The classical groups: their invariants and representations
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, NJ
Princeton Univ. Pr.
1997
|
| Edition: | 15. print., and 1. paperback print. |
| Series: | Princeton landmarks in mathematics and physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XIII, 320 S. |
| ISBN: | 0691057567 9780691057569 |
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MARC
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| 100 | 1 | |a Weyl, Hermann |d 1885-1955 |e Verfasser |0 (DE-588)118816624 |4 aut | |
| 245 | 1 | 0 | |a The classical groups |b their invariants and representations |c by Hermann Weyl |
| 250 | |a 15. print., and 1. paperback print. | ||
| 264 | 1 | |a Princeton, NJ |b Princeton Univ. Pr. |c 1997 | |
| 300 | |a XIII, 320 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Princeton landmarks in mathematics and physics | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Continuous groups | |
| 650 | 4 | |a Group theory | |
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Record in the Search Index
| _version_ | 1804135762797527040 |
|---|---|
| adam_text | TABLE
OF
CONTENTS
PAGE
Preface to the
Fibst Edition
................................................
vii
Preface to the Second Edition
.............................................. ix
Chapter I
INTRODUCTION
PAGE
1.
Fields, rings, ideals, polynomials
............................................... 1
2.
Vector space
................................................................... 6
3.
Orthogonal transformations, Euclidean vector geometry
........................ 11
4.
Groups, Klein s
Erlanger
program. Quantities
................................. 13
5.
Invariants and
covariante
...................................................... 23
Chapter II
VECTOR INVARIANTS
1.
Remembrance of things past
................................................... 27
2.
The main propositions of the theory of invariants
.............................. 29
A. First Main Theorem
3.
First example: the symmetric group
........................................... 36
4.
Capelli s identity
............................................................. 39
5.
Reduction of the first main problem by means of Capelli s identities
....... . . 42
6.
Second example: the unimodular group SL{n)
.............................. 45
7.
Extension theorem. Third example: the group of step transformations.
......... 47
8.
A general method for including
contravariant
arguments.
.................... 49
9.
Fourth example: the orthogonal group
—..................................... 52
B. A Close-Up of the Orthogonal Group
10.
Cayley s rational parametrization of the orthogonal group
...................... 56
11-
Formal orthogonal invariants
............................................... 62
12.
Arbitrary metric ground form
................................................ 65
13.
The infinitesimal standpoint
.................................................. 66
C. The Second Main Theorem
14.
Statement of the proposition for the unimodular group
........................ 70
15.
Capelii s formal congruence
................................................ 72
16.
Proof of the second main theorem for the unimodular group
................... 73
17.
The second main theorem for the unimodular group
........................... 75
Chapter III
MATRIC
ALGEBRAS AND GROUP RINGS
A. Theory of Fully Reducible
Matric
Algebras
1.
Fundamental notions concerning
matric
algebras. The
Schur
lemma
........... 79
2.
Preliminaries
.................... .............. ................. 84
3.
Representations of a simple algebra
............................... 87
4.
Wedderburn s theorem.
................................ 90
5. The fully reducible
matric algebra
and its commutator algebra
............ 93
B. The
Riso
of a Finite Group and Its Commctatob Algebra
β.
Stating the problem
. ............. ........................ 96
7.
Fuii
reducibiîity
of the group ring
........................... 101
xi
XU
TABLE
OF
CONTENTS
PAGE
8.
Formal lemmas
................................................................
Ю6
9.
Reciprocity between group ring and commutator algebra
........................ 107
10.
A generalization
...............................................................
112
Chapter IV
THE SYMMETRIC GROUP AND THE FULL LINEAR GROUP
1.
Representation of a finite group in an algebraically closed field
................. 115
2.
The Young symmetrizers. A combinatorial lemma
............................. 119
3.
The irreducible representations of the symmetric group
......................... 124
4.
Decomposition of tensor space
................................................. 127
5.
Quantities. Expansion
........................................................ 131
Chapter V
THE ORTHOGONAL GROUP
A. The Enveloping
Algebra
and the Orthogonal Ideal
1.
Vector invariants of the unimodular group again
............................... 137
2.
The enveloping algebra of the orthogonal group
................................ 140
3.
Giving the result its formal setting
............................................. 143
4.
The orthogonal prime ideal
..................................................... 144
5.
An abstract algebra related to the orthogonal group
............................ 147
B. The Ihrkducible Representations
6.
Decomposition by the trace operation
.......................................... 149
7.
The irreducible representations of the full orthogonal group
.................... 153
C. The Proper Orthogonal Gbodp
8.
Clifford s theorem
.............................................................. 159
9.
Representations of the proper orthogonal group
................................ 163
Chapter VI
THE SYMPLECTIC GROUP
1.
Vector invariants of the symplectic group
..................................... 165
2.
Parametrization and unitary restriction
........................................ 169
3.
Embedding algebra and representations of the symplectic group
................ 173
Chapter
VII
CHARACTERS
1.
Preliminaries about unitary transformations
.................................... 176
2.
Character for symmetrization or alternation alone
.............................. 181
3.
Averaging over a group
........................................................ 185
4.
The volume element of the unitary group
...................................... 194
5.
Computation of the characters
................................................. 198
6.
The characters of GL(n). Enumeration of
covariante
.......................... 201
7.
A purely algebraic approach
................................................... 208
8.
Characters of the symplectic group
............................................. 216
9.
Characters of the orthogonal group
............................................. 222
10.
Decomposition and X-multiplication
........................................... 229
11.
The
Poincaré
polynomial
....................................................... 232
TABLE
OF
CONTENTS
Х1Ц
Chapter
VIII
GENERAL
THEORY OF INVARIANTS
A. Algebraic
Рант
page
1.
Classic invariants and invariants of
quantice.
Gram s theorem
................ 239
2.
The symbolic method
......................................................... 243
3.
The binary quadratic
......................................................... 246
4.
Irrational methods
............................................................. 248
5.
Side remarks
.................................................................. 250
6.
Hubert s theorem on polynomial ideals
........................................ 251
7.
Proof of the first main theorem for QL(n)
...................................... 252
8.
The adjunction argument
..................................................... 254
B.
Differenti al
and Integral Methods
9.
Group germ and Lie algebras
.................................................. 258
10.
Differential equations for invariants. Absolute and relative invariants
......... 262
11.
The
unitarian
trick
........................................................... 265
12.
The connectivity of the classical groups
...................................... 268
13.
Spinors
...................................................................... 270
14.
Finite integrity basis for invariants of compact groups
........................ 274
15.
The first main theorem for finite groups
....................................... 275
16.
Invariant differentials and
Betti
numbers of a compact Lie group
............... 276
Chapter IX
MATRIC
ALGEBRAS RESUMED
1.
Automorphisms
.............................................................. 280
2.
A lemma on multiplication
..................................................... 283
3.
Products of simple algebras
................................................... 286
4.
Adjunction
................................................................. 288
Chapter X
SUPPLEMENTS
A. Supplement
то
Chapter II,
§§9-13,
and Chaptee VI,
§1,
Concerning
Infinitesimal
Vector Invariants
1.
An identity for infinitesimal orthogonal invariants
............................ 291
2.
First Main Theorem for the orthogonai group
............................... 293
3.
The same for the symplectic group
......................................... 294
B. Supplement to Chapter V,
§3,
and Chapter VI,
§§2
and
3,
Concerning the
Symplectic and Okthogonal Ideals
4.
A proposition on full reduction
............................................. 295
5.
The symplectic ideal
...................................................... 296
6.
The
fuli
and the proper orthogonal ideals
................................... 299
C. Supplement to Chapteb
VIII, §§7-8,
Concerning.
7.
A modified proof of the main theorem on invariants
......................... 300
D. Supplement to Chaptek IX,
§4,
About Extension of the Ground Field
8.
Effect of field extension on a division algebra
................................ 303
Erbata
and Addenda
....................................................... 307
Bibliography
............................................................... 308
Supplementaby BIBLIOGRAPHY, Mainly for the Years
1940-1945............... 314
Index
...................................................................... 317
|
| adam_txt |
TABLE
OF
CONTENTS
PAGE
Preface to the
Fibst Edition
.
vii
Preface to the Second Edition
. ix
Chapter I
INTRODUCTION
PAGE
1.
Fields, rings, ideals, polynomials
. 1
2.
Vector space
. 6
3.
Orthogonal transformations, Euclidean vector geometry
. 11
4.
Groups, Klein's
Erlanger
program. Quantities
. 13
5.
Invariants and
covariante
. 23
Chapter II
VECTOR INVARIANTS
1.
Remembrance of things past
. 27
2.
The main propositions of the theory of invariants
. 29
A. First Main Theorem
3.
First example: the symmetric group
. 36
4.
Capelli's identity
. 39
5.
Reduction of the first main problem by means of Capelli's identities
. . . 42
6.
Second example: the unimodular group SL{n)
. 45
7.
Extension theorem. Third example: the group of step transformations.
. 47
8.
A general method for including
contravariant
arguments.
. 49
9.
Fourth example: the orthogonal group
—. 52
B. A Close-Up of the Orthogonal Group
10.
Cayley's rational parametrization of the orthogonal group
. 56
11-
Formal orthogonal invariants
. 62
12.
Arbitrary metric ground form
. 65
13.
The infinitesimal standpoint
. 66
C. The Second Main Theorem
14.
Statement of the proposition for the unimodular group
. 70
15.
Capelii's formal congruence
. 72
16.
Proof of the second main theorem for the unimodular group
. 73
17.
The second main theorem for the unimodular group
. 75
Chapter III
MATRIC
ALGEBRAS AND GROUP RINGS
A. Theory of Fully Reducible
Matric
Algebras
1.
Fundamental notions concerning
matric
algebras. The
Schur
lemma
. 79
2.
Preliminaries
. . . 84
3.
Representations of a simple algebra
. 87
4.
Wedderburn's theorem.
. 90
5. The fully reducible
matric algebra
and its commutator algebra
. 93
B. The
Riso
of a Finite Group and Its Commctatob Algebra
β.
Stating the problem
. . . 96
7.
Fuii
reducibiîity
of the group ring
. 101
xi
XU
TABLE
OF
CONTENTS
PAGE
8.
Formal lemmas
.
Ю6
9.
Reciprocity between group ring and commutator algebra
. 107
10.
A generalization
.
112
Chapter IV
THE SYMMETRIC GROUP AND THE FULL LINEAR GROUP
1.
Representation of a finite group in an algebraically closed field
. 115
2.
The Young symmetrizers. A combinatorial lemma
. 119
3.
The irreducible representations of the symmetric group
. 124
4.
Decomposition of tensor space
. 127
5.
Quantities. Expansion
. 131
Chapter V
THE ORTHOGONAL GROUP
A. The Enveloping
Algebra
and the Orthogonal Ideal
1.
Vector invariants of the unimodular group again
. 137
2.
The enveloping algebra of the orthogonal group
. 140
3.
Giving the result its formal setting
. 143
4.
The orthogonal prime ideal
. 144
5.
An abstract algebra related to the orthogonal group
. 147
B. The Ihrkducible Representations
6.
Decomposition by the trace operation
. 149
7.
The irreducible representations of the full orthogonal group
. 153
C. The Proper Orthogonal Gbodp
8.
Clifford's theorem
. 159
9.
Representations of the proper orthogonal group
. 163
Chapter VI
THE SYMPLECTIC GROUP
1.
Vector invariants of the symplectic group
. 165
2.
Parametrization and unitary restriction
. 169
3.
Embedding algebra and representations of the symplectic group
. 173
Chapter
VII
CHARACTERS
1.
Preliminaries about unitary transformations
. 176
2.
Character for symmetrization or alternation alone
. 181
3.
Averaging over a group
. 185
4.
The volume element of the unitary group
. 194
5.
Computation of the characters
. 198
6.
The characters of GL(n). Enumeration of
covariante
. 201
7.
A purely algebraic approach
. 208
8.
Characters of the symplectic group
. 216
9.
Characters of the orthogonal group
. 222
10.
Decomposition and X-multiplication
. 229
11.
The
Poincaré
polynomial
. 232
TABLE
OF
CONTENTS
Х1Ц
Chapter
VIII
GENERAL
THEORY OF INVARIANTS
A. Algebraic
Рант
page
1.
Classic invariants and invariants of
quantice.
Gram's theorem
. 239
2.
The symbolic method
. 243
3.
The binary quadratic
. 246
4.
Irrational methods
. 248
5.
Side remarks
. 250
6.
Hubert's theorem on polynomial ideals
. 251
7.
Proof of the first main theorem for QL(n)
. 252
8.
The adjunction argument
. 254
B.
Differenti al
and Integral Methods
9.
Group germ and Lie algebras
. 258
10.
Differential equations for invariants. Absolute and relative invariants
. 262
11.
The
unitarian
trick
. 265
12.
The connectivity of the classical groups
. 268
13.
Spinors
. 270
14.
Finite integrity basis for invariants of compact groups
. 274
15.
The first main theorem for finite groups
. 275
16.
Invariant differentials and
Betti
numbers of a compact Lie group
. 276
Chapter IX
MATRIC
ALGEBRAS RESUMED
1.
Automorphisms
. 280
2.
A lemma on multiplication
. 283
3.
Products of simple algebras
. 286
4.
Adjunction
. 288
Chapter X
SUPPLEMENTS
A. Supplement
то
Chapter II,
§§9-13,
and Chaptee VI,
§1,
Concerning
Infinitesimal
Vector Invariants
1.
An identity for infinitesimal orthogonal invariants
. 291
2.
First Main Theorem for the orthogonai group
. 293
3.
The same for the symplectic group
. 294
B. Supplement to Chapter V,
§3,
and Chapter VI,
§§2
and
3,
Concerning the
Symplectic and Okthogonal Ideals
4.
A proposition on full reduction
. 295
5.
The symplectic ideal
. 296
6.
The
fuli
and the proper orthogonal ideals
. 299
C. Supplement to Chapteb
VIII, §§7-8,
Concerning.
7.
A modified proof of the main theorem on invariants
. 300
D. Supplement to Chaptek IX,
§4,
About Extension of the Ground Field
8.
Effect of field extension on a division algebra
. 303
Erbata
and Addenda
. 307
Bibliography
. 308
Supplementaby BIBLIOGRAPHY, Mainly for the Years
1940-1945. 314
Index
. 317 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Weyl, Hermann 1885-1955 |
| author_GND | (DE-588)118816624 |
| author_facet | Weyl, Hermann 1885-1955 |
| author_role | aut |
| author_sort | Weyl, Hermann 1885-1955 |
| author_variant | h w hw |
| building | Verbundindex |
| bvnumber | BV021832190 |
| callnumber-first | Q - Science |
| callnumber-label | QA385 |
| callnumber-raw | QA385 |
| callnumber-search | QA385 |
| callnumber-sort | QA 3385 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 260 |
| ctrlnum | (OCoLC)38336673 (DE-599)BVBBV021832190 |
| dewey-full | 512.2 512.86 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.2 512.86 |
| dewey-search | 512.2 512.86 |
| dewey-sort | 3512.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 15. print., and 1. paperback print. |
| format | Book |
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| id | DE-604.BV021832190 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:57:38Z |
| indexdate | 2024-07-09T20:45:40Z |
| institution | BVB |
| isbn | 0691057567 9780691057569 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015044161 |
| oclc_num | 38336673 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-355 DE-BY-UBR |
| owner_facet | DE-19 DE-BY-UBM DE-355 DE-BY-UBR |
| physical | XIII, 320 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Princeton Univ. Pr. |
| record_format | marc |
| series2 | Princeton landmarks in mathematics and physics |
| spelling | Weyl, Hermann 1885-1955 Verfasser (DE-588)118816624 aut The classical groups their invariants and representations by Hermann Weyl 15. print., and 1. paperback print. Princeton, NJ Princeton Univ. Pr. 1997 XIII, 320 S. txt rdacontent n rdamedia nc rdacarrier Princeton landmarks in mathematics and physics Hier auch später erschienene, unveränderte Nachdrucke Continuous groups Group theory Klassische Gruppe (DE-588)4164040-8 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Invariantentheorie (DE-588)4162209-1 gnd rswk-swf Gruppenring (DE-588)4158469-7 gnd rswk-swf Matrizenalgebra (DE-588)4139347-8 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 s Klassische Gruppe (DE-588)4164040-8 s DE-604 Gruppenring (DE-588)4158469-7 s Matrizenalgebra (DE-588)4139347-8 s Invariantentheorie (DE-588)4162209-1 s Darstellungstheorie (DE-588)4148816-7 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015044161&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Weyl, Hermann 1885-1955 The classical groups their invariants and representations Continuous groups Group theory Klassische Gruppe (DE-588)4164040-8 gnd Gruppentheorie (DE-588)4072157-7 gnd Darstellungstheorie (DE-588)4148816-7 gnd Invariantentheorie (DE-588)4162209-1 gnd Gruppenring (DE-588)4158469-7 gnd Matrizenalgebra (DE-588)4139347-8 gnd |
| subject_GND | (DE-588)4164040-8 (DE-588)4072157-7 (DE-588)4148816-7 (DE-588)4162209-1 (DE-588)4158469-7 (DE-588)4139347-8 |
| title | The classical groups their invariants and representations |
| title_auth | The classical groups their invariants and representations |
| title_exact_search | The classical groups their invariants and representations |
| title_exact_search_txtP | The classical groups their invariants and representations |
| title_full | The classical groups their invariants and representations by Hermann Weyl |
| title_fullStr | The classical groups their invariants and representations by Hermann Weyl |
| title_full_unstemmed | The classical groups their invariants and representations by Hermann Weyl |
| title_short | The classical groups |
| title_sort | the classical groups their invariants and representations |
| title_sub | their invariants and representations |
| topic | Continuous groups Group theory Klassische Gruppe (DE-588)4164040-8 gnd Gruppentheorie (DE-588)4072157-7 gnd Darstellungstheorie (DE-588)4148816-7 gnd Invariantentheorie (DE-588)4162209-1 gnd Gruppenring (DE-588)4158469-7 gnd Matrizenalgebra (DE-588)4139347-8 gnd |
| topic_facet | Continuous groups Group theory Klassische Gruppe Gruppentheorie Darstellungstheorie Invariantentheorie Gruppenring Matrizenalgebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015044161&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT weylhermann theclassicalgroupstheirinvariantsandrepresentations |