Table Of Content

Title Page: iii Copyright Page: iv Dedication Page: v TABLE OF CONTENTS Page: xi PREFACE TO THE FIRST EDITION Page: vii PREFACE TO THE SECOND EDITION Page: ix CHAPTER I INTRODUCTION Page: 1 1. Fields, rings, ideals, polynomials Page: 1 2. Vector space Page: 6 3. Orthogonal transformations, Euclidean vector geometry Page: 11 4. Groups, Klein's Erlanger program. Quantities Page: 13 5. Invariants and covariants Page: 23 CHAPTER II VECTOR INVARIANTS Page: 27 1. Remembrance of things past Page: 27 2. The main propositions of the theory of invariants Page: 29 A. FIRST MAIN THEOREM Page: 36 3. First example: the symmetric group Page: 36 4. Capelli's identity Page: 39 5. Reduction of the first main problem by means of Capelli's identities Page: 42 6. Second example: the unimodular group SL(n) Page: 45 7. Extension theorem. Third example: the group of step transformations Page: 47 8. A general method for including contravariant arguments Page: 49 9. Fourth example: the orthogonal group Page: 52 B. A CLOSE-UP OF THE ORTHOGONAL GROUP Page: 56 10. Cayley's rational parametrization of the orthogonal group Page: 56 11. Formal orthogonal invariants Page: 62 12. Arbitrary metric ground form Page: 65 13. The infinitesimal standpoint Page: 66 C. THB SECOND MAIN THEOREM Page: 70 14. Statement of the proposition for the unimodular group Page: 70 15. Capelli's formal congruence Page: 72 16. Proof of the second main theorem for the unimodular group Page: 73 17. The second main theorem for the unimodular group Page: 75 CHAPTER III MATRIC ALGEBRAS AND GROUP RINGS Page: 79 A. THEORY OF FULLY REDUCIBLE MATRIC ALGEBRAS Page: 79 1. Fundamental notions concerning matric algebras. The Schur lemma Page: 79 2. Preliminaries Page: 84 3. Representations of a simple algebra Page: 87 4. Wedderburn's theorem Page: 90 5. The fully reducible matric algebra and its commutator algebra Page: 93 B. THE RING OF A FINITE GROUP AND ITS COMMUTATOR ALGEBRA Page: 96 6. Stating the problem Page: 96 7. Full reducibility of the group ring Page: 101 8. Formal lemmas Page: 106 9. Reciprocity between group ring and commutator algebra Page: 107 10. A generalization Page: 112 CHAPTER IV THE SYMMETRIC GROUP AND THE FULL LINEAR GROUP Page: 115 1. Representation of a finite group in an algebraically closed field Page: 115 2. The Young symmetrizers. A combinatorial lemma Page: 119 3. The irreducible representations of the symmetric group Page: 124 4. Decomposition of tensor space Page: 127 5. Quantities. Expansion Page: 131 CHAPTER V THE ORTHOGONAL GROUP Page: 137 A. THE ENVELOPING ALGEBRA AND THE ORTHOGONAL IDJIAL Page: 137 1. Vector invariants of the unimodular group again Page: 137 2. The enveloping algebra of the orthogonal group Page: 140 3. Giving the result its formal setting Page: 143 4. The orthogonal prime ideal Page: 144 5. An abstract algebra related to the orthogonal group Page: 147 B. THE IRREDUCIBLE REPRESENTATIONS Page: 149 6. Decomposition by the trace operation Page: 149 7. The irreducible representations of the full orthogonal group Page: 163 C. THE PROPER ORTHOGONAL GROUP Page: 159 8. Clifford's theorem Page: 159 9. Representations of the proper orthogonal group Page: 163 CHAPTER VI THE SYMPLECTIC GROUP Page: 165 1. Vector invariants of the symplectic group Page: 165 2. Parametrization and unitary restriction Page: 169 3. Embedding algebra and representations of the symplectic group Page: 173 CHAPTER VII CHARACTERS Page: 176 1. Preliminaries about unitary transformations Page: 176 2. Character for symmetrization or alternation alone Page: 181 3. Averaging over a group Page: 185 4. The volume element of the unitary group Page: 194 5. Computation of the characters Page: 198 6. The characters of GL(n). Enumeration of covariants Page: 201 7. A purely algebraic approach Page: 208 8. Characters of the symplectic group Page: 216 9. Characters of the orthogonal group Page: 222 10. Decomposition and ×-multiplication Page: 229 11. The Poincaré polynomial Page: 232 CHAPTER VIII GENERAL THEORY OF INVARIANTS Page: 239 A. ALGEBRAIC PART Page: 239 1. Classic invariants and invariants of quantics. Gram’s theorem Page: 239 2. The symbolic method Page: 243 3. The binary quadratic Page: 246 4. Irrational methods Page: 248 6. Side remarks Page: 250 6. Hilbert’s theorem on polynomial ideals Page: 251 7. Proof of the first main theorem for GL(n) Page: 252 8. The adjunction argument Page: 254 B. DIFFERENTIAL AND INTEGRAL METHODS Page: 258 9. Group germ and Lie algebras Page: 258 10. Differential equations for invariants. Absolute and relative invariants Page: 262 11. The unitarian trick Page: 265 12. The connectivity of the classical groups Page: 268 13. Spinors Page: 270 14. Finite integrity basis for invariants of compact groups Page: 274 15. The first main theorem for finite groups Page: 275 16. Invariant differentials and Betti numbers of a compact Lie group Page: 276 CHAPTER IX MATRIC ALGEBRAS RESUMED Page: 280 1. Automorphisms Page: 280 2. A lemma on multiplication Page: 283 3. Products of simple algebras Page: 286 4. Adjunction Page: 288 CHAPTER X SUPPLEMENTS Page: 291 A. SUPPLEMENT TO CHAPTER II, §§9–13, AND CHAPTER VI, §1, CONCERNING INFINITESIMAL VECTOR INVARI Page: 291 1. An identity for infinitesimal orthogonal invariants Page: 291 2. First Main Theorem for the orthogonal group Page: 293 3. The same for the symplectic group Page: 294 B. SUPPLEMENT TO CHAPTER V, §3, AND CHAPTER VI, §§2 AND 3, CONCERNING THE SYMPLECTIC AND ORTHOGON Page: 295 4. A proposition on full reduction Page: 295 5. The symplectic ideal Page: 296 6. The full and the proper orthogonal ideals Page: 299 C. SUPPLEMENT TO CHAPTER VIII, §§7–8, CONCERNING. Page: 300 7. A modified proof of the main theorem on invariants Page: 300 D. SUPPLEMENT TO CHAPTER IX, §4, ABOUT EXTENSION OF THE GROUND FIELD Page: 303 8. Effect of field extension on a division algebra Page: 303 ERRATA AND ADDENDA Page: 307 BIBLIOGRAPHY Page: 308 SUPPLEMENTARY BIBLIOGRAPHY, MAINLY FOR THE YEARS 1940–1945 Page: 314 INDEX Page: 317

Description:

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. Hermann Weyl was among the greatest mathematicians of the twentieth century. He made fundamental contributions to most branches of mathematics, but he is best remembered as one of the major developers of group theory, a powerful formal method for analyzing abstract and physical systems in which symmetry is present. In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful." Weyl believed in the overall unity of mathematics and that it should be integrated into other fields. He had serious interest in modern physics, especially quantum mechanics, a field to which The Classical Groups has proved important, as it has to quantum chemistry and other fields. Among the five books Weyl published with Princeton, Algebraic Theory of Numbers inaugurated the Annals of Mathematics Studies book series, a crucial and enduring foundation of Princeton's mathematics list and the most distinguished book series in mathematics.