有限群的線性表示

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　Schurs　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?、?br />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;　Mackeys　criterion
7.1　Induction
7.2　The　character　of　an　induced　representation;
the　reciprocity　formula
7.3　Restriction　to　subgroups
7.4　Mackeys　irreducibility　criterion
8　Examples　of　induced　representations
8.　l　Normal　subgroups;　applications　to　the　degrees　of　the
ineducible　representations
8.2　Semidirect　products　by　an　ahelian　group
8.3　A　review　of　some　classes　of　finite　groups
8.4　Syiows　theorem
8.5　Linear　representations　of　superselvable　groups
9　Artins　theorem
9.1　The　ring　R(G)
9.2　Statement　of　Artins　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　Brauers　theorem
11　Applications　of　Brauers　theorem
11.1　Characterization　of　characters
11.2　A　theorem　of　Frobenius
11.3　A　converse　to　Brauers　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　Artins　theorem
12.6　Generalization　of　Brauers　theorem
12.7　Proof　of　theorem　28
13　Rationality　questions：　examples
13.　I　The　field　Q
13.2　The　field　R
Bibliography：　Part?、?br />Part?、?br />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　Brauers　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