Description
Book SynopsisAn Invitation to Representation Theory offers an introduction to groups and their representations, suitable for undergraduates. In this book, the ubiquitous symmetric group and its natural action on polynomials are used as a gateway to representation theory.
The subject of representation theory is one of the most connected in mathematics, with applications to group theory, geometry, number theory and combinatorics, as well as physics and chemistry. It can however be daunting for beginners and inaccessible to undergraduates. The symmetric group and its natural action on polynomial spaces provide a rich yet accessible model to study, serving as a prototype for other groups and their representations. This book uses this key example to motivate the subject, developing the notions of groups and group representations concurrently.
With prerequisites limited to a solid grounding in linear algebra, this book can serve as a first introduction to representation theory at the undergraduate level, for instance in a topics class or a reading course. A substantial amount of content is presented in over 250 exercises with complete solutions, making it well-suited for guided study.
Trade Review“The book under review is a nice introduction to the representation theory of the symmetric group. … The book is well structured and enriched with numerous exercises, many of which are solved or with hints for the solution.” (Enrico Jabara, zbMATH 1514.20002, 2023)
Table of Contents Preface
Introduction
Chapter 1. First Steps
Chapter 2. Polynomials, Subspaces, and Subrepresentations
Chapter 3. Intertwining Maps, Complete Reducibility, and Invariant Inner Products
Chapter 4. The Structure of the Symmetric Group
Chapter 5. Sn Decomposition of Polynomial Spaces for n= 1,2,3.
Chapter 6. The Group Algebra
Chapter 7. The Irreducible Representations of Sn: Characters
Chapter 8. The Irreducible Representations of Sn: Young Symmetrizers
Chapter 9. Cosets, Restricted and Induced Representations
Chapter 10. Direct Products of Groups, Young Subgroups and Permutation Modules
Chapter 11. Specht Modules
Chapter 12. Decomposition of Young Permutation Modules
Chapter 13. Branching Relations
Bibliography
Index
