由韦尔编著的《典型群》讨论了对称，全线性，正交和辛群，以及它们的不同的不变性和表示论，运用代数的基本观点阐释群的不同性质，恰到好处地运用分析和拓扑。书中也包括了矩阵代数，半群和交换子和自旋子，这些对于很好地理解量子力学的群理论结构很有帮助。目次：引入；向量不变量；矩阵代数和群环；对称群和完全线性群；正交群；对称群；特征；不变基本理论；矩阵代数综述；补充。

读者对象：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 EDITION

 PREFACE TO THE SECOND EDITION

CHAPTER Ⅰ INTRODUCTION

 I. 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. Capelli'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 Sehur 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 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

CHAPTER Ⅵ THE SYMPLECTIC GROUP

 I. Vector invariants of the symplectic group

 2. Parametrization and unitary restriction

 3. Embedding algebra and representations of the symplec_tic group

CHAPTER Ⅶ CHARACTERS

 I. 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 Poincare polynomial

 TABLE OF CONTENTS

CHAPTER Ⅷ GENERAL THEORY OF INVARIANTS

 A. ALGEBRAIC PAnT

 i. Classic invariants and invariants of quantics. Gram's thcorem

 2. The symbolic method

 3. The binary quadratic

 4. Irrational methods

 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

 I. Automorphisms

 2. A lemma on multiplication

 3. Products of simple algebras

 4. Adjunction

CHAPTER Ⅹ SUPPLEMENTS

 A. SUPPLEMENT TO CHAPTER Ⅱ, §§9-13, AND 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 sympleetic 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

BIBLIOGRAPHY

SUPPLEMENTARY BIBLIOGRAPHY, MAINLY FOR THE YEARS 1940--1945

INDEX