Representation theory of finite groups : algebra and arithmetic / Steven H. Weintraub.

Format:

Book

Author/Creator:

Weintraub, Steven H., author.

Series:

Graduate studies in mathematics ; v. 59.

Graduate studies in mathematics, 1065-7339 ; volume 59

Language:

English

Subjects (All):

Representations of groups.

Finite groups.

Physical Description:

1 online resource (226 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2003]

Language Note:

English

Summary:

``We explore widely in the valley of ordinary representations, and we take the reader over the mountain pass leading to the valley of modular representations, to a point from which (s)he can survey this valley, but we do not attempt to widely explore it. We hope the reader will be sufficiently fascinated by the scenery to further explore both valleys on his/her own.'' --from the Preface Representation theory plays important roles in geometry, algebra, analysis, and mathematical physics. In particular, representation theory has been one of the great tools in the study and classification of finite groups. There are some beautiful results that come from representation theory: Frobenius's Theorem, Burnside's Theorem, Artin's Theorem, Brauer's Theorem--all of which are covered in this textbook. Some seem uninspiring at first, but prove to be quite useful. Others are clearly deep from the outset. And when a group (finite or otherwise) acts on something else (as a set of symmetries, for example), one ends up with a natural representation of the group. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations. Methods and results particular to group representations, such as characters and induced representations, are developed in depth. Arithmetic comes into play when considering the field of definition of a representation, especially for subfields of the complex numbers. The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group. The book assumes only the material of a standard graduate course in algebra. It is suitable as a text for a year-long graduate course. The subject is of interest to students of algebra, number theory and algebraic geometry. The systematic treatment presented here makes the book also valuable as a reference.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Chapter 1. Introduction""; ""Chapter 2. Semisimple Rings and Modules""; ""Â2.1. Basic Notions""; ""Â2.2. Structure Theorems""; ""Â2.3. Idempotents and Blocks""; ""Â2.4. Behavior under Field Extensions""; ""Â2.5. Theorems of Burnside and Frobenius-Schur""; ""Chapter 3. Semisimple Group Representations""; ""Â3.1. Examples and General Results""; ""Â3.2. Representations of Abelian Groups""; ""Â3.3. Decomposition of the Regular Representation""; ""Â3.4. Applications of Frobenius's Theorem""; ""Â3.5. Characters""

""Â3.6. Idempotents and their Uses""""Â3.7. Subfields of the Complex Numbers""; ""Â3.8. Fields of Positive Characteristic""; ""Chapter 4. Induced Representations and Applications""; ""Â4.1. Induced Representations""; ""Â4.2. Mackey's Theorem""; ""Â4.3. Permutation Representations""; ""Â4.4. M-groups""; ""Â4.5. Theorems of Artin and Brauer""; ""Â4.6. Degrees of Irreducible Representations""; ""Chapter 5. Introduction to Modular Representations""; ""Chapter 6. General Rings and Modules""; ""Â6.1. Jordan-Holder and Krull-Schmidt Theorems""; ""Â6.2. The Jacobson Radical""

""Â6.3. Rings of Finite Length""""Â6.4. Finite-dimensional Algebras""; ""Chapter 7. Modular Group Representations""; ""Â7.1. General Results""; ""Â7.2. Characters and Brauer Characters""; ""Â7.3. Examples""; ""Appendix. Some Useful Results""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""M""; ""N""; ""R""; ""S""; ""T""; ""W""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references (page 209) and index.

Description based on print version record.

ISBN:

1-4704-2104-6