《有限群的線性表示》是2008年世界圖書出版公司出版的圖書，作者賽爾。該書是一部介紹有限群線性表示的教程。
基本介紹
- 中文名：有限群的線性表示
- 別名：Linear Representations of Finite Groups
- 作者：(法國)賽爾 (Serre.J.P)
- 出版社：世界圖書出版公司
- 出版時間：2008年10月1日
- 頁數：170 頁
- 開本：24 開
- 裝幀：平裝
- ISBN：9787506292597
- 正文語種：英語
內容簡介,目錄,
內容簡介
是一部介紹有限群線性表示的教程，作者是當今法國最突出的數學家之一，他對理論數學有全面的了解，尤以著述清晰、明了聞名。《有限群的線性表示》是他寫的為數不多的教科書之一，原文是法文(1971年版)，後出了德譯本和英譯本。《有限群的線性表示》是英譯本的重印本。它篇幅不大，但深入淺出的介紹了有限群的線性表示，並給出了在量子化學等方面的套用，便於廣大數學、物理、化學工作者初學時閱讀和參考。
目錄
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
