《The Classical Groups Their Invariants and Representations》

CHAPTER ⅠINTRODUCTION1

1．Fields,rings,ideals,polynomials1

2．Vector space6

3．Orthogonal transformations,Euclidean vector geometry11

4．Groups,Klein＇s Erlanger program．Quantities13

5．Invariants and covariants23

CHAPTER ⅡVECTOR INVARIANTS27

1．Remembrance of things past27

2．The main propositions of the theory of invariants29

A．FIRST MAIN THEOREM36

3．First example:the symmetric group36

4．Capelli＇s identity39

5．Reduction of the first main problem by means of Capelli＇s identities42

6．Second example:the unimodular group SL(n)45

7．Extension theorem．Third example:the group of step transformations47

8．A general method for including contravariant arguments49

9．Fourth example:the orthogonal group52

B．A CLOSE-UP OF THE ORTHOGONAL GROUP56

10．Cayley＇s rational parametrization of the orthogonal group56

11．Formal orthogonal invariants62

12．Arbitrary metric ground form65

13．The infinitesimal standpoint66

C．THE SECOND MAIN THEOREM70

14．Statement of the proposition for the unimodular group70

15．Capelli＇s formal congruence72

16．Proof of the second main theorem for the unimodular group73

17．The second main theorem for the unimodular group75

CHAPTER ⅢMATRIC ALGEBRAS AND GROUP RINGS79

A．THEORY OF FULLY REDUCIBLE MATRIC ALGEBRAS79

1．Fundamental notions concerning matric algebras．The Schur lemma79

2．Preliminaries84

3．Representations of a simple algebra87

4．Wedderburn＇8 theorem90

5．The fully reducible matric algebra and its commutator algebra93

B．THE RING OF A FINITE GROUP AND ITS COMMUTATOR ALGEBRA96

6．Stating the problem96

7．Full reducibility of the group ring101

8．Formal lemmas106

9．Reciprocity between group ring and commutator algebra107

10．A generalization112

CHAPTER ⅣTHE SYMMETRIC GROUP AND THE FULL LINEAR GROUP115

1．Representation of a finite group in an algebraically closed field115

2．The Young symmetrizers．A combinatorial lemma119

3．The irreducible representations of the symmetric group124

4．Decomposition of tensor space127

5．Quantities．Expansion131

CHAPTER ⅤTHE ORTHOGONAL GROUP137

A．THE ENVELOPING ALGEBRA AND THE ORTHOGONAL IDEAL137

1．Vector invariants of the unimodular group again137

2．The enveloping algebra of the orthogonal group140

3．Giving the result its formal setting143

4．The orthogonal prime ideal144

5．An abstract algebra related to the orthogonal group147

B．THE IRREDUCIBLE REPRESENTATIONS149

6．Decomposition by the trace operation149

7．The irreducible representations of the full orthogonal group153

C．THE PROPER ORTHOGONAL GROUP159

8．Clifford＇s theorem159

9．Representations of the proper orthogonal group163

CHAPTER ⅥTHE 8YMPLECTIC GROUP165

1．Vector invariants of the symplectic group165

2．Parametrization and unitary restriction169

3．Embedding algebra and representations of the symplectic group173

CHAPTER ⅦCHARACTERS176

1．Preliminaries about unitary transformations176

2．Character for symmetrization or alternation alone181

3．Averaging over a group185

4．The volume element of the unitary group194

5．Computation of the characters198

6．The characters of GL(n)．Enumeration of covariants201

7．A purely algebraic approach208

8．Characters of the symplectic group216

9．Characters of the orthogonal group222

10．Decomposition and X-multiplication229

11．The Poincaré polynomial232

CHAPTER ⅧGENERAL THEORY OF INVARIANTS239

A．ALGEBRAIC PART239

1．Classic invariants and invariants of qualities．Gram＇s theorem239

2．The symbolic method243

3．The binary quadratic246

4．Irrational methods248

5．Side remarks250

6．Hilbert＇s theorem on polynomial ideals251

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

8．The adjunction argument254

B．DIFFERENTIAL AND INTEGRAL METHODS258

9．Group germ and Lie algebras258

10．Differential equations for invariants．Absolute and relative invariants262

11．The unitarian trick265

12．The connectivity of the classical groups268

13．Spinors270

14．Finite integrity basis for invariants of compact groups274

15．The first main theorem for finite groups275

16．Invariant differentials and Betti numbers of a compact Lie group276

CHAPTER ⅨMATRIC ALGEBRAS RESUMED280

1．Automorphisms280

2．A lemma on multiplication283

3．Products of simple algebras286

4．Adjunction288

CHAPTER ⅩSUPPLEMENTS291

A．SUPPLEMENT TO CHAPTER II,9-13,AND CHAPTER VI,1,CONCERNING INFINITESIMAL VECTOR INVARIANTS291

1．An identity for infinitesimal orthogonal invariants291

2．First Main Theorem for the orthogonal group293

3．The same for the symplectic group294

B．SUPPLEMENT TO CHAPTER V,3,AND CHAPTER VI,2 AND 3,CONCERNING THE SYMPLECTIC AND ORTHOGONAL IDEALS295

4．A proposition on full reduction295

5．The symplectic ideal296

6．The full and the proper orthogonal？ ideals299

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

7．A modified proof of the main theorem on invariants300

D．SUPPLEMENT TO CHAPTER IX,4,ABOUT EXTENSION OF THE GROUND FIELD303

8．Effect of field extension on a division algebra303

ERRATA AND ADDENDA307

BIBLIOGRAPHY308

SUPPLEMENTARY BIBLIOGRAPHY,MAINLY FOR THE YEARS 1940-1945314

INDEX317