Fulton W., Harris J. — Representation Theory: A First Course :: Электронная библиотека попечительского совета мехмата МГУ

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Fulton W., Harris J. — Representation Theory: A First Course

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Название: Representation Theory: A First Course

Авторы: Fulton W., Harris J.

Аннотация:

The primary goal of these lectures is to introduce a beginner to the finite-dimensional representations of Lie groups and Lie algebras. Intended to serve non-specialists, the concentration of the text is on examples. The general theory is developed sparingly, and then mainly as useful and unifying language to describe phenomena already encountered in concrete cases. The book begins with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups. The focus then turns to Lie groups and Lie algebras and finally to the heart of the course: working out the finite dimensional representations of the classical groups. The goal of the last portion of the book is to make a bridge between the example-oriented approach of the earlier parts and the general theory.

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Рубрика: Математика/Алгебра/Теория представлений/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1991

Количество страниц: 551

Добавлена в каталог: 26.03.2005

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Предметный указатель

$\lambda$ -ring      380
Abelian groups (representations of)      8
Abelian Lie algebra      121
Abelian Lie group      94
Abelian variety      135
Abramsky — Jahn — King formula      411—412
Adams operators      380 449
Adjoint form (of a Lie group)      101
Adjoint representation      106
Admissible Coxeter diagram      327
Ado's Theorem      124 500—503
Algebraic group      95 374
Alternating group (representations of), $\mathfrak A_3$       9
Alternating group (representations of), $\mathfrak A_4$       20
Alternating group (representations of), $\mathfrak A_5$       29
Alternating group (representations of), $\mathfrak A_6$       63—67
Alternating map      472
Alternating representation      9
Artin's theorem      36
Automorphism group of a Lie algebra      498
Averaging      6 15 21
Bilinear form      40 97
Borel subalgebra      210 338 382
Borel subgroup      67 383 4'
Borel — Weil — Bott — Schmid theorem      392—393
Borel's fixed point theorem      384
Bracket      107—108 504
Branching formula      59 426
Brauer's theorem      36
Bruliat cell and decomposition      395—398
Burnside      24—25
Campbell — Hausdorff formula      117
Capelli's identity      507—508 514—515
Cartan      434
Cartan criterion for solvability      479
Cartan decomposition      198 437
Cartan matrix      334
Cartan multiplication      429
Cartan subalgebra      198 338 432 478—492
Cartan subgroup      369 373 381
Casimir operator      416 429 481
Cauchy's identity      457—458
Cayley algebra      362—365
Cayley operator      507
Center of Lie algebra      121
Character (of representation)      13 22 375 440 442
Character homomorphism      375
Character table      14
Character table, of $\mathfrak A_4$       20
Character table, of $\mathfrak A_5$       29
Character table, of $\mathfrak A_d$       66
Character table, of $\mathfrak G_3$       14
Character table, of $\mathfrak G_4$       19
Character table, of $\mathfrak G_5$       28
Character table, of $\mathfrak G_d$       49
Character table, of $\mathrm{GL}_2(\mathbb F_q)$       70
Character table, of $\mathrm{SL}_2(\mathbb F_q)$       71—73
Characteristic ideal      484
Characteristics (of Frobenius)      51
Chevalley groups      74
Chordal variety      192 230
Class function      13 22
Classical Lie algebras and groups      132 367—375
Clebsch      237
Clebsch — Gordon problem      8 424
Clifford      64
Clifford algebras      30 299—307 364—365
Commutator algebra      84
Commutator subalgebra of Lie algebra      122
Compact form      432—438
Complete reducibility      6 128 481—483
Complete symmetric polynomial      77 453
Complex Lie algebra      109
Complex Lie group      95
Complex representation      41 444—449
Complex torus      120
Complexification      430 438
Conjugate linear involution      436
Conjugate partition      45 454
Conjugate representation      64
Connected Lie group      94
Contraction maps      182 224 260—262 288 475—477
Convolution      38
Coroot      495—496
Coxeter diagram      327
Cube, rigid motions of      20
Degree (of representation)      3
Derivation      113 480 483—486
Derived series      122
Deruyts      237
Determinantal formula      58 404 406—411 454—470
Dihedral group      30 243
Dimension of Lie group      93
Direct sum (of representations)      4
Discriminant      48 400 454
Distinguished subalgebras      200
Dodecahedron, rigid motions of      29—30
Dominant weight      203 376
Dual (of representation)      4 110 233
Dual (of root system)      496
Dynkin      117
Dynkin diagrams      319—338
Eigenspace      162
Eigenvector      162
Eightfold way      179
Elementary subgroup      36
Elementary symmetric polynomial      77 454
Elliptic curve      133—135
Engel's theorem      125
Exceptional Lie algebras and groups      132 339—365
Exceptional Lie algebras and groups, —       361—362 392
Exceptional Lie algebras and groups,       362 365
Exceptional Lie algebras and groups,       339—359 362—364 391—392
Exponential map      115—120 369—370
Exterior algebra      475
Exterior powers of representations      4 31—32 472—477
External tensor product      24 427
Extra-special 2-groups      31
First fundamental theorem of invariant theory      504—513
Fixed point formula      14 384 393
Flag (complete and partial)      95—96 383—398
Flag manifold      73 383—398
Fourier inversion formula      17
Fourier transform      38
Freudenthal      359 361
Freudenthal multiplicity formula      415—419
Frobenius character formula      49 54—62
Frobenius reciprocity      35 37—38
Fundamental weights      205 287 295 376—378 528
Gelfand      426
General linear group      95 97 231—237
Giambelli's formula      404—411 455
Grassmannian      192 227—231 276—278 283 286 386—388
Grassmannian (Lagrangian and orthogonal)      386—387 390
Group algebra      36—39
Half-spin representations      306
Heisenberg group      31
Hermite Reciprocity      82 160 189 233
Hermitian inner product, form      6 11 16 98 99
hessian      157
Highest weight      175 203
Highest weight vector      167 175 202
Homogeneous spaces      382—398
Hook length (formula)      50 78 411—412
Hopf algebra      62
Icosahedron, rigid motions of      29—30
Ideal in Lie algebra      122
Immersed subgroup      95
Incidence correspondence      193
Indecomposable representation      6

Induced representation      32—36 37—38 393
Inner multiplicities      415
Inner product      16 23 79
Internal products      476
Invariant polynomials      504—513
Invariant subspace      6
Irreducible representation      4
Isogenous, isogeny      101
Isotropic      262 274 278 304 378 390
Jacobi identity      108 114
Jacobi — Trudy identity      455
Jordan algebra      365
Jordan decomposition      128—129 478 482—483
Killing form      202 206—210 240—241 272 478—479
King      411 424
Klimyk      428
Kostant      429
Kostant multiplicity formula      419—424
Kostka numbers      56—57 80 456—457 459
Level (of a root)      330
Levi decomposition, subalgebra      124 499—500
Lexicographic ordering of partitions      53
Lie algebra      108
Lie group      93
Lie subalgebra      109
Lie subgroup      94
Lie's theorem      126
Littlewood — Richardson number      58 79 82—83 424 427 455—456
Littlewood — Richardson rule      58 79 225—227 455—456
Lower central series      122
Map between Lie groups      93
Map between representations      3
Minuscule weight      423
Modification rules      426
Modular representation      7
Module (G-module, g-module)      3 481
Molien      24—25
Monomial symmetric polynomial      454
Morphism of Lie groups      93
Multilinear map      472
Multiplicities      7 17 199 375
Murnaghan — Nakayama rule      59
Natural real form      435 437
Newton polynomials      460
Nil radical      485
Nilpotent Lie algebra      122 124—125
Nilrepresentation      501
Nnipotent matrices      96
Octonians      362—365
One-parameter subgroup      115
Ordering of roots      202
Orthogonal group      96 97 268—269 300 301 367 374
Orthogonal Lie algebras      268—269
Orthonormal      16 17 22
Outer product      58 61
Pairing      4
Partition      18 44—45 421 453
Perfect Lie algebra      123
Perfect pairing      28
Permutation representation      5
Peter-Weyl theorem      440
Pfaffian      228
Pieri's formula      58—59 79—81 225—227 455 462
Plancherel formula      38
Plane conic      154—159
Plethysm      8 82 151—160 185—193 224—231
Pluecker embedding      227—228 389
Pluecker equations, relations      229 235
Poincare — Birchoff — Witt theorem      486
Positive definite      98 99 207
Positive roots      202 214 243 271
Power sums      48 459—460
Primitive root      204 215 243 271—272
Projection (formulas)      15 21 23
Projective space      153
Quadric      189—190 228 274—278 285—286 313 388 391
Quaternionic representational      444—449
Quaternions      99 312
Racah      422 425 428
Radical or a Lie algebra      123 483—481
Rank (of a partition)      51
Rank (of Lie algebra or root system)      321 488
Rational normal curve      153—160
Real form      430 442
Real representation      5 17 444—449
Real simple Lie algebras and groups      430—439
Reductive Lie algebra      131
Regular element      487—488
Regular representation      5 17
Representation      3 95 100 109
Representation defined over a field      41
Representation of a Lie algebra      109
Representation ring of finite group      22
Representation ring of Lie group or algebra      375—382
Representations of , , ,       414
Representations of       350—359 412—414
Representations of $\mathfrak{sl}_2\mathbb C$       146—160
Representations of $\mathfrak{sl}_3\mathbb C$       161—193
Representations of $\mathfrak{sl}_4\mathbb C$ and $\mathfrak{sl}_n\mathbb C$       217—231
Representations of $\mathfrak{so}_3\mathbb C$       273
Representations of $\mathfrak{so}_4\mathbb C$       274—277
Representations of $\mathfrak{so}_5\mathbb C$       277—281
Representations of $\mathfrak{so}_6\mathbb C$       282—286
Representations of $\mathfrak{so}_7\mathbb C$       294—296
Representations of $\mathfrak{so}_8\mathbb C$       312—315
Representations of $\mathfrak{so}_{2n+1}\mathbb C$       294—296 307 407—409
Representations of $\mathfrak{so}_{2n}\mathbb C$       286—292 305—306 409—411
Representations of $\mathfrak{sp}_4\mathbb C$       244—252
Representations of $\mathfrak{sp}_6\mathbb C$       256—259
Representations of $\mathfrak{sp}_{2n}\mathbb C$       259—266 404—407
Representations of $\mathrm{GL}_n\mathbb C$       231—237
Restricted representation      32 80 381—382 425—428
Right action      38—39
Root      165 198 240 270 332—334 489
Root lattice      166 213 242 273 372—374
Root space      165 198
Root system      320
Schur functor      76 222—227
Schur polynomial      49 77 223 399 454—462
Schur's lemma      7
Semisimple Lie algebra      123 131 209 480
Semistandard tableau      56 236 456 461
Serre      337
Severi      392
Shuffle      474
Simple Lie algebra      122 131—132
Simple root      204 324
Simply reducible group      227
Skew hook      59
Skew Schur functor, function      82—83
Skew symmetric bilinear form      238
Skew Young diagram      82
Snapper conjecture      60
Solvable Lie algebra      122 125 479—480
Specht module      60
Special linear group      95—97
Special linear Lie algebra      211—212
Special unitary group      98
Spin groups      102 299—300 307—312 368—372
Spin representations      30 281 291 295 306 446 448
Spinor      306
Spinor variety      390
Split conjugacy class      64
Split form      432—438 445
Standard representation      9 151 176 244 257 273 352
Standard tableau      57 81 457
Steinberg's formula      425
String (of roots)      201 324
Subrepresentation      4
Symmetric algebra      475

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