群與格引論：有限群與正定有理格

《群與格引論有限群與正定有理格》是2010年高等教育出版社出版的書籍，作者是格里斯(RobertL.Griess)。

基本介紹

• 書名：群與格引論:有限群與正定有理格
• 作者：格里斯(RobertL.Griess)
• ISBN：9787040292053
• 定價：58.00元
• 出版社：高等教育出版社
• 出版時間：2010年5月1日
• 開本：16開

內容簡介

《群與格引論:有限群與正定有理格(國內英文版)》內容簡介：The launch of this Advanced Lectures in Mathematics series is aimed at keeping　mathematicians informed of the latest developments in mathematics, as well as　to aid in the learning of new mathematical topics by students all over the world.

Each volume consists of either an expository monograph or a collection of signifi-　cant introductions to important topics. This series emphasizes the history and　sources of motivation for the topics under discussion, and also gives an overview　of the current status of research in each particular field. These volumes are the　first source to which people will turn in order to learn new subjects and to dis-　cover the latest results of many cutting-edge fields in mathematics.

作者簡介

作者：（美國）格里斯（Robert L.Griess）

圖書目錄

1 Introduction

1.1 Outline of the book

1.2 Suggestions for further reading

1.3 Notations, background, conventions

2 Bilinear Forms, Quadratic Forms and Their Isometry Groups.

2.1 Standard results on quadratic forms and reflections

2.1.1 Principal ideal domains （PIDs）

2.2 Linear algebra

2.2.1 Interpretation of nonsingularity

2.2.2 Extension of scalars

2.2.3 Cyclicity of the values of a rational bilinea.r form

2.2.4 Gram matrix

2.3 Discriminant group

2.4 Relations between a lattice and sublattices

2.5 Involutions on quadratic spaces

2.6 Standard results on quadratic forms and reflections, II

2.6.1 Involutions on lattices

2.7 Scaled isometries: norm doublers and triplers

3 General Results on Finite Groups and Invariant Lattices

3.1 Discreteness of rational lattices

3.2 Finiteness of the isometry group

3.3 Construction of a G-invariant bilinear form

3.4 Semidirect products and wreath products

3.5 Orthogonal decomposition of lattices

4 Root Lattices of Types A, D, E

4.1 Background from Lie theory

4.2 Root lattices, their duals and their isometry groups

4.2.1 Definition of the AN lattices

4.2.2 Definition of the Dn lattices

4.2.3 Definition of the En lattices

4.2.4 Analysis of the A,n root lattices

4.2.5 Analysis of the Dn root lattices

4.2.6 More on the isometry groups of type Dn

4.2.7 Analysis of the En root lattices

5 Hermite and Minkowski Functions

5.1 Small ranks and small determinants

5.1.1 Table for the Minkowski and Hermite functions

5.1.2 Classifications of small rank, small determinant lattices

5.2 Uniqueness of the lattices E6, E7 and Es

5.3 More small ranks and small determinants

6 Constructions of Lattices by Use of Codes

6.1 Definitions and basic results

6.1.1 A construction of the Es-lattice with the binary [8, 4, 4] cod

6.1.2 A construction of the Es-lattice with the ternary [4, 2, 3] cod

6.2 The proofs

6.2.1 About power sets, boolean sums and quadratic forms

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

6.2.3 Reed-Muller codes

6.2.4 Uniqueness of the tetracode

6.2.5 The automorphism group of the tetracode

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

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

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

6.3.1 The A6-1attice

7 Group Theory and Representations

7.1 Finite groups

7.2 Extraspecial p-groups

7.2.1 Extraspecial groups and central products

7.2.2 A normal form in an extraspecial group

7.2.3 A classification of extraspecial groups

7.2.4 An application to automorphism groups of extraspecial groups

7.3 Group representations

7.3.1 Representations of extraspecial p-groups

7.3.2 Construction of the BRW groups

7.3.3 Tensor products

7.4 Representation of the BRW group G

7.4.1 BRW groups as group extensions

8 Overview of the Barnes-Wall Lattices

8.1 Some properties of the series

8.2 Commutator density

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

8.2.2 Extraspecial groups and commutator density

9 Construction and Properties of the Barnes-Wall Lattices

9.1 The Barnes-Wall series and their minimal vectors

9.2 Uniqueness for the BW lattices

9.3 Properties of the BRW groups

9.4 Applications to coding theory

9.5 More about minimum vectors

10 Even unimodular lattices in small dimensions

10.1 Classifications of even unimodular lattices

10.2 Constructions of some Niemeier lattices

10.2.1 Construction of a Leech lattice

10.3 Basic theory of the Golay code

10.3.1 Characterization of certain Reed-Muller codes

10.3.2 About the Golay code

10.3.3 The octad Triangle and dodecads

10.3.4 A uniqueness theorem for the Golay code

10.4 Minimal vectors in the Leech lattice

10.5 First proof of uniqueness of the Leech lattice

10.6 Initial results about the Leech lattice

10.6.1 An automorphism which moves the standard frame ...

10.7 Turyn-style construction of a Leech lattice

10.8 Equivariant unimodularizations of even lattices

11 Pieces of Eight

11.1 Leech trios and overlattices

11.2 The order of the group O（A）

11.3 The simplicity of M24

11.4 Sublattices of Leech and subgroups of the isometry group

11.5 Involutions on the Leech lattice

References

Index

Appendix A The Finite Simple Groups