Representation theory: a first course
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2004
|
| Ausgabe: | Springer study ed. |
| Schriftenreihe: | Graduate texts in mathematics
129 : Readings in mathematics |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 551 S. graph. Darst. |
| ISBN: | 0387974954 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804134163935133696 |
|---|---|
| adam_text | Titel: Representation theory
Autor: Fulton, William
Jahr: 2004
Contents
Preface v
Using This Book ix
Part I: Finite Groups 1
1. Representations of Finite Groups 3
§1.1: Definitions 3
§1.2: Complete Reducibility; Schur s Lemma 5
§1.3: Examples: Abelian Groups; 8
2. Characters 12
§2.1: Characters 12
§2.2: The First Projection Formula and Its Consequences 15
§2.3: Examples: S4 and 2I4 18
§2.4: More Projection Formulas; More Consequences 21
3. Examples; Induced Representations; Group Algebras; Real
Representations 26
§3.1: Examples: S5 and 9I5 26
§3.2: Exterior Powers of the Standard Representation of Bd 31
§3.3: Induced Representations 32
§3.4: The Group Algebra 36
§3.5: Real Representations and Representations over Subfields of C 39
4. Representations of S„: Young Diagrams and Frobenius s
Character Formula
§4.1: Statements of the Results
§4.2: Irreducible Representations of
§4.3: Proof of Frobenius s Formula
5. Representations of and GL2(F9)
§5.1: Representations of
§5.2: Representations of GL2(F,) and SL2(F,)
6. WeyFs Construction
§6.1: Schur Functors and Their Characters
§6.2: The Proofs
Part IX: Lie Groups and Lie Algebras
7. Lie Groups
§7.1: Lie Groups: Definitions
§7.2: Examples of Lie Groups
§7.3: Two Constructions
8. Lie Algebras and Lie Groups
§8.1: Lie Algebras: Motivation and Definition
§8.2: Examples of Lie Algebras
§8.3: The Exponential Map
9. Initial Classification of Lie Algebras
§9.1: Rough Classification of Lie Algebras
§9.2: Engel s Theorem and Lie s Theorem
§9.3: Semisimple Lie Algebras
§9.4: Simple Lie Algebras
10. Lie Algebras in Dimensions One, Two, and Three
§10.1: Dimensions One and Two
§10.2: Dimension Three, Rank 1
§10.3: Dimension Three, Rank 2
§10.4: Dimension Three, Rank 3
11. Representations of sI2C
§11.1: The Irreducible Representations
§11.2: A Little Plethysm
§11.3: A Little Geometric Plethysm
Contents
xiii
12. Representations of sl3C, Part I 161
13. Representations of sI3C, Part II: Mainly Lots of Examples 175
§13.1: Examples 175
§13.2: Description of the Irreducible Representations 182
§13.3: A Little More Plethysm 185
§13.4: A Little More Geometric Plethysm 189
Part III: The Classical Lie Algebras and Their Representations 195
14. The General Set-up: Analyzing the Structure and Representations
of an Arbitrary Semisimple Lie Algebra 197
§14.1: Analyzing Simple Lie Algebras in General 197
§14.2: About the Killing Form 206
15. sl4C and sl„C 211
§15.1: Analyzing 211
§15.2: Representations of sI4C and sI„C 217
§15.3: Weyl s Construction and Tensor Products 222
§15.4: Some More Geometry 227
§15.5: Representations of GL„C 231
16. Symplectic Lie Algebras 238
§16.1: The Structure of Sp2„C and sp2llC 238
§16.2: Representations of sp4C 244
17. sp6C and sp2„C 253
§17.1: Representations of sp6C 253
§17.2: Representations of sp2-C in General 259
§17.3: Weyl s Construction for Symplectic Groups 262
18. Orthogonal Lie Algebras 267
§18.1: SOMC and somC 267
§18.2: Representations of so3C, so4C, and so5C 273
19. so6C, so7C, and sowC 282
§19.1: Representations of so6C 282
§19.2: Representations of the Even Orthogonal Algebras 286
§19.3: Representations of ao7C 292
§19.4: Representations of the Odd Orthogonal Algebras 294
§19.5: Weyl s Construction for Orthogonal Groups 296
xiv
Contents
299
20. Spin Representations of somC
§20.1: Clifford Algebras and Spin Representations of eo„C 299
§20.2: The Spin Groups SpinMC and SpinmR 30
§20.3: Spin8C and Triality
312
Part IV: Lie Theory 317
21. The Classification of Complex Simple Lie Algebras 319
§21.1: Dynkin Diagrams Associated to Semisimple Lie Algebras 319
§21.2: Classifying Dynkin Diagrams f 325
§21.3: Recovering a Lie Algebra from Its Dynkin Diagram 330
22. g2 and Other Exceptional Lie Algebras 339
§22.1: Construction of g2 from Its Dynkin Diagram 339
§22.2: Verifying That g2 is a Lie Algebra ,, 346
§22.3: Representations of q2 350
§22.4: Algebraic Constructions of the Exceptional Lie Algebras 359
23. Complex Lie Groups; Characters 366
§23.1: Representations of Complex Simple Groups 366
§23.2: Representation Rings and Characters 375
§23.3: Homogeneous Spaces 382
§23.4: Bruhat Decompositions . 395
24. Weyl Character Formula 399
§24.1: The Weyl Character Formula 399
§24.2: Applications to Classical Lie Algebras and Groups 403
25. More Character Formulas 415
§25.1: Freudenthars Multiplicity Formula 415
§25.2: Proof of (WCF); the Kostant Multiplicity Formula 419
§25.3: Tensor Products and Restrictions to Subgroups 424
26. Real Lie Algebras and Lie Groups 430
§26.1: Classification of Real Simple Lie Algebras and Groups 430
§26.2: Second Proof of Weyl s Character Formula 440
§26.3. Real, Complex, and Quaternionic Representations 444
Appendices 451
A. On Symmetric Functions 453
§A.l: Basic Symmetric Polynomials and Relations among Them 453
§A.2. Proofs of the Determinantal Identities 462
§A.3: Other Determinantal Identities 465
Contents xv
B. On Multilinear Algebra 471
§B.l: Tensor Products 471
§B.2: Exterior and Symmetric Powers 472
§B.3: Duals and Contractions 475
C. On Semisimplicity 478
§0.1: The Killing Form and Cartan s Criterion 478
§C.2: Complete Reducibility and the Jordan Decomposition 481
§C.3: On Derivations 483
D. Cartan Subalgebras 487
§D.l: The Existence of Cartan Subalgebras 487
§D.2: On the Structure of Semisimple Lie Algebras 489
§D.3: The Conjugacy of Cartan Subalgebras 491
§D.4: On the Weyl Group 493
E. Ado s and Levi s Theorems 499
§E.l: Levi s Theorem 499
§E.2: Ado s Theorem 500
F. Invariant Theory for the Classical Groups 504
§F.l: The Polynomial Invariants 504
§F.2: Applications to Symplectic and Orthogonal Groups 511
§F.3: Proof of Capelli s Identity 514
Hints, Answers, and References 516
Bibliography 536
Index of Symbols 543
Index 547
|
| any_adam_object | 1 |
| author | Fulton, William 1939- Harris, Joe 1951- |
| author_GND | (DE-588)136272541 (DE-588)112574718 |
| author_facet | Fulton, William 1939- Harris, Joe 1951- |
| author_role | aut aut |
| author_sort | Fulton, William 1939- |
| author_variant | w f wf j h jh |
| building | Verbundindex |
| bvnumber | BV020831955 |
| classification_rvk | SK 180 SK 260 |
| classification_tum | MAT 173f MAT 202f MAT 225f |
| ctrlnum | (OCoLC)254573793 (DE-599)BVBBV020831955 |
| dewey-full | 512/.2 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.2 512/.55 |
| dewey-search | 512/.2 512/.55 |
| dewey-sort | 3512 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Springer study ed. |
| format | Book |
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| id | DE-604.BV020831955 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T20:20:15Z |
| institution | BVB |
| isbn | 0387974954 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013837146 |
| oclc_num | 254573793 |
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| physical | XV, 551 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics Undergraduate texts in mathematics : Readings in mathematics |
| spelling | Fulton, William 1939- Verfasser (DE-588)136272541 aut Representation theory a first course William Fulton ; Joe Harris Springer study ed. New York [u.a.] Springer 2004 XV, 551 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 129 : Readings in mathematics Undergraduate texts in mathematics : Readings in mathematics Lie-Algebra - Darstellungstheorie Lie-Gruppe - Darstellungstheorie Gruppe Mathematik (DE-588)4022379-6 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Darstellung Mathematik (DE-588)4128289-9 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Darstellung Mathematik (DE-588)4128289-9 s Lie-Algebra (DE-588)4130355-6 s DE-604 Lie-Gruppe (DE-588)4035695-4 s Darstellungstheorie (DE-588)4148816-7 s 1\p DE-604 2\p DE-604 Gruppe Mathematik (DE-588)4022379-6 s 3\p DE-604 Harris, Joe 1951- Verfasser (DE-588)112574718 aut Graduate texts in mathematics 129 : Readings in mathematics (DE-604)BV000000067 129 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013837146&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Fulton, William 1939- Harris, Joe 1951- Representation theory a first course Graduate texts in mathematics Lie-Algebra - Darstellungstheorie Lie-Gruppe - Darstellungstheorie Gruppe Mathematik (DE-588)4022379-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Darstellung Mathematik (DE-588)4128289-9 gnd Lie-Algebra (DE-588)4130355-6 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4022379-6 (DE-588)4148816-7 (DE-588)4128289-9 (DE-588)4130355-6 (DE-588)4035695-4 |
| title | Representation theory a first course |
| title_auth | Representation theory a first course |
| title_exact_search | Representation theory a first course |
| title_full | Representation theory a first course William Fulton ; Joe Harris |
| title_fullStr | Representation theory a first course William Fulton ; Joe Harris |
| title_full_unstemmed | Representation theory a first course William Fulton ; Joe Harris |
| title_short | Representation theory |
| title_sort | representation theory a first course |
| title_sub | a first course |
| topic | Lie-Algebra - Darstellungstheorie Lie-Gruppe - Darstellungstheorie Gruppe Mathematik (DE-588)4022379-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Darstellung Mathematik (DE-588)4128289-9 gnd Lie-Algebra (DE-588)4130355-6 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Lie-Algebra - Darstellungstheorie Lie-Gruppe - Darstellungstheorie Gruppe Mathematik Darstellungstheorie Darstellung Mathematik Lie-Algebra Lie-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013837146&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT fultonwilliam representationtheoryafirstcourse AT harrisjoe representationtheoryafirstcourse |