A Tour of Representation Theory [draft]

620 Pages·2017·3.201 MB·English

by  Martin Lorenz

Martin Lorenz
A Tour of Representation Theory
(preliminary version)
January11,2017

Contents
NotationsandConventions ..........................................XVII
PartI Algebras
1 RepresentationsofAlgebras..................................... 3
1.1 Algebras.................................................. 3
1.1.1 TheCategoryofk-Algebras ........................... 3
1.1.2 SomeImportantAlgebras ............................. 5
1.1.3 Modules............................................ 12
1.1.4 EndomorphismAlgebrasandMatrices .................. 15
Exercises ................................................. 18
1.2 Representations ............................................ 23
1.2.1 TheCategoryRepAandFirstExamples................. 24
1.2.2 ChangingtheAlgebraortheBaseField ................. 26
1.2.3 IrreducibleRepresentations............................ 29
1.2.4 CompositionSeries .................................. 31
1.2.5 EndomorphismAlgebrasandSchur’sLemma ............ 34
1.2.6 IndecomposableRepresentations ....................... 36
Exercises ................................................. 37
1.3 PrimitiveIdeals ............................................ 40
1.3.1 Degree-1Representations ............................. 40
1.3.2 CommutativeAlgebras ............................... 41
1.3.3 ConnectionswithPrimeandMaximalIdeals ............. 42
1.3.4 TheJacobson-ZariskiTopology ........................ 43
1.3.5 TheJacobsonRadical ................................ 46
Exercises ................................................. 47
1.4 Semisimplicity............................................. 49
1.4.1 CompletelyReducibleRepresentations .................. 49
1.4.2 SocleandHomogeneousComponents................... 52
1.4.3 SemisimpleAlgebras................................. 54
VI Contents
1.4.4 SomeConsequencesofWedderburn’sStructureTheorem .. 57
1.4.5 Finite-DimensionalIrreducibleRepresentations........... 58
1.4.6 SemisimplicityandtheJacobsonradical ................. 60
1.4.7 AbsolutelyIrreducibleRepresentations.................. 61
Exercises ................................................. 61
1.5 Characters ................................................ 63
1.5.1 DefinitionandBasicProperties......................... 64
1.5.2 SpacesofTraceForms................................ 65
1.5.3 IrreducibleCharacters ................................ 68
1.5.4 TheGrothendieckGroupR(A) ........................ 69
Exercises ................................................. 73
2 FurtherTopicsonAlgebrasandtheirRepresentations ............. 77
2.1 Projectives ................................................ 77
2.1.1 DefinitionandBasicProperties......................... 77
2.1.2 Hattori-StallingsTraces............................... 80
2.1.3 TheGrothendieckGroupsK (A)andP(A)............. 82
0
2.1.4 Finite-DimensionalAlgebras .......................... 86
Exercises ................................................. 92
2.2 FrobeniusandSymmetricAlgebras ........................... 94
2.2.1 DefinitionofFrobeniusandSymmetricAlgebras.......... 94
2.2.2 FrobeniusForm,DualBasesandNakayamaAutomorphism 95
2.2.3 CasimirElements,CasimirOperatorandHigmanTrace.... 98
2.2.4 TraceFormulas...................................... 99
2.2.5 SymmetricAlgebras.................................. 100
2.2.6 SemisimpleAlgebrasasSymmetricAlgebras............. 102
2.2.7 IntegralityandDivisibility............................. 103
2.2.8 Separability......................................... 105
Exercises ................................................. 106
PartII Groups
3 GroupsandGroupAlgebras .................................... 111
3.1 GroupAlgebrasandRepresentationsofGroups:Generalities...... 111
3.1.1 GroupAlgebras ..................................... 111
3.1.2 RepresentationsofGroupsandGroupAlgebras........... 114
3.1.3 ChangingtheGroup.................................. 115
3.1.4 CharactersofFinite-DimensionalGroupRepresentations... 118
3.1.5 FiniteGroupAlgebrasasSymmetricAlgebras............ 121
Exercises ................................................. 121
3.2 FirstExamples............................................. 122
3.2.1 FiniteAbelianGroups ................................ 122
3.2.2 Degree-1Representations ............................. 123
3.2.3 TheDihedralGroupD ............................... 124
4
Contents VII
3.2.4 SomeRepresentationsoftheSymmetricGroupS ........ 126
n
3.2.5 PermutationRepresentations........................... 127
Exercises ................................................. 129
3.3 MoreStructure............................................. 129
3.3.1 Comultiplication,Counit,andAntipode ................. 129
3.3.2 Invariants........................................... 131
3.3.3 APlethoraofRepresentations.......................... 133
3.3.4 CharactersandSymmetricPolynomials ................. 137
Exercises ................................................. 141
3.4 SemisimpleGroupAlgebras ................................. 144
3.4.1 TheSemisimplicityCriterion .......................... 144
3.4.2 TheOrthogonalityRelations........................... 145
3.4.3 TheCaseoftheComplexNumbers ..................... 148
3.4.4 PrimitiveCentralIdempotentsofGroupAlgebras ......... 148
Exercises ................................................. 150
3.5 SomeApplicationstoInvariantTheory ........................ 153
3.5.1 SymmetricandAntisymmetricTensors.................. 153
3.5.2 Polarization......................................... 157
3.5.3 Molien’sTheorem ................................... 158
Exercises ................................................. 160
3.6 FurtherExamples .......................................... 162
3.6.1 ExteriorPowersoftheStandardRepresentationofS ..... 162
n
3.6.2 TheGroupsS andS ................................ 165
4 5
3.6.3 TheAlternatingGroupsA andA ..................... 167
4 5
Exercises ................................................. 170
3.7 SomeClassicalTheorems.................................... 171
3.7.1 DivisibilityTheoremsofFrobenius,SchurandItô......... 171
3.7.2 Burnside’spaqb-Theorem ............................. 173
3.7.3 TheBrauer-FowlerTheorem........................... 175
3.7.4 CliffordTheory...................................... 177
Exercises ................................................. 179
4 TheSymmetricGroups ......................................... 181
4.1 Gelfand-ZetlinAlgebrasandJucys-MurphyElements ............ 182
4.1.1 CentralizerSubalgebras............................... 183
4.1.2 GeneratorsoftheGelfand-ZetlinAlgebra................ 184
Exercises ................................................. 185
4.2 TheBranchingGraph ....................................... 185
4.2.1 RestrictingIrreducibleRepresentations.................. 186
4.2.2 TheGraphB ........................................ 186
4.2.3 Gelfand-ZetlinBases ................................. 187
4.2.4 MaximalityofGZ .................................. 188
n
Exercises ................................................. 189
4.3 TheYoungGraph .......................................... 190
4.3.1 PartitionsandYoungDiagrams......................... 190
VIII Contents
4.3.2 ThegraphYandtheGraphIsomorphismTheorem ........ 191
4.3.3 SomeConsequencesoftheGraphIsomorphismTheorem .. 192
4.3.4 PathsinYandStandardYoungTableaux................. 193
4.3.5 TheHook-LengthFormula ............................ 194
Exercises ................................................. 197
4.4 ProofoftheGraphIsomorphismTheorem...................... 198
4.4.1 Content ............................................ 198
4.4.2 Weights ............................................ 203
4.4.3 Cont(n)=Spec(n).................................. 206
Exercises ................................................. 208
4.5 TheIrreducibleRepresentations .............................. 209
4.5.1 RealizationoverQ ................................... 209
4.5.2 Young’sOrthogonalForm............................. 212
4.5.3 SkewShapes........................................ 213
4.5.4 RepresentationsAssociatedtoSkewShapes.............. 215
4.5.5 TheMurnaghan-NakayamaRule:StatementandExamples . 218
4.5.6 ProofoftheMurnaghan-NakayamaRule ................ 221
Exercises ................................................. 226
4.6 Schur-WeylDuality......................................... 226
4.6.1 TheDoubleCentralizerTheoremforSemisimpleAlgebras . 227
4.6.2 TheDoubleCentralizerTheoremforS andGL(V) ...... 229
n
4.6.3 SchurFunctors ...................................... 230
Exercises ................................................. 234
PartIII LieAlgebras
5 LieAlgebrasandEnvelopingAlgebras ........................... 237
5.1 TheBasicsaboutLieAlgebras ............................... 238
5.1.1 LieAlgebrasandtheirHomomorphisms................. 238
5.1.2 GeneralLinearLieAlgebrasandRepresentationsofLie
Algebras ........................................... 239
5.1.3 SubalgebrasandIdeals................................ 239
5.1.4 ExamplesofLieAlgebras............................. 241
5.1.5 Derivations ......................................... 242
Exercises ................................................. 244
5.2 TypesofLieAlgebras....................................... 246
5.2.1 NilpotentandSolvableLieAlgebras .................... 246
5.2.2 SimpleandSemisimpleLieAlgebras ................... 248
Exercises ................................................. 249
5.3 ThreeTheoremsaboutLinearLieAlgebras..................... 249
5.3.1 Engel’sTheorem..................................... 249
5.3.2 Lie’sTheorem....................................... 251
5.3.3 Cartan’sCriterion.................................... 253
Exercises ................................................. 256
Contents IX
5.4 EnvelopingAlgebras........................................ 257
5.4.1 TheEnvelopingAlgebraofaLieAlgebra................ 257
5.4.2 ThePoincaré-Birkhoff-WittTheorem ................... 259
5.4.3 MoreStructure ...................................... 264
Exercises ................................................. 267
5.5 GeneralitiesonRepresentationsofLieAlgebras................. 269
5.5.1 InvariantsandtheTrivialRepresentation................. 269
5.5.2 Homomorphisms .................................... 270
5.5.3 TensorProducts ..................................... 271
5.5.4 Tensor,ExteriorandSymmetricPowers ................. 271
5.5.5 g-Algebras.......................................... 272
5.5.6 AdjointActions ..................................... 273
5.5.7 TheRepresentationRingofaLieAlgebra ............... 275
Exercises ................................................. 276
5.6 TheNullstellensatzforEnvelopingAlgebras.................... 277
5.6.1 TheStatement....................................... 277
5.6.2 TheProof........................................... 278
5.6.3 LocallyClosedPrimes................................ 280
5.6.4 CentralCharactersandPrimUgforNilpotentg .......... 281
5.6.5 RationalIdealsandtheDixmier-MœglinEquivalence...... 283
Exercises ................................................. 289
5.7 Representationsofsl ....................................... 290
2
5.7.1 TheRepresentationsV(m) ............................ 290
5.7.2 Finite-DimensionalRepresentationsofsl ............... 292
2
5.7.3 WeightLadders...................................... 293
5.7.4 TheCasimirElement ................................. 294
5.7.5 ProofofTheorem5.37................................ 295
5.7.6 FormalCharactersandtheRepresentationRingofsl ..... 296
2
5.7.7 TheCenterofU(sl ) ................................. 299
2
5.7.8 PrimeandPrimitiveIdealsofU(sl ) .................... 299
2
Exercises ................................................. 302
6 SemisimpleLieAlgebras........................................ 305
6.1 CharacterizationsofSemisimplicity ........................... 305
6.1.1 TheKillingForm .................................... 306
6.1.2 StartoftheProofofTheorem6.1....................... 307
6.1.3 SomeConsequencesofTheorem6.1 .................... 308
Exercises ................................................. 309
6.2 CompleteReducibility ...................................... 309
6.2.1 CasimirElements .................................... 309
6.2.2 ProofofWeyl’sTheorem.............................. 311
6.2.3 AbstractJordanDecomposition ........................ 312
Exercises ................................................. 313
6.3 CartanSubalgebrasandtheRootSpaceDecomposition........... 314
6.3.1 CartanSubalgebras................................... 314
X Contents
6.3.2 RootSpaceDecomposition............................ 316
6.3.3 Simplicity .......................................... 319
6.3.4 EmbeddingintoEuclideanSpace....................... 320
Exercises ................................................. 322
6.4 TheClassicalLieAlgebras................................... 322
6.4.1 CheckingforSemisimplicity........................... 323
6.4.2 TheSpecialLinearLieAlgebrasl (TypeA ).......... 323
n+1 n
6.4.3 TheClassicalLieAlgebrasofTypesB ,C andD ....... 326
n n n
Exercises ................................................. 329
7 RootSystems .................................................. 331
7.1 AbstractRootSystems ...................................... 331
7.1.1 TheCrystallographicRestriction ....................... 332
7.1.2 RootSystemsofRanks≤2 ........................... 333
7.1.3 AutomorphismGroupandWeylGroup.................. 333
7.1.4 TheClassicalRootSystems ........................... 335
Exercises ................................................. 339
7.2 BasesofaRootSystem ..................................... 339
7.2.1 Examples:TheClassicalRootSystems.................. 339
7.2.2 Existence........................................... 341
7.2.3 Uniqueness ......................................... 342
Exercises ................................................. 345
7.3 Classification .............................................. 345
7.3.1 Irreducibility........................................ 345
7.3.2 CartanMatrixandDynkinDiagram..................... 347
7.3.3 ClassificationTheorem ............................... 347
Exercises ................................................. 349
7.4 LatticesAssociatedtoaRootSystem .......................... 350
7.4.1 RootandWeightLattice .............................. 350
7.4.2 DominantWeights ................................... 352
7.4.3 TheActionoftheWeylGroup ......................... 353
7.4.4 MultiplicativeInvariantsofWeightLattices .............. 354
7.4.5 Anti-invariants ...................................... 356
Exercises ................................................. 358
8 RepresentationsofSemisimpleLieAlgebras ...................... 359
8.1 Reminders ................................................ 359
8.1.1 TheSetting ......................................... 359
8.1.2 TriangularDecomposition............................. 361
8.1.3 TheCaseofsl ...................................... 361
2
Exercises ................................................. 362
8.2 Finite-DimensionalRepresentations ........................... 362
8.2.1 WeightSpaces....................................... 362
8.2.2 WeightsofFinite-DimensionalRepresentations........... 363
Exercises ................................................. 364
Contents XI
8.3 Finite-DimensionalIrreducibleRepresentations ................. 364
8.3.1 HighestWeightRepresentations........................ 364
8.3.2 VermaModules...................................... 365
8.3.3 TheMainResult..................................... 368
8.3.4 FundamentalRepresentationsofsl .................. 370
n+1
8.3.5 WeightsandWeightDiagrams ......................... 371
Exercises ................................................. 371
8.4 TheRepresentationRing .................................... 372
8.4.1 GroupStructure ..................................... 372
8.4.2 FormalCharacters ................................... 373
8.4.3 RingStructure....................................... 373
8.4.4 FundamentalCharactersandRepresentationRingofsl . 374
n+1
Exercises ................................................. 375
8.5 TheCenteroftheEnvelopingAlgebra ......................... 375
8.5.1 InvariantPolynomialFunctions ........................ 375
8.5.2 ElementaryAutomorphismsofg ....................... 377
8.5.3 InvariantsintheSymmetricAlgebra .................... 379
8.5.4 CentralCharacters ................................... 382
8.5.5 TheHarish-ChandraHomomorphism ................... 382
8.5.6 TheHarish-ChandraIsomorphism ...................... 383
8.5.7 TheShephard-Todd-ChevalleyTheorem................. 386
Exercises ................................................. 387
8.6 Weyl’sCharacterFormula ................................... 387
8.6.1 Dimensions ......................................... 388
8.6.2 CharactersandDimensionsforsl ................... 390
n+1
8.6.3 FormalCharactersRevisited........................... 394
8.6.4 ProofofWeyl’sCharacterFormula ..................... 395
Exercises ................................................. 397
8.7 Representationsofsl(V) .................................... 398
8.7.1 TheActionofgl(V)onV⊗n .......................... 398
8.7.2 SchurFunctorsRevisited.............................. 399
8.7.3 SλV asaHighestWeightRepresentation ................ 400
Exercises ................................................. 402
PartIV HopfAlgebras
9 Coalgebras,BialgebrasandHopfAlgebras........................ 405
9.1 Coalgebras ................................................ 405
9.1.1 TheCategoryofk-Coalgebras ......................... 405
9.1.2 InitialExamples ..................................... 407
9.1.3 SweedlerandGraphicalNotation....................... 408
9.1.4 Convolution......................................... 412
9.1.5 FromCoalgebrastoAlgebrasandBack.................. 414
Exercises ................................................. 417
XII Contents
9.2 Comodules................................................ 419
9.2.1 Comodules ......................................... 419
9.2.2 LocalFiniteness ..................................... 421
9.2.3 ThePassagebetweenComodulesandModules ........... 422
Exercises ................................................. 424
9.3 BialgebrasandHopfAlgebras................................ 425
9.3.1 Bialgebras .......................................... 425
9.3.2 FurtherExamplesofBialgebras ........................ 427
9.3.3 HopfAlgebras....................................... 429
9.3.4 PropertiesoftheAntipode............................. 430
9.3.5 DualsofBialgebrasandHopfAlgebras.................. 431
9.3.6 SomeExamplesofHopfAlgebras ...................... 434
Exercises ................................................. 438
10 RepresentationsandActions .................................... 441
10.1 RepresentationsofHopfAlgebras............................. 441
10.1.1 GeneralConstructions ................................ 441
10.1.2 HopfModules....................................... 447
Exercises ................................................. 449
10.2 FirstApplications .......................................... 451
10.2.1 FinitenessofHopfAlgebras ........................... 451
10.2.2 SomePropertiesofFinite-DimensionalHopfAlgebras..... 453
10.2.3 InnerFaithfulRepresentations ......................... 454
10.2.4 Non-DivisibilityResults .............................. 456
10.2.5 TheChevalleyProperty ............................... 458
Exercises ................................................. 459
10.3 TheRepresentationRingofaHopfAlgebra .................... 460
10.3.1 RingStructure....................................... 460
10.3.2 TheCharacterMap................................... 461
10.3.3 TheRepresentationRingofaFiniteGroupAlgebra ....... 462
10.3.4 AdditionalStructure.................................. 464
Exercises ................................................. 466
10.4 ActionsandCoactionsofHopfAlgebrasonAlgebras ............ 467
10.4.1 ModuleandComoduleAlgebras ....................... 467
10.4.2 Examples........................................... 469
10.4.3 AdjointActionandChevalleyProperty.................. 471
10.4.4 FiniteGenerationofInvariants ......................... 472
Exercises ................................................. 475
11 AffineAlgebraicGroups ........................................ 477
11.1 AffineGroupSchemes ...................................... 477
11.1.1 GroupFunctorsfromHopfAlgebras .................... 477
11.1.2 AffineGroupSchemes................................ 478
11.1.3 CategoricalView .................................... 479
11.1.4 SomeExamplesofAffineGroupSchemes ............... 479