The Classical Groups: Their Invariants and Representations, Numéro 1,Partie 1Princeton University Press, 1946 - 320 pages In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics. |
Table des matières
CHAPTER | 1 |
Vector space | 8 |
Invariants and covariants | 23 |
The main propositions of the theory of invariants | 29 |
the symmetric group | 36 |
Reduction of the first main problem by means of Capellis identities | 42 |
the unimodular group SLn | 49 |
the orthogonal group | 56 |
A purely algebraic approach | 208 |
Characters of the symplectic group | 216 |
Characters of the orthogonal group | 222 |
Decomposition and Xmultiplication | 229 |
The Poincaré polynomial | 232 |
GENERAL THEORY OF INVARIANTS A ALGEBRAIC PART PAGE 1 Classic invariants and invariants of quantics Grams theorem | 239 |
The symbolic method | 243 |
The binary quadratic | 246 |
Formal orthogonal invariants | 62 |
Statement of the proposition for the unimodular group | 70 |
CHAPTER III | 79 |
Representations of a simple algebra | 90 |
Stating the problem | 96 |
Formal lemmas | 106 |
Reciprocity between group ring and commutator algebra | 107 |
A generalization | 112 |
CHAPTER IV | 115 |
The Young symmetrizers A combinatorial lemma | 119 |
The irreducible representations of the symmetric group | 124 |
Decomposition of tensor space | 127 |
Quantities Expansion | 131 |
CHAPTER V | 137 |
The enveloping algebra of the orthogonal group | 140 |
Giving the result its formal setting | 143 |
The orthogonal prime ideal | 144 |
An abstract algebra related to the orthogonal group | 147 |
B THE IRREDUCIBLE REPRESENTATIONS 6 Decomposition by the trace operation | 149 |
The irreducible representations of the full orthogonal group | 153 |
THE PROPER Orthogonal Group 8 Cliffords theorem | 159 |
Representations of the proper orthogonal group | 163 |
CHAPTER VI | 165 |
Parametrization and unitary restriction | 169 |
Embedding algebra and representations of the symplectic group | 173 |
CHAPTER VII | 176 |
Character for symmetrization or alternation alone | 181 |
Averaging over a group | 185 |
The volume element of the unitary group | 194 |
Computation of the characters | 198 |
The characters of GLn Enumeration of covariants | 201 |
Irrational methods | 248 |
Side remarks | 250 |
Hilberts theorem on polynomial ideals | 251 |
Proof of the first main theorem for GLn | 252 |
The adjunction argument | 254 |
B DIFFERENTIAL AND INTEGRAL METHODS 9 Group gerin and Lie algebras | 258 |
Differential equations for invariants Absolute and relative invariants | 262 |
The unitarian trick | 265 |
The connectivity of the classical groups | 268 |
Spinors | 270 |
Finite integrity basis for invariants of compact groups | 274 |
The first main theorem for finite groups | 275 |
Invariant differentials and Betti numbers of a compact Lie group | 276 |
CHAPTER IX | 280 |
A lemma on multiplication | 283 |
Products of simple algebras | 286 |
Adjunction | 288 |
CHAPTER X | 291 |
First Main Theorem for the orthogonal group | 293 |
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 4 A propos... | 295 |
The symplectic ideal | 296 |
The full and the proper orthogonal ideals | 299 |
SUPPLEMENT TO CHAPTER VIII 78 CONCERNING 7 A modified proof of the main theorem on invariants | 300 |
SUPPLEMENT TO CHAPTER IX 4 ABOUT EXTENSION OF THE GROUND FIELD 8 Effect of field extension on a division algebra | 303 |
ERRATA AND ADDENDA | 307 |
BIBLIOGRAPHY | 308 |
SUPPLEMENTARY BIBLIOGRAPHY MAINLY FOR THE YEARS 19401945 | 314 |
317 | |