The Theory of Group Characters and Matrix Representations of Groups (AMS Chelsea Publishing)

Originally written in 1940, this book remains a classical source on representations and characters of finite and compact groups. The book starts with necessary information about matrices, algebras, and groups. Then the author proceeds to representations of finite groups. Of particular interest in this part of the book are several chapters devoted to representations and characters of symmetric groups and the closely related theory of symmetric polynomials. The concluding chapters present the representation theory of classical compact Lie groups, including a detailed description of representations of the unitary and orthogonal groups. The book, which can be read with minimal prerequisites (an undergraduate algebra course), allows the reader to get a good understanding of beautiful classical results about group representations. Cover Date-line of AMS edition Title page Date-line of original edition Preface CONTENTS CORRIGENDUM I. MATRICES 1.1. Linear transformations 1.2. Matrices 1.3. The transform of a matrix 1.4. Rectangular matrices and vectors 1.5. The characteristic equation of a matrix 1.6. The classical canonical form of a matrix 1.7. The classical canonical form; multiple characteristic roots 1.8. Various properties of matrices 1.9. Unitary and orthogonal matrices II. ALGEBRAS 2.1. Definition of an algebra over the complex numbers 2.2. Change of basis and the regular matrix representation 2.3. Simple matrix algebras 2.4. Examples of associative algebras 2.5. Linear sets and sub-algebras 2.6. Modulus, idempotent and nilpotent elements 2.7. The reduced characteristic equation 2.8. Reduction of an algebra relative to an idempotent 2.9. The trace of an element III. GROUPS 3.1. Definition of a group 3.2. Subgroups 3.3. Examples of groups 3.4. Permutation groups 3.5. The alternating group 3.6. Classes of conjugate elements 3.7. Conjugate and self-conjugate subgroups 3.8. The representations of an abstract group as a permutation group IV. THE EROBENIUS ALGEBRA 4.1. Groups and algebras 4.2. The group characters 4.3. Matrix representations and group matrices 4.4. Characteristic units 4.5. The relations between the characters of a group and those of a subgroup V. THE SYMMETRIC GROUP 5.1. Partitions 5.2. Frobenius's formula for the characters of the symmetric group 5.3. Characters and lattices 5.4. Primitive characteristic units and Young tableaux VI. IMMANANTS AND $S$-FUNCTIONS 6.1. Immanants of a matrix 6.2. Schur functions 6.3. Properties of $S$-functions 6.4. Generating functions and further properties of $S$-functions 6.5. Relations between immanants and $S$-functions VII. $S$-FUNCTIONS OF SPECIAL SERIES 7.1. The function $\Phi(q,x)$ 7.2. The functions $(1—x)^{-N}$ and $(1—x^{-r})^{-m}$ 7.3. $S$-functions associated with $f(x^r)$ VIII. THE CALCULATION OF THE CHARACTERS OF THE SYMMETRIC GROUP 8.1. Frobenius's formula $S$-functions of special series Recurrence relations Congruences Classes for which the orders of the cycles have a common factor Graphs and lattices Orthogonal properties IX. GROUP CHARACTERS AND THE STRUCTURE OF GROUPS 9.1. The compound character associated with a subgroup 9.2. Deduction of the characters of a subgroup from those of the group 9.3. Determination of subgroups: necessary criteria that a compound character should correspond to a permutation representation of the group 9.4. The properties of groups and character tables 9.5. Transitivity 9.6. Invariant subgroups X. CONTINUOUS MATRIX GROUPS AND INVARIANT MATRICES 10.1. Invariant matrices 10.2. The classical canonical form of an invariant matrix 10.3. Application to invariant theory XI. GROUPS OF UNITARY MATRICES 11.1. Introductory 11.2. Fundamental formula for integration over the group manifold 11.3. Simplification of integration formulae for class functions 11.4. Verification of the orthogonal properties of the characters of the unitary group 11.5. Orthogonal matrices and the rotation groups 11.6. Relations between the characters of $D$ and $D'$ 11.7. Integration formulae connected with $D$ and $D'$ 11.8. The characters of the orthogonal group 11.9. Alternative forms for the characters of the orthogonal group 11.10. The difference characters of the rotation group 11.11. The spin representations of the orthogonal group 11.12. Complex orthogonal matrices and groups of matrices with a quadratic invariant APPENDIX Tables of Characters of the Symmetric Groups Tables of Characters of Transitive Subgroups. Alternating Groups General Cyclic Group of Order n Other Transitive Subgroups Some Recent Developments BIBLIOGRAPHY SUPPLEMENTARY BIBLIOGRAPHY INDEX Starts with necessary information about matrices, algebras, and groups. This title then proceeds to representations of finite groups. It includes several chapters dealing with representations and characters of symmetric groups and the closely related theory of symmetric polynomials.