图书信息

作者简介

内容简介

目录

图书信息

出版社: 世界图书出版公司; 第1版 (2008年10月1日)

外文书名: Linear Representations of Finite Groups

平装: 170页

正文语种: 英语

开本: 24

ISBN: 9787506292597

条形码: 9787506292597

尺寸: 22.2 x 14.6 x 1.4 cm

重量: 240 g

作者简介

作者：(法国)赛尔 (Serre.J.P)

内容简介

《有限群的线性表示》是一部非常经典的介绍有限群线性表示的教程，原版曾多次修订重印，作者是当今法国最突出的数学家之一，他对理论数学有全面的了解，尤以著述清晰、明了闻名。《有限群的线性表示》是他写的为数不多的教科书之一，原文是法文(1971年版)，后出了德译本和英译本。《有限群的线性表示》是英译本的重印本。它篇幅不大，但深入浅出的介绍了有限群的线性表示，并给出了在量子化学等方面的应用，便于广大数学、物理、化学工作者初学时阅读和参考。

目录

Part Ⅰ

Representations and Characters

1 Generalities on linear representations

1.1 Definitions

1.2 Basic examples

1.3 Submpmsentations

1.4 Irreducible representations

1.5 Tensor product of two representations

1.6 Symmetric square and alternating square

2 Character theory

2.1 The character of a representation

2.2 Schur's lemma; basic applications

2.3 0rthogonality relations for characters

2.4 Decomposition of the regular representation

2.5 Number of irreducible representations

2.6 Canonical decomposition of a representation

2.7 Explicit decomposition of a representation

3 Subgroups, products, induced representations

3.1 Abelian subgroups

3.2 Product of two groups

3.3 Induced representations

4 Compact groups

4.1 Compact groups

4.2 lnvariant measure on a compact group

4.3 Linear representations of compact groups

5 Examples

5.1 The cyclic Group

5.2 The group

5.3 The dihedral group

5.4 The group

5.5 The group

5.6 The group

5.7 The alternating group

5.8 The symmetric group

5.9 The group of the cube

Bibliography: Part Ⅰ

Part Ⅱ

Representations in Characteristic Zero

6 The group algebra

6.1 Representations and modules

6.2 Decomposition of C[G]

6.3 The center of C[G]

6.4 Basic properties of integers

6.5 lntegrality properties of characters. Applications

7 Induced representations; Mackey's criterion

7.1 Induction

7.2 The character of an induced representation;

the reciprocity formula

7.3 Restriction to subgroups

7.4 Mackey's irreducibility criterion

8 Examples of induced representations

8. l Normal subgroups; applications to the degrees of the

in'educible representations

8.2 Semidirect products by an ahelian group

8.3 A review of some classes of finite groups

8.4 Syiow's theorem

8.5 Linear representations of superselvable groups

9 Artin's theorem

9.1 The ring R(G)

9.2 Statement of Artin's theorem

9.3 First proof

9.4 Second proof of (i) = (ii)

10 A theorem of Brauer

10.1 p-regular elements;p-elementary subgroups

10.2 Induced characters arising from p-elementary

subgroups

10.3 Construction of characters

10.4 Proof of theorems 18 and 18'

10.5 Brauer's theorem

11 Applications of Brauer's theorem

11.1 Characterization of characters

11.2 A theorem of Frobenius

11.3 A converse to Brauer's theorem

11.4 The spectrum of A R(G)

12 Rationality questions

12.1 The rings RK(G) and RK(G)

12.2 Schur indices

12.3 Realizability over cyclotomic fields

12.4 The rank of RK(G)

12.5 Generalization of Artin's theorem

12.6 Generalization of Brauer's theorem

12.7 Proof of theorem 28

13 Rationality questions: examples

13. I The field Q

13.2 The field R

Bibliography: Part Ⅱ

Part Ⅲ

Introduction to Brauer Theory

14 The groups RK(G), R(G), and Pk(G)

14.1 The rings RK(G) and R,(G)

14.2 The groups Pk(G) and P^(G)

14.3 Structure of Pk(G)

14.4 Structure of PA(G)

14.5 Dualities

14.6 Scalar extensions

15 The cde triangle

15.1 Definition of c: Pk(G) ——Rk(G)

15.2 Definition of d: Rs(G) —— Rk(G)

15.3 Definition of e: Pk(G) —— RK(G)

15.4 Basic properties of the cde triangle

15.5 Example: p'-gmups

15.6 Example: p-groups

15.7 Example: products ofp'-groups and p-groups

16 Theorems

16.1 Properties of the cde triangle

16.2 Characterization of the image of e

16.3 Characterization of projective A [G ]-modules

by their characters

16.4 Examples of projective A [G ]-modules: irreducible

representations of defect zero

17 Proofs

17. I Change of groups

17.2 Brauer's theorem in the modular case

17.3 Proof of theorem 33

17.4 Proof of theorem 35

17.5 Proof of theorem 37

17.6 Proof of theorem 38

18 Modular characters

18.1 The modular character of a representation

18.2 Independence of modular characters

18.3 Reformulations

18.4 A section ford

18.5 Example: Modular characters of the symmetric group

18.6 Example: Modular characters of the alternating group

19 Application to Artin representations

19.1 Artin and Swan representations

19.2 Rationality of the Artin and Swan representations

19.3 An invariant

Appendix

Bibliography: Part Ⅲ

Index of notation

Index of terminology