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CHAPTER Ⅰ INTRODUCTION 1

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 covariants 23

CHAPTER Ⅱ VECTOR INVARIANTS 27

1．Remembrance of things past 27

2．The main propositions of the theory of invariants 29

A．FIRST MAIN THEOREM 36

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 56

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 70

14．Statement of the proposition for the unimodular group 70

15．Capelli＇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 79

A．THEORY OF FULLY REDUCIBLE MATRIC ALGEBRAS 79

1．Fundamental notions concerning matric algebras．The Schur lemma 79

2．Preliminaries 84

3．Representations of a simple algebra 87

4．Wedderburn＇8 theorem 90

5．The fully reducible matric algebra and its commutator algebra 93

B．THE RING OF A FINITE GROUP AND ITS COMMUTATOR ALGEBRA 96

6．Stating the problem 96

7．Full reducibility of the group ring 101

8．Formal lemmas 106

9．Reciprocity between group ring and commutator algebra 107

10．A generalization 112

CHAPTER Ⅳ THE SYMMETRIC GROUP AND THE FULL LINEAR GROUP 115

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 137

A．THE ENVELOPING ALGEBRA AND THE ORTHOGONAL IDEAL 137

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 IRREDUCIBLE REPRESENTATIONS 149

6．Decomposition by the trace operation 149

7．The irreducible representations of the full orthogonal group 153

C．THE PROPER ORTHOGONAL GROUP 159

8．Clifford＇s theorem 159

9．Representations of the proper orthogonal group 163

CHAPTER Ⅵ THE 8YMPLECTIC GROUP 165

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 176

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 covariants 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

CHAPTER Ⅷ GENERAL THEORY OF INVARIANTS 239

A．ALGEBRAIC PART 239

1．Classic invariants and invariants of qualities．Gram＇s theorem 239

2．The symbolic method 243

3．The binary quadratic 246

4．Irrational methods 248

5．Side remarks 250

6．Hilbert＇s theorem on polynomial ideals 251

7．Proof of the first main theorem for GL(n) 252

8．The adjunction argument 254

B．DIFFERENTIAL AND INTEGRAL METHODS 258

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 280

1．Automorphisms 280

2．A lemma on multiplication 283

3．Products of simple algebras 286

4．Adjunction 288

CHAPTER Ⅹ SUPPLEMENTS 291

A．SUPPLEMENT TO CHAPTER II,9-13,AND CHAPTER VI,1,CONCERNING INFINITESIMAL VECTOR INVARIANTS 291

1．An identity for infinitesimal orthogonal invariants 291

2．First Main Theorem for the orthogonal 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 ORTHOGONAL IDEALS 295

4．A proposition on full reduction 295

5．The symplectic ideal 296

6．The full and the proper orthogonal？ ideals 299

C．SUPPLEMENT TO CHAPTER VIII,7-8,CONCERNING． 300

7．A modified proof of the main theorem on invariants 300

D．SUPPLEMENT TO CHAPTER IX,4,ABOUT EXTENSION OF THE GROUND FIELD 303

8．Effect of field extension on a division algebra 303

ERRATA AND ADDENDA 307

BIBLIOGRAPHY 308

SUPPLEMENTARY BIBLIOGRAPHY,MAINLY FOR THE YEARS 1940-1945 314

INDEX 317