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Tables of Contents for Group Representations
Chapter/Section Title
Page #
Page Count
Preface
vii
Part I: G-Algebras and Puig's Theory
1
378
Introduction to G-Algebras
3
108
Definitions and examples
4
5
The relative trace map
9
8
Idempotents in G-algebras
17
4
Skew group rings over G-algebras
21
6
Defect groups in G-algebras
27
10
General facts
28
6
Characterizations via Brauer morphism
34
1
Additional information
35
2
Permutation G-algebras
37
3
The Brauer morphism
40
17
General information
41
2
Brauer morphism and permutation G-algebras
43
5
Brauer pairs
48
6
The group algebra case
54
3
Weak homomorphisms and direct embeddings
57
3
Inflated G-algebras
60
3
G-algebras and strongly graded algebras
63
3
Trace of radicals
66
11
Brauer's first main theorem for G-algebras
77
13
The main theorem
77
4
Applications to permutation G-algebras
81
2
Additional results
83
7
Matrix G-algebras
90
6
Applications
96
15
A congruence modulo commutator submodule
97
3
Some properties of idempotents
100
6
Additional information
106
5
Interior G-Algebras
111
70
Definitions and elementary properties
112
18
Characterizations of defect groups and applications
130
5
Restriction and induction of interior G-algebras
135
10
Subsiduary results
145
3
Blocks, vertices and interior G-algebras
148
4
Inflated interior G-algebras
152
2
Skew group rings over interior G-algebras
154
11
Morita contexts and direct embeddings
165
7
Source algebras
172
9
Puig's Theory: Part A
181
60
Points in semiperfect rings
182
10
Elementary properties of pointed groups
192
19
General facts
192
15
Specialization to EndR(V)
207
4
Simple G-algebras and multiplicity modules
211
14
Preliminaries
211
4
Multiplicity modules for simple G-algebras
215
7
Multiplicity modules for pointed groups
222
3
Defect pointed groups
225
16
Introduction
225
1
General theory
226
12
Applications to EndR(V)
238
3
Puig's Theory: Part B
241
50
Puig correspondence
242
10
Multiplicity modules and defect pointed groups
252
14
General facts
252
3
Simple multiplicity modules
255
5
Free multiplicity modules
260
2
Barker's theorem
262
4
Green correspondence for pointed groups
266
4
An internal characterization of source algebras
270
2
Restrictions, exomorphisms and direct embeddings
272
5
Direct embeddings and local control
277
14
Puig's Theory: Part C
291
62
Pointed groups on interior G-algebras
292
8
Pointed groups and ideals
300
8
Multiplicity algebras of maximal local pointed groups
308
4
Applications to trace of radicals
312
9
Points of R-simple interior G-algebras
321
3
Pointed groups and Brauer pairs
324
2
Preparatory results for the group algebra case
326
6
Pointed groups on group algebras
332
21
General facts
332
5
Point correspondences
337
8
Some additional results
345
8
Bilinear Forms on G-Algebras
353
26
Preliminary results
354
13
General facts
354
5
The group algebra case
359
4
Comparison with a bilinear form of Green
363
4
The Broue - Robinson's theorem
367
6
Applications to the group algebra
373
6
Part II: Block Theory
379
510
Preliminaries
381
76
Blocks and their defect groups
381
17
Block decompositions
382
2
Defect groups and their characterizations
384
4
Defect groups and normal p-subgroups
388
2
Induced block decompositions
390
2
Blocks of central separable group algebras
392
1
Changing the characteristic
393
1
Field coefficients
394
4
Defect groups are Sylow intersections
398
3
Nagao's theorem
401
4
Defect groups and vertices
405
5
Blocks of defect zero
410
7
The Jacobson radical of Z(FG)
417
3
Some properties of idempotents
420
11
Elementary technical lemmas
421
5
Theorems of Iizuka, Watanabe and Osima
426
3
An application
429
2
The principal block
431
7
The principal block idempotent
438
6
Kernels of blocks
444
7
Exponents of defect groups of blocks
451
6
Block Inductions and Brauer's Theorems
457
58
Brauer's first main theorem for semilocal coefficient rings
458
4
Central characters and block inductions
462
35
Central characters and blocks
462
4
Brauer's block induction
466
16
Extended block induction
482
6
Central characters and block inductions in the semilocal case
488
9
Theorems of Conlon, Green, Nagao and Watanabe
497
12
Brauer's second main theorem
509
2
Brauer's third main theorem
511
4
Counting Blocks With a Given Defect Group
515
20
Preliminary results
515
5
Robinson's theorem
520
3
The F-dimension of J(Z(FG))
523
3
Existence of p-blocks with a given defect group
526
9
Blocks and Normal Subgroups
535
92
Preliminaries
536
9
Block covers
545
10
Block covers and central characters
555
5
Extended and Brauer roots
560
4
Block covers and defect groups
564
9
Brauer's extended first main theorem
573
3
Defect groups and normal subgroups
576
4
Brauer correspondence for covering blocks
580
6
Regular and weakly regular blocks
586
5
The Fong correspondence
591
10
Preliminary results
591
5
The Fong correspondence
596
5
A decomposition theorem and applications
601
15
Preliminaries
601
6
A decomposition theorem
607
5
Groups with primary and quasi-primary blocks
612
4
Blocks of groups and factor groups
616
11
Preliminary results
616
4
Main theorems
620
7
Blocks With Normal Defect Groups
627
44
Inertial subalgebras
628
15
Preliminaries
628
6
Existence and conjugacy of inertial subalgebras
634
9
Blocks of crossed products
643
17
Preliminaries
643
6
Main theorems
649
11
Structure of blocks with normal defect groups
660
11
Blocks and Characters: Part A
671
72
Some general facts
672
10
General background
672
5
Central endomorphisms
677
3
Partitions of bases
680
2
D-blocks
682
11
Blocks of characters: general coefficient fields
693
10
Blocks of characters: splitting coefficient fields
703
14
Two applications of block orthogonality
717
2
Central characters corresponding to induced blocks
719
4
Characters and extended block induction
723
3
Characters of height zero
726
1
Blocks, characters and α-covering groups
727
11
Generalities
727
10
An application
737
1
Characters and blocks with normal defect groups
738
5
Blocks and Characters: Part B
743
72
Characterizations of defects of blocks
744
7
Recognizing characters in the same block
751
8
General coefficient fields
751
2
Splitting coefficient fields
753
6
Some open problems
759
4
An integral matrix
763
9
Introduction
763
1
An integral matrix
764
8
Counting irreducible characters in blocks
772
20
The Brauer - Feit theorem
772
3
Inequalities for block-theoretic invariants
775
2
Characterizations of the case k(B) = 2
777
3
Situations in which k(B) is known precisely
780
4
Central separable blocks
784
8
Generalized decomposition numbers
792
15
Heights of R-generalized characters
807
8
Blocks and Characters: Part C
815
74
Block partitions, p-sections and multiplicities
816
18
Block partititons of conjugacy classes
816
3
Multiplicities
819
8
The trivial p-section
827
6
Lower defect groups
833
1
Multiplicities of defect groups
834
10
Preliminary results
834
4
The main result
838
4
Applications
842
2
Subsections
844
4
Characters, normal subgroups and block covers
848
10
Subpairs and Brauer nets
858
18
Generalities
858
8
Centralizer, normalizer and extremal subpairs
866
4
Major subpairs
870
2
Brauer nets
872
4
Vertices of virtually irreducible lattices
876
3
Blocks, characters and subgroups
879
10
General results
879
5
Characters and block induction
884
5
Bibliography
889
64
Notation
953
8
Index
961