Representing finite groups: a semisimple introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2012
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 371 S. |
| ISBN: | 9781461412304 |
Internformat
MARC
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| 245 | 1 | 0 | |a Representing finite groups |b a semisimple introduction |c Ambar N. Sengupta |
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c 2012 | |
| 300 | |a XV, 371 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-027202226 | ||
Datensatz im Suchindex
| _version_ | 1804152056640962560 |
|---|---|
| adam_text | Contents
Concepts
and Constructs
1
1.1
Representations of Groups
................... 2
1.2
Representations and Their Morphisms
............. 4
1.3
Direct Sums and Tensor Products
............... 4
1.4
Change of Field
......................... 5
1.5
Invariant Subspaces and Quotients
...............
(i
1.6
Dual Representations
...................... 7
1.7
Irreducible Representations
................... 10
1.8 Schürt
Lemma
.......................... 12
1.9
The Frobonius
Schur
Indicator
................. 14
1.10
Character of a Representation
................. 1С
1.11
Diagonalizability
......................... 20
1.12
Unitarity
............................. 22
1.13
Rival Reads
........................... 24
1.14
Afterthoughts: Lattices
..................... 25
Exercises
.............................. 28
A Reckoning
............................... 37
Basic Examples
39
2.1
Cyclic Groups
.......................... 40
2.2
Dihedral Groups
......................... 43
2.3
The Symmetric Group 54
.................... 48
2.4
Quaternionic Units
....................... 52
2.5
Afterthoughts: Geometric Groups
............... 54
Exercises
.............................. 56
XI
XII CONTENTS
3
The Group Algebra
59
3.1
Definition of the Group Algebra
................ 60
3.2
Representations of
G
and ¥[G]
................. 61
3.3
The Center
............................ 63
3.4
Deconstructing ¥[S3]
...................... 65
3.5
When ¥[G) Is
Semisimple.................... 73
3.6
Afterthoughts: Invariants
.................... 77
Exercises
.............................. 79
4
More Group Algebra
83
4.1
Looking Ahead
.......................... 84
4.2
Submodules
and Idempotents
.................. 86
4.3
Deconstructing
¥ G),
the Module
................ 89
4.4
Deconstructing ¥[G], the Algebra
............... 91
4.5
As Simple as Matrix Algebras
................. 97
4.6
Putting ¥[G] Back Together
..................102
4.7
The Mother of All Representations
...............107
4.8
The Center
...........................109
4.9
Representing Abelian Groups
..................111
4.10
Indecomposable Idempotents
..................112
4.11
Beyond Our Borders
.......................114
Exercises
..............................115
5
Simply Semisimple
125
5.1 Schur
s
Lemma
..........................126
5.2
Semisimple Modules
.......................127
5.3
Deconstructing Semisimple Modules
..............131
5.4
Simple Modules for Semisimple Rings
.............134
5.5
Deconstructing Semisimple Rings
...............136
5.6
Simply Simple
..........................140
5.7
Commutants
and Double
Commutants
.............141
5.8
Artin-Wedderburn Structure
..................144
5.9
A Module as the Sum of Its Parts
...............145
510
Readings on Rings
........................147
5.11
Afterthoughts: Clifford Algebras
................147
Exercises
..............................151
CONTENTS
ХШ
6
Representations of Sn
157
6.1
Permutations and Partitions
.................. 157
6.2
Complements and Young Tableaux
............... 161
6.3
Symmetries of Partitions
....................165
6.4
Conjugacy Classes to Young Tableaux
............. 168
6.5
Young Tableaux to Young Symmetrizers
............ 169
6.6
Youngtabs to Irreducible Representations
........... 170
6.7
Youngtab Apps
......................... 174
6.8
Orthogonality
.......................... 179
6.9
Deconstructing ¥[Sn]
...................... 180
6.10
Integrality
............................ 183
6.11
Rivals and Rebels
........................ 184
6.12
Afterthoughts: Reflections
................... 184
Exercises
.............................. 188
7
Characters
189
7.1
The Regular Character
....................190
7.2
Character Orthogonality
....................194
7.3
Character Expansions
......................203
7.4
Comparing Z-Bases
.......................206
7.5
Character Arithmetic
......................208
7.6
Computing Characters
.....................211
7.7
Return of the Group Determinant
........-......214
7.8
Orthogonality of Matrix Elements
..............217
7.9
Solving Equations in Groups
..................219
7.10
Character References
......................228
7.11
Afterthoughts: Connections
.......-...........229
Exercises
..............................230
8
Induced Representations
235
8.1
Constructions
..........................235
8.2
The Induced Character
.....................238
8.3
Induction Workout
........................239
8.4
Universality
...........................242
8.5
Universal Consequences
.....................243
8.6
Reciprocity
............................245
8.7
Afterthoughts: Numbers
....................247
Exercises
..............................248
XIV
CONTENTS
9
Commutant
Duality
249
9-1
The Commutant
.........................249
9.2
The Double
Commutant ....................
251
9.3
Commutant
Decomposition of a Module
............254
9.4
The Matrix Version
.......................260
Exercises
..............................264
10
Character Duality
267
10.1
The
Commutant
for 5n on V®n
................267
10.2
Schur-Weyl Duality
.......................269
10.3
Character Duality, the High Road
...............270
10.4
Character Duality by Calculations
...............271
Exercises
..............................278
11
Representations of
ГУ
(TV)
281
11.1
The
Haar
Integral
........................ 282
11.2
The Weyl Integration Formula
................. 283
11.3
Character Orthogonality
................... 284
11.4
Weights
.......................... 285
11.5
Characters of U{N)
...................... 286
11.6
Weyl Dimension Formula
.................... 290
11.7
From Weights to Representations
................ 291
11.8
Characters of Sn from Characters of U(N)
...........294
Exercises
..............................299
12
Postscript: Algebra
301
12.1
Groups and Less
.........................301
12.2
Rings and More
.........................306
12.3
Fields
...............................314
12.4
Modules over Rings
........................315
12.5
Free Modules and Bases
.....................319
12.6
Power Series and Polynomials
..................323
12.7
Algebraic Integers
.........................329
12.8
Linear Algebra
..........................330
12.9
Tensor Products
.........................335
12.10
Extension of Base Ring
.......-.............338
CONTENTS
XV
12.11
Determinants and TV
ас
es
of Matrices
............. 339
12.12
Exterior Powers
......................... 340
12.13
Eigenvalues and Eigenvectors
.................. 345
12.14
Topology, Integration, and Hilbeit Spaces
........... 346
Bibliography
353
Index
361
|
| any_adam_object | 1 |
| author | Sengupta, Ambar 1963- |
| author_GND | (DE-588)120724952 |
| author_facet | Sengupta, Ambar 1963- |
| author_role | aut |
| author_sort | Sengupta, Ambar 1963- |
| author_variant | a s as |
| building | Verbundindex |
| bvnumber | BV041755980 |
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| dewey-full | 512/.23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.23 |
| dewey-search | 512/.23 |
| dewey-sort | 3512 223 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T01:04:39Z |
| institution | BVB |
| isbn | 9781461412304 |
| language | English |
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| physical | XV, 371 S. |
| publishDate | 2012 |
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| spelling | Sengupta, Ambar 1963- Verfasser (DE-588)120724952 aut Representing finite groups a semisimple introduction Ambar N. Sengupta New York, NY [u.a.] Springer 2012 XV, 371 S. txt rdacontent n rdamedia nc rdacarrier Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4614-1231-1 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027202226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sengupta, Ambar 1963- Representing finite groups a semisimple introduction Darstellungstheorie (DE-588)4148816-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| subject_GND | (DE-588)4148816-7 (DE-588)4014651-0 |
| title | Representing finite groups a semisimple introduction |
| title_auth | Representing finite groups a semisimple introduction |
| title_exact_search | Representing finite groups a semisimple introduction |
| title_full | Representing finite groups a semisimple introduction Ambar N. Sengupta |
| title_fullStr | Representing finite groups a semisimple introduction Ambar N. Sengupta |
| title_full_unstemmed | Representing finite groups a semisimple introduction Ambar N. Sengupta |
| title_short | Representing finite groups |
| title_sort | representing finite groups a semisimple introduction |
| title_sub | a semisimple introduction |
| topic | Darstellungstheorie (DE-588)4148816-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| topic_facet | Darstellungstheorie Endliche Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027202226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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