第1版 (2008年10月1日)
外文书名: Linear Representations of Finite Groups
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
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作者简介
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
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