangle and dodecads 113

10.3.4 A uniqueness theorem for the Golay code 116

10.4 Minimal vectors in the Leech lattice 116

10.5 First proof of uniqueness of the Leech lattice 117

10.6 Initial results about the Leech lattice 118

10.6.1 An automorphism which moves the standard frame 118

10.7 Turyn-style construction of a Leech lattice 119

10.8 Equivariant unimodularizations of even lattices 121

11 Pieces of Eight 125

11.1 Leech trios and overlattices 125

11.2 The order of the group O(∧) 128

11.3 The simplicity of M24 130

11.4 Sublattices of Leech and subgroups of the isometry group 132

11.5 Involutions on the Leech lattice 134

References 137

Index 143

Appendix A The Finite Simple Groups 149

Appendix B Reprints of Selected Articles 153

B.1 Pieces of Eight:Semiselfdual Lattices and a New Foundation for the Theory of Conway and Mathieu Groups 155

B.2 Pieces of 2d:Existence and Uniqueness for Barnes-Wall and Ypsilanti Lattices 181

B.3 Involutions on the Barnes-Wall Lattices and Their Fixed Point Sublattices,Ⅰ 223

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