Representations of Finite and Compact Groups (Graduate Studies in Mathematics ; V. 10) (Graduate Studies in Mathematics ; V. 10)

Barry Simon, I.B.M. Professor of Mathematics and Theoretical Physics at the California Institute of Technology, is the author of several books, including such classics as Methods of Mathematical Physics (with M. Reed) and Functional Integration and Quantum Physics. This new book, based on courses given at Princeton, Caltech, ETH-Zurich, and other universities, is an introductory textbook on representation theory. According to the author, "Two facets distinguish my approach. First, this book is relatively elementary, and second, while the bulk of the books on the subject is written from the point of view of an algebraist or a geometer, this book is written with an analytical flavor". The exposition in the book centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semisimple Lie groups. It culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups. Extremely well tailored both for a one-year course in representation theory and for independent study, this book is an excellent introduction to the subject which, according to the author, is unique in having "so much innate beauty so close to the surface". Introduction CHAPTER I. Groups and Counting Principles 1 Groups 2 G-spaces 3 Direct and semidirect products 4 Finite groups of rotations 5 The Platonic groups 6 The Sylow theorems 7 Counting and group structure CHAPTER II. Fundamentals of Group Representations 1 Definition and unitarity 2 Irreducibility arid complete reduction 3 The group algebra and the regular representations 4 Schur's lemma 5 Tensor products 6 Complex conjugate representations; Quaternionic representations 7 One-dimensional representations CHAPTER III. Abstract Theory of Representations of Finite Groups 1 Orthogonality relations and the first fundamental relation 2 Characters, class functions, and conjugacy classes 3 One-dimensional representations 4 The dimension theorem 5 The theorem of Frobenius and Schur Appendix to III.5-Representations on real and quaternionic vector spaces 6 Representations and group structure 7 Projections in the group algebra 8 Fourier analysis 9 Direct products 10 Restrictions 11 Subgroups of index 2 12 Examples CHAPTER IV. Representations of Concrete Finite Groups. I: Abelian and Clifford Groups 1 The structure of finite abelian groups 2 Representations of abelian groups 3 The Clifford group CHAPTER V. Representations of Concrete Finite Groups. II: Semidirect Products and Induced Representations 1 Frobenius theory of semidirect products 2 Examples of the semidirect product theory 3 Induced representations 4 The Frobenius character formula 5 The Frobenius reciprocity theorem 6 Mackey irreducibility criterion 7 Semidirect products, revisited CHAPTER VI. Representations of Concrete Finite Groups. III: The Symmetric Groups 1 Permutations and classes 2 Young frames and Young tableaux 3 Projections in A(S_n): Classification of representations 4 Branching relations 5 The Frobenius character formula 6 Consequences of the character formula CHAPTER VII. Compact Groups 1 C∞-manifolds: A review 2 Lie groups and Lie algebras 3 Haar measure on Lie groups 4 Matrix groups 5 The classical groups 6 Homotopy and covering groups 7 Spin groups 8 The structure of compact groups 9 Representations of compact groups: Abstract theory 10 The Peter-Weyl theorem CHAPTER VIII. The Structure of Compact Semisimple Groups 1 Maximal tori 2 The Killing form 3 Representations of tori 4 Representations of SU(2) and sl(2, C) 5 Roots and root spaces 6 Fundamental systems and their classification 7 Regular and singular elements 8 The Weyl group 9 The classical groups CHAPTER IX. The Representations of Compact Semisimple Groups 1 Geometry of the Cartan-Stiefel diagram 2 The geometry of integral forms 3 The Weyl integration formula 4 Maximal weights 5 The classification theorem and the Weyl character formula 6 Consequences of the Weyl character formula 7 Representation theory: The algebraic approach 8 Representations of the classical groups 9 Determinant formulas for the classical characters 10 Real and quaternionic representations of the classical groups 11 Tensors, permutations, and the Frobenius character formula Appendix A Multilinear algebra Appendix B The analysis of self-adjoint Hilbert-Schmidt operators Bibliography Index

Barry Simon is the author of many well-known books, including such classics as Methods of Mathematical Physics (with M. Reed) and Functional Integration and Quantum Physics. This book, based on courses given at Princeton, Caltech, ETH-Zurich, and other universities, is an introductory textbook on representation theory. Two facets distinguish the approach. First, the book is relatively elementary, and second, while the bulk of the books on the subject is written from the point of view of an algebraist or a geometer, this book is written with an analytical flavor. The exposition centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semisimple Lie groups. It culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups. Extremely well tailored both for a one-year course in representation theory and for independent study, this book is an excellent introduction to the subject which is unique in having so much innate beauty so close to the surface.

Centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semi simple Lie groups. This book culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups.