| 《princeton landmarks in mathematics》系列之一的经典教材 适用于数学专业的本科生，研究生和相关的科研人员 |
| table of contents preface to the first ei)ition preface to the second edition chapter i introduction 1. fields, rings, ideals, polynomials 2. vector space 3. orthogonal transformations, euclidean vector geometry 4. groups, klein's erlanger program. quantities 5. invariants and covariants chapter ii vector invariants 1. remembrance of things past 2. the main propositions of the theory of invariants a. first main theorem 3. first example: the symmetric group 4. capeui's identity 5. reduction of the first main problem by means of capelli's identities 6. second example: the unimodular group sl(n) 7. extension theorem. third example: the group of step transformations .8. a general method for including contravariant arguments 9. fourth example: the orthogonal group b. a close-up of the orthogonal group 10. cayley's rational parametrization of the orthogonal group 11, formal orthogonal invariants 12. arbitrary metric ground form 13. the infinitesimal standpoint c. the second main theorem 14. statement of the proposition for the unimodular group 15. capelli's formal congruence 16. proof of the second main theorem for the unimodular group 17. the second main theorem for the unimodular group chapter iii matric algebras and group rings a. theory of fully reducible matric algebras 1. fundamental notions concerning matric algebras. the schur lemma 2,.preliminaries 3. representations of a simple algebra 4. wedderburn's theorem 5. the fully reducible matric algebra and its commutator algebra b. the ring of a finite group and its commutator algesra 6. stating the problem 7. full reducibility of the group ring 8. formal lemmas 9. reciprocity between group ring and commutator algebra 10. a generalization chapter iv the symmetric group and the full linear group 1. representation of a finite group in an algebraically closed field 2. the young symmetrizers. a combinatorial lemma 3. the irreducible representations of the symmetric group 4. decomposition of tensor space 5. quantities. expansion chapter v the orthogonal group a. the enveloping algebra and the orthogonal ideal 1. vector invariants of the unimodular group again 2. the enveloping algebra of the orthogonal group. 3. giving the result its formal setting 4. the orthogonal prime ideal 5. an abstract algebra related to the orthogonal group b. the irreducible representations 6. decomposition by the trace operation 7. the irreducible representations of the full orthogonal group c. the proper orthogonal group 8. clifford's theorem 9. representations of the proper orthogonal group crapter vi the symplectic group 1. vector invariants of the sympleetic group 2. parametrization and unitary restriction 3. embedding algebra and representations of the symplectic group charter vii characters 1. preliminaries about unitary transformations 2. character for symmetrisation or alternation alone 3. averaging over a group 4. the volume element of the unitary group 5. computation of the characters 6. the characters of gl(n). enumeration of covariants 7. a purely algebraic approach 8. characters of the symplectic group 9. characters of the orthogonal group 10. decomposition and ~-multiplication 11. the poineare polynomial chapter viii general theory of invariants a. algebraic part 1. classic invariants and invariants of quantics. gram's theorem 2. the symbolic method 3. the binary quadratic 4. irrational metllods 5. side remarks 6. hilbert's theorem on polynomial ideals 7. proof of the first main theorem for gl(n) 8. the adjunction argument b. differential and integral methods 9. group germ and lie algebras 10. differential equations for invariants. absolute and relative invariants. 11. the unitarian trick 12. the connectivity of the classical groups 13. spinors 14. finite integrity basis for invariants of compact groups 15. the first main theorem for finite groups 16. invariant differentials and betti numbers of a compact lie group chapter ix matric algebras resumed 1. automorphisms 2. a lemma on multiplication.. 3. products of simple algebras. 4. adjunction chapter x supplements a. supplement to chapter ii, §§9-13, ann chapter vi, §1, concerning infinitesimal vector invariants 1. an identity for infinitesimal orthogonal invariants. 2. first main theorem for the orthogonal group 3. the same for the symplectic group b. supplement to chapter v, §3, and chapter vi, §§2 and 3, concerning the symplectic and orthogonal ideals 4. a proposition on full reduction 5. the symplectic ideal 6. the full and the proper orthogonal ideals. c. supplement to chapter viii, §§7-8, concerning. 7. a modified proof of the main theorem on invariants. d. supplement to chapter ix, §4, about extension of the ground field 8. effect of field extension on a division algebra errata and addenda bib liooraphy supplementary bibliography, mainly for the years 1940--1945 index |
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