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

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Ќазвание: Representation Theory: A First Course

јвторы: Fulton W., Harris J.

јннотаци€:

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.


язык: en

–убрика: ћатематика/јлгебра/“еори€ представлений/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1991

 оличество страниц: 551

ƒобавлена в каталог: 26.03.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\lambda$-ring      380
Abelian groups (representations of)      8
Abelian Lie algebra      121
Abelian Lie group      94
Abelian variety      135
Abramsky Ч Jahn Ч King formula      411Ч412
Adams operators      380 449
Adjoint form (of a Lie group)      101
Adjoint representation      106
Admissible Coxeter diagram      327
Ado's Theorem      124 500Ч503
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), $\mathfrak A_6$      63Ч67
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 Ч Weil Ч Bott Ч Schmid theorem      392Ч393
Borel's fixed point theorem      384
Bracket      107Ч108 504
Branching formula      59 426
Brauer's theorem      36
Bruliat cell and decomposition      395Ч398
Burnside      24Ч25
Campbell Ч Hausdorff formula      117
Capelli's identity      507Ч508 514Ч515
Cartan      434
Cartan criterion for solvability      479
Cartan decomposition      198 437
Cartan matrix      334
Cartan multiplication      429
Cartan subalgebra      198 338 432 478Ч492
Cartan subgroup      369 373 381
Casimir operator      416 429 481
Cauchy's identity      457Ч458
Cayley algebra      362Ч365
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 $\mathrm{SL}_2(\mathbb F_q)$      71Ч73
Characteristic ideal      484
Characteristics (of Frobenius)      51
Chevalley groups      74
Chordal variety      192 230
Class function      13 22
Classical Lie algebras and groups      132 367Ч375
Clebsch      237
Clebsch Ч Gordon problem      8 424
Clifford      64
Clifford algebras      30 299Ч307 364Ч365
Commutator algebra      84
Commutator subalgebra of Lie algebra      122
Compact form      432Ч438
Complete reducibility      6 128 481Ч483
Complete symmetric polynomial      77 453
Complex Lie algebra      109
Complex Lie group      95
Complex representation      41 444Ч449
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 260Ч262 288 475Ч477
Convolution      38
Coroot      495Ч496
Coxeter diagram      327
Cube, rigid motions of      20
Degree (of representation)      3
Derivation      113 480 483Ч486
Derived series      122
Deruyts      237
Determinantal formula      58 404 406Ч411 454Ч470
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      29Ч30
Dominant weight      203 376
Dual (of representation)      4 110 233
Dual (of root system)      496
Dynkin      117
Dynkin diagrams      319Ч338
Eigenspace      162
Eigenvector      162
Eightfold way      179
Elementary subgroup      36
Elementary symmetric polynomial      77 454
Elliptic curve      133Ч135
Engel's theorem      125
Exceptional Lie algebras and groups      132 339Ч365
Exceptional Lie algebras and groups, $e_6$ Ч $e_8$      361Ч362 392
Exceptional Lie algebras and groups, $f_4$      362 365
Exceptional Lie algebras and groups, $g_2$      339Ч359 362Ч364 391Ч392
Exponential map      115Ч120 369Ч370
Exterior algebra      475
Exterior powers of representations      4 31Ч32 472Ч477
External tensor product      24 427
Extra-special 2-groups      31
First fundamental theorem of invariant theory      504Ч513
Fixed point formula      14 384 393
Flag (complete and partial)      95Ч96 383Ч398
Flag manifold      73 383Ч398
Fourier inversion formula      17
Fourier transform      38
Freudenthal      359 361
Freudenthal multiplicity formula      415Ч419
Frobenius character formula      49 54Ч62
Frobenius reciprocity      35 37Ч38
Fundamental weights      205 287 295 376Ч378 528
Gelfand      426
General linear group      95 97 231Ч237
Giambelli's formula      404Ч411 455
Grassmannian      192 227Ч231 276Ч278 283 286 386Ч388
Grassmannian (Lagrangian and orthogonal)      386Ч387 390
Group algebra      36Ч39
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      382Ч398
Hook length (formula)      50 78 411Ч412
Hopf algebra      62
Icosahedron, rigid motions of      29Ч30
Ideal in Lie algebra      122
Immersed subgroup      95
Incidence correspondence      193
Indecomposable representation      6
Induced representation      32Ч36 37Ч38 393
Inner multiplicities      415
Inner product      16 23 79
Internal products      476
Invariant polynomials      504Ч513
Invariant subspace      6
Irreducible representation      4
Isogenous, isogeny      101
Isotropic      262 274 278 304 378 390
Jacobi identity      108 114
Jacobi Ч Trudy identity      455
Jordan algebra      365
Jordan decomposition      128Ч129 478 482Ч483
Killing form      202 206Ч210 240Ч241 272 478Ч479
King      411 424
Klimyk      428
Kostant      429
Kostant multiplicity formula      419Ч424
Kostka numbers      56Ч57 80 456Ч457 459
Level (of a root)      330
Levi decomposition, subalgebra      124 499Ч500
Lexicographic ordering of partitions      53
Lie algebra      108
Lie group      93
Lie subalgebra      109
Lie subgroup      94
Lie's theorem      126
Littlewood Ч Richardson number      58 79 82Ч83 424 427 455Ч456
Littlewood Ч Richardson rule      58 79 225Ч227 455Ч456
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      24Ч25
Monomial symmetric polynomial      454
Morphism of Lie groups      93
Multilinear map      472
Multiplicities      7 17 199 375
Murnaghan Ч Nakayama rule      59
Natural real form      435 437
Newton polynomials      460
Nil radical      485
Nilpotent Lie algebra      122 124Ч125
Nilrepresentation      501
Nnipotent matrices      96
Octonians      362Ч365
One-parameter subgroup      115
Ordering of roots      202
Orthogonal group      96 97 268Ч269 300 301 367 374
Orthogonal Lie algebras      268Ч269
Orthonormal      16 17 22
Outer product      58 61
Pairing      4
Partition      18 44Ч45 421 453
Perfect Lie algebra      123
Perfect pairing      28
Permutation representation      5
Peter-Weyl theorem      440
Pfaffian      228
Pieri's formula      58Ч59 79Ч81 225Ч227 455 462
Plancherel formula      38
Plane conic      154Ч159
Plethysm      8 82 151Ч160 185Ч193 224Ч231
Pluecker embedding      227Ч228 389
Pluecker equations, relations      229 235
Poincare Ч Birchoff Ч Witt theorem      486
Positive definite      98 99 207
Positive roots      202 214 243 271
Power sums      48 459Ч460
Primitive root      204 215 243 271Ч272
Projection (formulas)      15 21 23
Projective space      153
Quadric      189Ч190 228 274Ч278 285Ч286 313 388 391
Quaternionic representational      444Ч449
Quaternions      99 312
Racah      422 425 428
Radical or a Lie algebra      123 483Ч481
Rank (of a partition)      51
Rank (of Lie algebra or root system)      321 488
Rational normal curve      153Ч160
Real form      430 442
Real representation      5 17 444Ч449
Real simple Lie algebras and groups      430Ч439
Reductive Lie algebra      131
Regular element      487Ч488
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      375Ч382
Representations of $e_6$, $e_7$, $e_8$, $f_4$      414
Representations of $g_2$      350Ч359 412Ч414
Representations of $\mathfrak{sl}_2\mathbb C$      146Ч160
Representations of $\mathfrak{sl}_3\mathbb C$      161Ч193
Representations of $\mathfrak{sl}_4\mathbb C$ and $\mathfrak{sl}_n\mathbb C$      217Ч231
Representations of $\mathfrak{so}_3\mathbb C$      273
Representations of $\mathfrak{so}_4\mathbb C$      274Ч277
Representations of $\mathfrak{so}_5\mathbb C$      277Ч281
Representations of $\mathfrak{so}_6\mathbb C$      282Ч286
Representations of $\mathfrak{so}_7\mathbb C$      294Ч296
Representations of $\mathfrak{so}_8\mathbb C$      312Ч315
Representations of $\mathfrak{so}_{2n+1}\mathbb C$      294Ч296 307 407Ч409
Representations of $\mathfrak{so}_{2n}\mathbb C$      286Ч292 305Ч306 409Ч411
Representations of $\mathfrak{sp}_4\mathbb C$      244Ч252
Representations of $\mathfrak{sp}_6\mathbb C$      256Ч259
Representations of $\mathfrak{sp}_{2n}\mathbb C$      259Ч266 404Ч407
Representations of $\mathrm{GL}_n\mathbb C$      231Ч237
Restricted representation      32 80 381Ч382 425Ч428
Right action      38Ч39
Root      165 198 240 270 332Ч334 489
Root lattice      166 213 242 273 372Ч374
Root space      165 198
Root system      320
Schur functor      76 222Ч227
Schur polynomial      49 77 223 399 454Ч462
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 131Ч132
Simple root      204 324
Simply reducible group      227
Skew hook      59
Skew Schur functor, function      82Ч83
Skew symmetric bilinear form      238
Skew Young diagram      82
Snapper conjecture      60
Solvable Lie algebra      122 125 479Ч480
Specht module      60
Special linear group      95Ч97
Special linear Lie algebra      211Ч212
Special unitary group      98
Spin groups      102 299Ч300 307Ч312 368Ч372
Spin representations      30 281 291 295 306 446 448
Spinor      306
Spinor variety      390
Split conjugacy class      64
Split form      432Ч438 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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