Symmetry, representations, and invariants:
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| Previous Title: | Goodman, Roe Representations and invariants of the classical groups |
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| Main Authors: | , |
| Format: | Book |
| Language: | English |
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Springer
2009
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| Series: | Graduate texts in mathematics
255 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XX, 716 S. graph. Darst. |
| ISBN: | 9780387798516 |
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| 245 | 1 | 0 | |a Symmetry, representations, and invariants |c Roe Goodman ; Nolan R. Wallach |
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| 490 | 1 | |a Graduate texts in mathematics |v 255 | |
| 650 | 4 | |a Algebra | |
| 650 | 4 | |a Invariants | |
| 650 | 4 | |a Lie groups | |
| 650 | 4 | |a Representations of groups | |
| 650 | 4 | |a Symmetry (Mathematics) | |
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Titel: Symmetry, representations and invariants
Autor: Goodman, Roe
Jahr: 2009
Contents
Preface xv
Organization and Notation xix
1 Lie Groups and Algebraic Groups 1
1.1 The Classical Groups 1
1.1.1 General and Special Linear Groups 1
1.1.2 Isometry Groups of Bilinear Forms 3
1.1.3 Unitary Groups 7
1.1.4 Quaternionic Groups 8
1.1.5 Exercises 11
1.2 The Classical Lie Algebras 13
1.2.1 General and Special Linear Lie Algebras 13
1.2.2 Lie Algebras Associated with Bilinear Forms 14
1.2.3 Unitary Lie Algebras 15
1.2.4 Quaternionic Lie Algebras 16
1.2.5 Lie Algebras Associated with Classical Groups 17
1.2.6 Exercises 17
1.3 Closed Subgroups of GL(n,R) 18
1.3.1 Topological Groups 18
1.3.2 Exponential Map 19
1.3.3 Lie Algebra of a Closed Subgroup of GL(n, M) 23
1.3.4 Lie Algebras of the Classical Groups 25
1.3.5 Exponential Coordinates on Closed Subgroups 28
1.3.6 Differentials of Homomorphisms 30
1.3.7 Lie Algebras and Vector Fields 31
1.3.8 Exercises 34
1.4 Linear Algebraic Groups 35
1.4.1 Definitions and Examples 35
1.4.2 Regular Functions 36
1.4.3 Lie Algebra of an Algebraic Group 39
v
vi Contents
1.4.4 Algebraic Groups as Lie Groups 43
1.4.5 Exercises 45
1.5 Rational Representations 47
1.5.1 Definitions and Examples 47
1.5.2 Differential of a Rational Representation 49
1.5.3 The Adjoint Representation 53
1.5.4 Exercises 54
1.6 Jordan Decomposition 55
1.6.1 Rational Representations of C 55
1.6.2 Rational Representations of Cx 57
1.6.3 Jordan-Chevalley Decomposition 58
1.6.4 Exercises 61
1.7 Real Forms of Complex Algebraic Groups 62
1.7.1 Real Forms and Complex Conjugations 62
1.7.2 Real Forms of the Classical Groups 65
1.7.3 Exercises 67
1.8 Notes 68
2 Structure of Classical Groups 69
2.1 Semisimple Elements 69
2.1.1 Toral Groups 70
2.1.2 Maximal Torus in a Classical Group 72
2.1.3 Exercises 76
2.2 Unipotent Elements 77
2.2.1 Low-Rank Examples 77
2.2.2 Unipotent Generation of Classical Groups 79
2.2.3 Connected Groups 81
2.2.4 Exercises 83
2.3 Regular Representations of SL(2, C) 83
2.3.1 Irreducible Representations of sl(2,C) 84
2.3.2 Irreducible Regular Representations of SL(2, C) 86
2.3.3 Complete Reducibility of SL(2,C) 88
2.3.4 Exercises 90
2.4 The Adjoint Representation 91
2.4.1 Roots with Respect to a Maximal Torus 91
2.4.2 Commutation Relations of Root Spaces 95
2.4.3 Structure of Classical Root Systems 99
2.4.4 Irreducibility of the Adjoint Representation 104
2.4.5 Exercises 106
2.5 Semisimple Lie Algebras 108
2.5.1 Solvable Lie Algebras 108
2.5.2 Root Space Decomposition 114
2.5.3 Geometry of Root Systems 118
2.5.4 Conjugacy of Cartan Subalgebras 122
2.5.5 Exercises 125
Contents vii
2.6 Notes 125
3 Highest-Weight Theory 127
3.1 Roots and Weights 127
3.1.1 Weyl Group 128
3.1.2 Root Reflections 132
3.1.3 Weight Lattice 138
3.1.4 Dominant Weights 141
3.1.5 Exercises 145
3.2 Irreducible Representations 147
3.2.1 Theorem of the Highest Weight 148
3.2.2 Weights of Irreducible Representations 153
3.2.3 Lowest Weights and Dual Representations 157
3.2.4 Symplectic and Orthogonal Representations 158
3.2.5 Exercises 161
3.3 Reductivity of Classical Groups 163
3.3.1 Reductive Groups 163
3.3.2 Casimir Operator 166
3.3.3 Algebraic Proof of Complete Reducibility 169
3.3.4 The Unitarian Trick 171
3.3.5 Exercises 173
3.4 Notes 174
4 Algebras and Representations 175
4.1 Representations of Associative Algebras 175
4.1.1 Definitions and Examples 175
4.1.2 Schur's Lemma 180
4.1.3 Jacobson Density Theorem 181
4.1.4 Complete Reducibility 182
4.1.5 Double Commutant Theorem 184
4.1.6 Isotypic Decomposition and Multiplicities 184
4.1.7 Characters 187
4.1.8 Exercises 191
4.2 Duality for Group Representations 195
4.2.1 General Duality Theorem 195
4.2.2 Products of Reductive Groups 197
4.2.3 Isotypic Decomposition of Q[G] 199
4.2.4 Schur-Weyl Duality 200
4.2.5 Commuting Algebra and Highest-Weight Vectors 203
4.2.6 Abstract Capelli Theorem 204
4.2.7 Exercises 206
4.3 Group Algebras of Finite Groups 206
4.3.1 Structure of Group Algebras 206
4.3.2 Schur Orthogonality Relations 208
4.3.3 Fourier Inversion Formula 209
viii Contents
4.3.4 The Algebra of Central Functions 211
4.3.5 Exercises 214
4.4 Representations of Finite Groups 215
4.4.1 Induced Representations 216
4.4.2 Characters of Induced Representations 217
4.4.3 Standard Representation of „ 218
4.4.4 Representations of k on Tensors 221
4.4.5 Exercises 222
4.5 Notes 224
5 Classical Invariant Theory 225
5.1 Polynomial Invariants for Reductive Groups 226
5.1.1 The Ring of Invariants 226
5.1.2 Invariant Polynomials for 6,, 228
5.1.3 Exercises 234
5.2 Polynomial Invariants 237
5.2.1 First Fundamental Theorems for Classical Groups 238
5.2.2 Proof of a Basic Case 242
5.2.3 Exercises 246
5.3 Tensor Invariants 246
5.3.1 Tensor Invariants for GL(V) 247
5.3.2 Tensor Invariants for O(V) and Sp(V) 248
5.3.3 Exercises 254
5.4 Polynomial FFT for Classical Groups 256
5.4.1 Invariant Polynomials as Tensors 256
5.4.2 Proof of Polynomial FFT for GL( V) 258
5.4.3 Proof of Polynomial FFT for O(V) and Sp( V) 258
5.5 Irreducible Representations of Classical Groups 259
5.5.1 Skew Duality for Classical Groups 259
5.5.2 Fundamental Representations 268
5.5.3 Cartan Product 272
5.5.4 Irreducible Representations of GL(V) 273
5.5.5 Irreducible Representations of O(V) 275
5.5.6 Exercises 277
5.6 Invariant Theory and Duality 278
5.6.1 Duality and the Weyl Algebra 278
5.6.2 GL(k)-GL( ) Schur-Weyl Duality 282
5.6.3 O(n)sp(k) Howe Duality 284
5.6.4 Spherical Harmonics 287
5.6.5 Sp(«)-so(2A:) Howe Duality 290
5.6.6 Exercises 292
5.7 Further Applications of Invariant Theory 293
5.7.1 Capelli Identities 293
5.7.2 Decomposition of S(S2(V)) under GL(V) 295
5.7.3 Decomposition of S(f\2(V)) under GL(V) 296
Contents ix
5.7.4 Exercises 298
5.8 Notes 298
6 Spinors 301
6.1 Clifford Algebras 301
6.1.1 Construction of Cliff (V) 301
6.1.2 Spaces of Spinors 303
6.1.3 Structure of Cliff (V) 307
6.1.4 Exercises 310
6.2 Spin Representations of Orthogonal Lie Algebras 311
6.2.1 Embedding so(V) in Cliff(V) 312
6.2.2 Spin Representations 314
6.2.3 Exercises 315
6.3 Spin Groups 316
6.3.1 Action ofO(V) on Cliff(V) 316
6.3.2 Algebraically Simply Connected Groups 321
6.3.3 Exercises 322
6.4 Real Forms of Spin(n,C) 323
6.4.1 Real Forms of Vector Spaces and Algebras 323
6.4.2 Real Forms of Clifford Algebras 325
6.4.3 Real Forms of Pin(n) and Spin(n) 325
6.4.4 Exercises 327
6.5 Notes 327
7 Character Formulas 329
7.1 Character and Dimension Formulas 329
7.1.1 Weyl Character Formula 329
7.1.2 Weyl Dimension Formula 334
7.1.3 Commutant Character Formulas 337
7.1.4 Exercises 339
7.2 Algebraic Group Approach to the Character Formula 342
7.2.1 Symmetric and Skew-Symmetric Functions 342
7.2.2 Characters and Skew-Symmetric Functions 344
7.2.3 Characters and Invariant Functions 346
7.2.4 Casimir Operator and Invariant Functions 347
7.2.5 Algebraic Proof of the Weyl Character Formula 352
7.2.6 Exercises 353
7.3 Compact Group Approach to the Character Formula 354
7.3.1 Compact Form and Maximal Compact Torus 354
7.3.2 Weyl Integral Formula 356
7.3.3 Fourier Expansions of Skew Functions 358
7.3.4 Analytic Proof of the Weyl Character Formula 360
7.3.5 Exercises 361
7.4 Notes 362
x Contents
8 Branching Laws 363
8.1 Branching for Classical Groups 363
8.1.1 Statement of Branching Laws 364
8.1.2 Branching Patterns and Weight Multiplicities 366
8.1.3 Exercises 368
8.2 Branching Laws from Weyl Character Formula 370
8.2.1 Partition Functions 370
8.2.2 Kostant Multiplicity Formulas 371
8.2.3 Exercises 372
8.3 Proofs of Classical Branching Laws 373
8.3.1 Restriction from GL(n) to GL(n - 1) 373
8.3.2 Restriction from Spin(2n + 1) to Spin(2n) 375
8.3.3 Restriction from Spin(2«) to Spin(2« - 1) 378
8.3.4 Restriction from Sp(n) to Sp(« - 1) 379
8.4 Notes 384
9 Tensor Representations of GL(V) 387
9.1 Schur-Weyl Duality 387
9.1.1 Duality between GL(») and 6* 388
9.1.2 Characters of 6k 391
9.1.3 Frobenius Formula 394
9.1.4 Exercises 396
9.2 Dual Reductive Pairs 399
9.2.1 Seesaw Pairs 399
9.2.2 Reciprocity Laws 401
9.2.3 Schur-Weyl Duality and GL(*)-GL(n) Duality 405
9.2.4 Exercises 406
9.3 Young Symmetrizers and Weyl Modules 407
9.3.1 Tableaux and Symmetrizers 407
9.3.2 Weyl Modules 412
9.3.3 Standard Tableaux 414
9.3.4 Projections onto Isotypic Components 416
9.3.5 Littlewood-Richardson Rule 418
9.3.6 Exercises 421
9.4 Notes 423
10 Tensor Representations of O(V) and Sp(V) 425
10.1 Commuting Algebras on Tensor Spaces 425
10.1.1 Centralizer Algebra 426
10.1.2 Generators and Relations 432
10.1.3 Exercises 434
10.2 Decomposition of Harmonic Tensors 435
10.2.1 Harmonic Tensors 435
10.2.2 Harmonic Extreme Tensors 436
10.2.3 Decomposition of Harmonics for Sp(V) 440
Contents xi
10.2.4 Decomposition of Harmonics for O(2/ + 1) 442
10.2.5 Decomposition of Harmonics for O(2Z) 446
10.2.6 Exercises 451
10.3 Riemannian Curvature Tensors 451
10.3.1 The Space of Curvature Tensors 453
10.3.2 Orthogonal Decomposition of Curvature Tensors 455
10.3.3 The Space of Weyl Curvature Tensors 458
10.3.4 Exercises 460
10.4 Invariant Theory and Knot Polynomials 461
10.4.1 The Braid Relations 461
10.4.2 Orthogonal Invariants and the Yang-Baxter Equation 463
10.4.3 The Braid Group 464
10.4.4 The Jones Polynomial 469
10.4.5 Exercises 475
10.5 Notes 476
11 Algebraic Groups and Homogeneous Spaces 479
11.1 General Properties of Linear Algebraic Groups 479
11.1.1 Algebraic Groups as Affine Varieties 479
11.1.2 Subgroups and Homomorphisms 481
11.1.3 Group Structures on Affine Varieties 484
11.1.4 Quotient Groups 485
11.1.5 Exercises 490
11.2 Structure of Algebraic Groups 491
11.2.1 Commutative Algebraic Groups 491
11.2.2 Unipotent Radical 493
11.2.3 Connected Algebraic Groups and Lie Groups 496
11.2.4 Simply Connected Semisimple Groups 497
11.2.5 Exercises 500
11.3 Homogeneous Spaces 500
11.3.1 G-Spaces and Orbits 500
11.3.2 Flag Manifolds 501
11.3.3 Involutions and Symmetric Spaces 506
11.3.4 Involutions of Classical Groups 507
11.3.5 Classical Symmetric Spaces 510
11.3.6 Exercises 516
11.4 Borel Subgroups 519
11.4.1 Solvable Groups 519
11.4.2 Lie-Kolchin Theorem 520
11.4.3 Structure of Connected Solvable Groups 522
11.4.4 Conjugacy of Borel Subgroups 524
11.4.5 Centralizer of a Torus 525
11.4.6 Weyl Group and Regular Semisimple Conjugacy Classes . 526
11.4.7 Exercises 530
11.5 Further Properties of Real Forms 531
xii Contents
11.5.1 Groups with a Compact Real Form 531
11.5.2 Polar Decomposition by a Compact Form 536
11.6 Gauss Decomposition 538
11.6.1 Gauss Decomposition of GL(w.C) 538
11.6.2 Gauss Decomposition of an Algebraic Group 540
11.6.3 Gauss Decomposition for Real Forms 541
11.6.4 Exercises 543
11.7 Notes 543
12 Representations on Spaces of Regular Functions 545
12.1 Some General Results 545
12.1.1 Isotypic Decomposition of 0[X] 546
12.1.2 Frobenius Reciprocity 548
12.1.3 Function Models for Irreducible Representations 549
12.1.4 Exercises 550
12.2 Multiplicity-Free Spaces 551
12.2.1 Multiplicities and B-Orbits 552
12.2.2 B-Eigenfunctions for Linear Actions 553
12.2.3 Branching from GL(n) to GL(n - 1) 554
12.2.4 Second Fundamental Theorems 557
12.2.5 Exercises 563
12.3 Regular Functions on Symmetric Spaces 566
12.3.1 Iwasawa Decomposition 566
12.3.2 Examples of Iwasawa Decompositions 575
12.3.3 Spherical Representations 585
12.3.4 Exercises 587
12.4 Isotropy Representations of Symmetric Spaces 588
12.4.1 A Theorem of Kostant and Rallis 589
12.4.2 Invariant Theory of Reflection Groups 590
12.4.3 Classical Examples 593
12.4.4 Some Results from Algebraic Geometry 597
12.4.5 Proof of the Kostant-Rallis Theorem 601
12.4.6 Some Remarks on the Proof 605
12.4.7 Exercises 607
12.5 Notes 609
A Algebraic Geometry 611
A. 1 Affine Algebraic Sets 611
A. 1.1 Basic Properties 611
A.1.2 Zariski Topology 615
A.1.3 Products of Affine Sets 616
A.1.4 Principal Open Sets 617
A. 1.5 Irreducible Components 617
A. 1.6 Transcendence Degree and Dimension 619
A. 1.7 Exercises 621
Contents xiii
A.2 Maps of Algebraic Sets 622
A.2.1 Rational Maps 622
A.2.2 Extensions of Homomorphisms 623
A.2.3 Image of a Dominant Map 626
A.2.4 Factorization of a Regular Map 626
A.2.5 Exercises 627
A.3 Tangent Spaces 628
A.3.1 Tangent Space and Differentials of Maps 628
A.3.2 Vector Fields 630
A.3.3 Dimension 630
A.3.4 Differential Criterion for Dominance 632
A.4 Projective and Quasiprojective Sets 635
A.4.1 Basic Definitions 635
A.4.2 Products of Projective Sets 637
A.4.3 Regular Functions and Maps 638
B Linear and Multilinear Algebra 643
B.I Jordan Decomposition 643
B. 1.1 Primary Projections 643
B. 1.2 Additive Jordan Decomposition 644
B.1.3 Multiplicative Jordan Decomposition 645
B.2 Multilinear Algebra 645
B.2.1 Bilinear Forms 646
B.2.2 Tensor Products 647
B.2.3 Symmetric Tensors 650
B.2.4 Alternating Tensors 653
B.2.5 Determinants and Gauss Decomposition 654
B.2.6 Pfaffians and Skew-Symmetric Matrices 656
B.2.7 Irreducibility of Determinants and Pfaffians 659
C Associative Algebras and Lie Algebras 661
C.I Some Associative Algebras 661
C.I.I Filtered and Graded Algebras 661
C.1.2 Tensor Algebra 663
C.I.3 Symmetric Algebra 663
C.1.4 Exterior Algebra 666
C.I.5 Exercises 668
C.2 Universal Enveloping Algebras 668
C.2.1 Lie Algebras 668
C.2.2 Poincare-Birkhoff-Witt Theorem 670
C.2.3 Adjoint Representation of Enveloping Algebra 672
xiv Contents
D Manifolds and Lie Groups 675
D.I C°° Manifolds 675
D.I.I Basic Definitions 675
D. 1.2 Tangent Space 680
D. 1.3 Differential Forms and Integration 683
D. 1.4 Exercises 686
D.2 Lie Groups 687
D.2.1 Basic Definitions 687
D.2.2 Lie Algebra of a Lie Group 688
D.2.3 Homogeneous Spaces 691
D.2.4 Integration on Lie Groups and Homogeneous Spaces 692
D.2.5 Exercises 696
References 697
Index of Symbols 705
Subject Index 709
Symmetry is a key ingredient in many mathematical, physical, and biological theories.
Using representation theory and invariant theory to analyze the symmetries that arise
from group actions, and with strong emphasis on the geometry and basic theory of Lie
groups and Lie algebras, Symmetry, Representations, and invariants is a significant
reworking of an earlier highly-acclaimed work by the authors. The result is a
comprehensive introduction to Lie theory, representation theory, invariant theory, and
algebraic groups, in a new presentation that is more accessible to students and includes
a broader range of applications.
The philosophy of the earlier book is retained, i.e., presenting the principal theorems of
representation theory for the classical matrix groups as motivation for the general theory
of reductive groups. The wealth of examples and discussion prepares the reader for the
complete arguments now given in the general case.
Key Features of Symmetry, Representations, and Invariants:
•
Early chapters suitable for honors undergraduate or beginning graduate courses,
requiring only linear algebra, basic abstract algebra, and advanced calculus
•
Applications to geometry (curvature tensors), topology (Jones polynomial via
symmetry), and combinatorics (symmetric group and Young tableaux)
•
Self-contained chapters, appendices, comprehensive bibliography
•
More than
350
exercises (most with detailed hints for solutions) further explore main
concepts
•
Serves as an excellent main text for a one-year course in Lie group theory
•
Benefits physicists as well as mathematicians as a reference work |
| any_adam_object | 1 |
| author | Goodman, Roe 1938- Wallach, Nolan R. 1940- |
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| dewey-ones | 512 - Algebra |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035506427 |
| illustrated | Illustrated |
| indexdate | 2024-09-24T00:16:22Z |
| institution | BVB |
| isbn | 9780387798516 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017562560 |
| oclc_num | 382399201 |
| open_access_boolean | |
| owner | DE-20 DE-706 DE-384 DE-703 DE-11 DE-91G DE-BY-TUM DE-29T DE-188 DE-824 DE-739 DE-83 DE-355 DE-BY-UBR DE-19 DE-BY-UBM |
| owner_facet | DE-20 DE-706 DE-384 DE-703 DE-11 DE-91G DE-BY-TUM DE-29T DE-188 DE-824 DE-739 DE-83 DE-355 DE-BY-UBR DE-19 DE-BY-UBM |
| physical | XX, 716 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Goodman, Roe 1938- Verfasser (DE-588)123134943 aut Symmetry, representations, and invariants Roe Goodman ; Nolan R. Wallach New York, NY Springer 2009 XX, 716 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 255 Algebra Invariants Lie groups Representations of groups Symmetry (Mathematics) Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Invariante (DE-588)4128781-2 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 s Darstellungstheorie (DE-588)4148816-7 s Invariante (DE-588)4128781-2 s DE-604 Wallach, Nolan R. 1940- Verfasser (DE-588)133231690 aut Erscheint auch als Online-Ausgabe 978-0-387-79852-3 Frühere Ausg. u.d.T. Goodman, Roe Representations and invariants of the classical groups Graduate texts in mathematics 255 (DE-604)BV000000067 255 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017562560&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017562560&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Goodman, Roe 1938- Wallach, Nolan R. 1940- Symmetry, representations, and invariants Graduate texts in mathematics Algebra Invariants Lie groups Representations of groups Symmetry (Mathematics) Darstellungstheorie (DE-588)4148816-7 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd Invariante (DE-588)4128781-2 gnd |
| subject_GND | (DE-588)4148816-7 (DE-588)4295326-1 (DE-588)4128781-2 |
| title | Symmetry, representations, and invariants |
| title_auth | Symmetry, representations, and invariants |
| title_exact_search | Symmetry, representations, and invariants |
| title_full | Symmetry, representations, and invariants Roe Goodman ; Nolan R. Wallach |
| title_fullStr | Symmetry, representations, and invariants Roe Goodman ; Nolan R. Wallach |
| title_full_unstemmed | Symmetry, representations, and invariants Roe Goodman ; Nolan R. Wallach |
| title_old | Goodman, Roe Representations and invariants of the classical groups |
| title_short | Symmetry, representations, and invariants |
| title_sort | symmetry representations and invariants |
| topic | Algebra Invariants Lie groups Representations of groups Symmetry (Mathematics) Darstellungstheorie (DE-588)4148816-7 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd Invariante (DE-588)4128781-2 gnd |
| topic_facet | Algebra Invariants Lie groups Representations of groups Symmetry (Mathematics) Darstellungstheorie Lineare algebraische Gruppe Invariante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017562560&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017562560&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT goodmanroe symmetryrepresentationsandinvariants AT wallachnolanr symmetryrepresentationsandinvariants |