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Methods of representation theory : with applications to finite groups and orders

Methods of representation theory : with applications to finite groups and orders (2회 대출)

자료유형
단행본
개인저자
Curtis, Charles W. Reiner, Irving.
서명 / 저자사항
Methods of representation theory : with applications to finite groups and orders / Charles W. Curtis, Irving Reiner.
발행사항
New York :   Wiley ,   c1981-c1987.  
형태사항
2 v. ; 24 cm.
총서사항
Pure and applied mathematics, 0079-8185.
ISBN
0471189944 (v. 1) 0471888710 (v. 2)
일반주기
"A Wiley-Interscience publication."  
v. 1. 1981. xxi, 819 p.  
서지주기
Includes bibliographical references and indexes.
일반주제명
Representations of groups. Finite groups.
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100 1 ▼a Curtis, Charles W.
245 1 0 ▼a Methods of representation theory : ▼b with applications to finite groups and orders / ▼c Charles W. Curtis, Irving Reiner.
260 ▼a New York : ▼b Wiley , ▼c c1981-c1987.
300 ▼a 2 v. ; ▼c 24 cm.
490 1 ▼a Pure and applied mathematics, ▼x 0079-8185.
500 ▼a "A Wiley-Interscience publication."
500 ▼a v. 1. 1981. xxi, 819 p.
504 ▼a Includes bibliographical references and indexes.
650 0 ▼a Representations of groups.
650 0 ▼a Finite groups.
700 1 ▼a Reiner, Irving.
830 0 ▼a Pure and applied mathematics (John Wiley & Sons : unnumbered)

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컨텐츠정보

목차

CONTENTS
Notation = xix
Introduction = 1
1. Background material on algebras and groups = 1
  1A. Algebras over commutative rings. Integral closure = 1
  1B. Background from group theory = 6
2. The functors Hom and Projective, injective, and flat modules = 13
  2A. Homomorphisms = 14
  2B. Tenser products = 23
  2C. Categories and functors = 27
  2D. Projective, infective, and flat modules = 29
3. Semisimple rings and modules. The Wedderburn and Morita Theorems = 40
  3A. Finiteness conditions = 40
  3B. Semisimple modules and rings = 42
  3C. Semisimple algebras over fields. The theorems of Burnside and Frobenius-Schur = 50
  3D. The Morita theorems = 55
  3E. Tensor products of simple algebras and modules. The Skolem-Noether Theorem = 64
4. Dedekind domains = 73
  4A. Localization = 73
  4B. Ideal theory = 76
  4C. Valuations, completions, localizations = 81
  4D. Modules over Dedekind domains = 84
  4E. Duals of lattices = 89
  4F. Ideal class groups : global fields = 92
  4G. Primary decompositions = 93
  4H. Cyclotomic fields = 94
5. Radicals = 101
  5A. Basic definitions = 101
  5B. Radicals of artinian rings = 109
  5C. Local rings = 111
6. Idempotents, indecomposable modules, and the KruIl-Schmidt-Azumaya Theorem. Projective covers and injective hulls = 119
  6A. Idempotents = 119
  6B. The Krull-Schmidt-Azumaya Theorem = 127
  6C. Projective covers = 131
  6D. Injective hulls = 134
7. Separable algebras and splitting fields = 142
  7A. Separable algebras and modules = 142
  7B. Splitting fields = 149
  7C. Splitting fields for division algebras = 154
  7D. Reduced norms = 158
8. Ext, Tor : cohomology of groups = 171
  8A. Ext, Tor = 171
  8B. Cohomology of groups = 179
  8C. The Schur Criterion for split extensions = 184
  8D. Tate cohomology groups = 187
Chapter 1. Group representations and character theory = 195
  9. Orthogonality relations and central idempotents = 195
    9A. Frobenius and symmetric algebras = 195
    9B. Characters and central idempotents in split semisimple algebras : orthogonality relations = 202
    9C. Orthogonality relations for characters of finite groups = 206
    9D. The character table = 214
    9E. Burnside's pa qb -Theorem = 221
  10. Induced modules = 227
    10A. Definition of induced modules. Frobenius Reciprocity = 227
    10B. Mackey's Subgroup Theorem and Tensor Product Theorem = 235
    10C. The Intertwining Number Theorem = 243
    10D. Contragredient modules = 245
    10E. Outer tensor products = 249
  11. Decomposition of induced modules. Clifford theory and Hecke algebras = 259
    11A. Clifford's Theorem = 259
    11B. Applications of Clifford's Theorem to character theory = 262
    11C. Decomposition of induced modules from normal subgroups = 267
    11D. Hecke algebras and induced modules = 279
    11E. Projective representations and central extensions = 291
  12. Tensor algebras = 308
    12A. Tensor algebras = 308
    12B. Adams operators on the ring of virtual characters = 313
    12C. Symmetric and skew-symmetric squares of induced modules = 318
  13. Tensor induction and transfer = 331
    13A. Tensor induction = 331
    13B. Transfer and the determinant map = 337
    13C. Normal p-complements and the transfer = 341
  14. Special classes and exceptional characters = 343
    14A. Frobenius groups = 344
    14B. Special classes = 348
    14C. Exceptional characters. Suzuki's CA-group Theorem = 353
    14D. Characters of As . Examples of exceptional characters = 363
    14E. The Brauer-Suzuki Theorem on generalized quaternion Sylow 2-groups = 366
    14F. Centralizers of involutions and special classes = 370
  15. The Artin and Brauer Induction Theorems = 377
    15A. Artin's Induction Theorem revisited = 377
    15B. Character rings and the Brauer Induction Theorem = 380
    15C. Applications of the Brauer Induction Theorem 384
    15D. Extensions of invariant characters = 388
    15E. A criterion for existence of normal complements = 392
    15F. A converse to Brauer's Theorem = 395
    15G. The Aramata-Brauer Induction Theorem = 396
Chapter 2. Introduction to modular representations = 401
  16. The decomposition map = 402
    16A. Notation and terminology = 402
    16B. Grothendieck groups = 403
    16C. Reduction mod % and the decomposition map = 408
    16D. Behavior of Grothendieck groups under extension of ground field = 414
  17. Brauer characters = 417
    17A. Splitting fields = 417
    17B. Brauer characters = 419
  18. The Cartan-Brauer triangle = 427
    18A. The Cartan map and the Cartan-Brauer triangle = 428
    18B. Properties of the Cartan-Brauer triangle (K sufficiently large) = 432
    18C. Orthogonality relations for Brauer characters = 437
  19. Vertices and sources = 448
    19A. Relative projective and injective modules over group rings = 449
    19B. Vertices and sources of indecomposable lattices = 453
    19C. The Green Indecomposability Theorem = 459
  20. The Green correspondence. Applications to character theory = 470
    20A. The Green correspondence = 470
    20B. Applications to character theory = 475
  21. The induction theorem for arbitrary fields = 491
    21A. The Witt-Berman Induction Theorem = 491
    21B. The induction theorem over fields of characteristic p>0 = 500
    21C. The Cartan-Brauer triangle (general case) = 503
  22. Modular representations of p-solvable groups = 513
Chapter 3. Integral representations : Orders and lattices = 520
  23. Lattices and orders = 522
  24. Jordan-Zassenhaus Theorem = 534
  25. Extensions of lattices = 538
  26. Maximal and hereditary orders = 559
    26A. Existence of maximal orders in separable algebras = 559
    26B. Maximal orders are hereditary = 565
    26C. Structure theorems for maximal and hereditary orders = 571
  27. Group rings and maximal orders = 581
  28. Twisted group rings and crossed product orders = 588
  29. Annihilator of Ext = 603
    29A. Annihilator of Ext, Higman ideal = 603
    29B. Projectile endomorphisms = 609
Chapter 4. Local and global theory of integral representations = 617
  30. Local theory = 618
    30A. Reduction mod pk = 621
    30B. Extension of the ground ring = 631
    30C. Representations mod pk = 637
  31. Genus = 642
    31A. Basic properties = 643
    31B. Id e ` les and class groups = 651
    31C. Roiter's Theorem on genera = 659
  32. Projective lattices over group rings : Swan's Theorem = 670
    32A. Local case = 671
    32B. Global case = 676
    32C. Characters afforded by projectile lattices = 679
  33. Finite representation type = 686
    33A. Jones' Theorem. Jacobinski's criterion for group rings = 687
    33B. Dade's Theorem = 691
    33C. Commutative orders = 695
  34. Examples of integral representations = 711
    34A. Extensions of lattices = 712
    34B. Cyclic p-groups = 719
    34C. Cyclic groups of order p2 = 730
    34D. An order in a matrix algebra = 742
    34E. Dihedral and metacyclic groups = 747
  35. Invertible ideals = 755
  36. The Krull-Schmidt-Azumaya Theorem over discrete valuation rings = 767
  37. Bass and Gorenstein orders = 776
Bibliography = 795
Index 813


2

CONTENTS
Chapter 5. Algebraic K-theory = 1
  38. Grothendieck groups = 2
    38A. Grothendieck groups. Frobenius functors = 2
    38B. Grothendieck groups and projective class groups = 14
    38C. Regular rings = 19
    38D. Localization sequences = 31
  39. Grothendieck groups of integral group rings = 44
    39A. Localization sequences = 45
    39B. Explicit calculations = 54
  40. Whitehead groups = 61
    40A. Introduction = 61
    40B. Localization sequences = 65
    40C. Elementary matrices = 73
    40D. Unimodular rows and stably free modules = 77
  41. Basic elements, stable range, and cancellation = 83
  42. Mayer-Vietoris sequences = 101
  43. K-theory of polynomial rings = 112
  44. Relative K-theory = 120
  45. S K1 of orders = 138
    45A. Reduced norms = 138
    45B. Maximal orders = 142
    45C. Finiteness of S K1 = 151
    45D. Profinite groups = 156
  46. Whitehead groups of integral group rings = 163
  47. Milnor's K2 -group = 184
    47A. Steinberg groups and K2 = 184
    47B. Relative K-theory = 190
    47C. Symbols = 197
  48. S K1 of integral group rings = 210
Chapter 6. Class groups of integral group rings and orders = 216
  49. Locally free class groups = 217
    49A. Basic-formulas = 217
    49B. Functorial properties and the kernel group = 229
    49C. Frobenius functor properties for class groups of group rings = 238
  50. Class groups of integral group rings = 243
    50A. Cyclic groups of squarefree order = 243
    50B. The kernel group for p-groups = 254
    50C. Metacyclic groups = 259
    50D. Dihedral and quaternion 2-groups = 266
    50E. An involution on class groups and kernel groups = 274
    50F. Cyclic p-groups = 283
    50G. Twisted group rings and crossed-product orders = 291
  51. Jacobinski's Cancellation Theorem and the Eichler condition = 303
    51A. The Eichler condition = 304
    51B. The Eichler-Swan Theorem = 306
    51C. Locally free cancellation = 322
  52. The Horn description of the class group = 329
  53. The Swan subgroup of the class group = 343
    53A. The Swan subgroup = 343
    53B. Rings of integers in tame extensions = 351
    53C. Generalized Swan subgroups = 353
  54. p-Adic logarithms and Taylor's Theorem = 356
  55. Picard groups = 369
    55A. Basic properties = 369
    55B. Picard groups of orders = 376
    55C. Locally free Picard groups = 382
    55D. Radical reduction = 391
    55E. Picard groups of group rings = 396
Chapter 7. The theory of blocks = 406
  56. Introduction to block theory = 407
    56A. Background and notation for block theory = 407
    56B. Definition of p-blocks for a finite group G = 412
    56C. A criterion for P.I.M.'s to belong to the same p-block = 414
    56D. Central characters and blocks of KG-modules = 416
    56E. The defect of a block = 422
  57. The defect group of a p-block = 429
    57A. G-algebras, the trace map. and defect groups = 429
    57B. Defect groups as vertices = 437
    57C. Defect groups as Sylow intersections = 440
  58. The Brauer Correspondence = 445
    58A. The Brauer map = 445
    58B. Brauer's First Main Theorem = 448
    58C. The Brauer Correspondence = 451
  59. Applications of blocks to character theory = 462
    59A. The Nagao Decomposition = 463
    59B. Brauer's Second Main Theorem = 467
  60. p-Sections and characters in blocks = 471
    60A. Block orthogonality and p'sections = 471
    60B. Determination of the principal block using block orthogonality = 473
    60C. Applications to the classification of transitive permutation groups of degree p = 478
  61. Refinements of the Brauer Correspondence = 484
    61A. Blocks and normal subgroups = 484
    61B. An extension of Brauer's First Main Theorem = 489
    61C. Brauer's Third Main Theorem = 494
  62. Blocks with cyclic defect groups = 495
    62A. Preliminary results from homological algebra = 496
    62B. Functorial properties of the Green Correspondence = 499
    62C. Uniserial algebras and blocks of finite representation type = 504
    62D. Modular representations in blocks wish cyclic defect groups = 512
    62E. Periodic projective resolutions in blocks with cyclic defect groups = 522
  63. Applications to group theory = 530
    63A. The kernel of the principal block = 530
    63B. The Brauer-Suzuki Theorem on quaternion Sylow 2-subgroups = 532
    63C. Glauberman's Z* -Theorem = 545
Chapter 8. The representation theory of finite groups of Lie type = 549
  64. Root systems and finite reflection groups = 550
    64A. Finite groups generated by reflections. Root systems = 550
    64B. Coxeter groups = 561
    64C. Parabolic subgroups of finite Coxeter groups = 570
  65. Finite groups with BN-pairs = 576
    65A. The Bruhat decomposition = 576
    65B. Examples of BN-pairs = 580
    65C. Parabolic subgroups of finite groups with BN-pairs = 583
  66. Homology representations of finite groups with BN-pairs = 586
    66A. Homology representations of finite groups = 586
    66B. The Coxeter poset of a finite g.g.r. = 600
    66C. The combinatorial building and the Steinberg representation of a finite group with a BN-pair = 605
  67. The Hecke algebra H(G. B) and the decomposition of ( IB \G = 609
    67A. The structure of the Hecke algebra H(G. B) = 609
    67B. The sign representation of H and the Steinberg representation of G = 614
    67C. Representations of the Hecke algebra H for a BN-pair of rank 2 = 619
    67D. The Feit-Higman Theorem on generalized polygons = 623
    67E. The reflection representation of the Hecke algebra H = 630
  68. Generic algebras and finite Coxeter groups = 635
    68A. Generic algebras and the Deformation Theorem = 635
    68B. Parametrization of characters in ( IB \G = 643
    68C. Generic degrees = 648
  69. Finite groups with split BN-pairs = 653
    69A. The Levi Decomposition = 653
    69B. Intersections of parabolic subgroups = 662
  70. Cuspidal characters = 666
    70A. Generalized restriction and induction = 666
    70B. The philosophy of cusp forms = 676
    70C. Formulas for character values = 681
  71. A Duality Operation in eh CG. = 688
    71A. Definition and basic properties of DG = 689
    71B. The effects of DG on character degrees = 692
    71C. The values of the Steinberg character = 697
  72. Modular representations of finite groups of Lie type = 700
    72A. The Ballard-Lusztig Theorem on characters of P.I.M.'s = 700
    72B. The simple kG-modules = 706
Chapter 9. Rationality questions = 719
  73. Unitary, orthogonal, and symplectic CG-modules = 720
    73A. Rationality questions over the real field R = 720
    73B. Induction theorems for real-rained characters = 727
  74. The Schur Index = 732
    74A. General theory = 732
    74B. Schur indices for group algebras = 740
    74C. The Benard-Schacher Theorem = 746
  75. Representations and characters of the symmetric group = 762
    75A. Specht modules and simple F Sn -modules = 762
    75B. Solomon's Theorem and the irreducible characters of Sn = 774
  76. The Artin exponent = 782
Chapter 10. Indecomposable modules = 790
  77. Representations of graphs and Gabriel's Theorem = 790
    77A. Representations of graphs and Coxeter functors = 790
    77B. Representation categories of finite type (Gabriel's Theorem) = 799
  78. Auslander-Reiten sequences = 806
    78A. The Heller loop-space operator = 807
    78B. Auslander-Reiten sequences for group algebras = 815
    78C. Auslander-Reiten sequences for algebras = 822
  79. Algebras of finite representation type = 830
Chapter 11. The Burnside ring and the representation ring of a finite group = 837
  80. Permutation representations and Burnside rings = 838
    80A. Burnside rings = 838
    80B. G-sets and induction maps = 846
    80C. Tenser induction and algebraic maps = 852
    80D. Coition's Induction Theorem = 859
  81. Representation rings = 868
    81A. Preliminary results = 869
    8lB. Conlon's Theorems = 878
    81C. Species = 891
    81D. Dual elements in the Green algebra = 898
    81E. Semisimplicity of representation algebras = 906
    81F. Nilpotent elements in representation algebras = 912
Bibliography = 921
Notation index = 943
Subject index = 947



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