经典群

TABLE OF CONTENTS

PREFACE TO THE FIRST EDITION

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. Frost MAIN THEOREM

3. First example the symmetric group

4. Capelli's identity

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

6. Second example the unimodular group ,.qL(n)

7. Extension theorem. Third example the group of step transformations

8. A general method for including eontravariant 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 ALGEBRA

6. Stating the problem

7. Full reducibility of the group ring

TABLE OF CONTENTS

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 lsmma

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

CHAPTER VI

THE SYMPLECTIC GROUP

1. Vector invariants of the symplectic group

2. Parametrization and unitary restriction

3. Embedding algebra and representations of the symplectic group

CHAPTER VII

CHARACTERS

1. Preliminaries about unitary transformations

2. Character for symmetrization 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 X-multiplication

11. The Poinear~ polynomial

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《典型群(英文版)》是由世界图书出版公司出版的。 《典型群(英文版)》内容简介：Ever since the year 1925, when I succeeded in determining the characters of the semi-simple continuous groups by a combination of E. Cartan's infini-tesimal methods and I. Schur's integral procedure, I have looked toward thegoal of deriving the decisive results for the most important of these groups bydirect algebraic construction, in particular for the full group of all non-singu-lar linear transformations and for the orthogonal group. Owing mainly toR. Brauer's intervention and collaboration during the last few years, it nowappears that I have in my hands all the tools necessary for this purpose. Thetask may be characterized precisely as follows with respect to the assignedgroup of linear transformations in the underlying vector space, to decomposethe space of tensors of given rank into its irreducible invariant subspaces.【作者简介】作者：（德国）韦尔（Hermann Weyl）