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An Introduction to the Representation Theory of Groups 1st Edition by Emmanuel Kowalski ISBN 1470418576 9781470418571

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Author(s): Emmanuel Kowalski
ISBN(s): 9781470418571, 1470418576
Edition: 1
File Details: PDF, 4.68 MB
Year: 2014
Language: english
SKU: EB-6806564 Category: Tag:

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ISBN 10: 1470418576 
ISBN 13: 9781470418571
Author: Emmanuel Kowalski

Representation theory is an important part of modern mathematics, not only as a subject in its own right but also as a tool for many applications. It provides a means for exploiting symmetry, making it particularly useful in number theory, algebraic geometry, and differential geometry, as well as classical and modern physics. The goal of this book is to present, in a motivated manner, the basic formalism of representation theory as well as some important applications. The style is intended to allow the reader to gain access to the insights and ideas of representation theory–not only to verify that a certain result is true, but also to explain why it is important and why the proof is natural. The presentation emphasizes the fact that the ideas of representation theory appear, sometimes in slightly different ways, in many contexts. Thus the book discusses in some detail the fundamental notions of representation theory for arbitrary groups. It then considers the special case of complex representations of finite groups and discusses the representations of compact groups, in both cases with some important applications. There is a short introduction to algebraic groups as well as an introduction to unitary representations of some noncompact groups. The text includes many exercises and examples.

An Introduction to the Representation Theory of Groups 1st Table of contents:

Chapter 1: Group Theory Fundamentals (Review/Refresher)

  • 1.1 Definition and Examples of Groups
  • 1.2 Subgroups and Cosets
  • 1.3 Normal Subgroups and Quotient Groups
  • 1.4 Homomorphisms and Isomorphisms
  • 1.5 Group Actions and Permutation Representations
  • 1.6 Conjugacy Classes
  • 1.7 Direct Products and Semi-Direct Products

Chapter 2: Introduction to Vector Spaces and Linear Algebra

  • 2.1 Vector Spaces over Arbitrary Fields (with emphasis on C)
  • 2.2 Linear Transformations and Matrices
  • 2.3 Eigenvalues and Eigenvectors
  • 2.4 Inner Product Spaces (Hermitian spaces)
  • 2.5 Bases and Dimension

Chapter 3: Basic Concepts of Representation Theory

  • 3.1 Definitions of Group Representations (Matrix and Module Representations)
  • 3.2 Examples of Representations (Permutation, Regular, Trivial)
  • 3.3 Equivalence of Representations
  • 3.4 Reducibility and Irreducibility of Representations
  • 3.5 Maschke’s Theorem (for finite groups over C)
  • 3.6 Decompositions of Representations

Chapter 4: Character Theory

  • 4.1 Definition and Properties of Characters
  • 4.2 Irreducible Characters and Orthogonality Relations
  • 4.3 Character Tables
  • 4.4 Computing Character Tables for Small Groups (e.g., S3​,D4​,A4​)
  • 4.5 Applications of Character Tables (e.g., decomposing representations, checking irreducibility)
  • 4.6 Products of Representations and Characters

Chapter 5: Representations of Specific Groups

  • 5.1 Representations of Abelian Groups
  • 5.2 Representations of Symmetric Groups (Sn​)
    • Young Diagrams and Young Tableaux
    • Specht Modules
  • 5.3 Representations of Alternating Groups (An​)
  • 5.4 Representations of Dihedral Groups (Dn​)
  • 5.5 Representations of Matrix Groups (e.g., GLn​(F),SLn​(F) – possibly simplified cases)

Chapter 6: Induced Representations

  • 6.1 Definition of Induced Representations
  • 6.2 Frobenius Reciprocity Theorem
  • 6.3 Mackey’s Irreducibility Criterion
  • 6.4 Applications of Induced Representations

Chapter 7: Tensor Products and Exterior Powers

  • 7.1 Tensor Products of Vector Spaces
  • 7.2 Tensor Products of Representations
  • 7.3 Symmetric and Exterior Powers of Representations

Chapter 8: Representations of Compact Lie Groups (Introduction)

  • 8.1 Basic Concepts of Lie Groups
  • 8.2 Haar Measure and Orthogonality for Compact Groups
  • 8.3 Peter-Weyl Theorem
  • 8.4 Representations of SU(2) and SO(3) (Connection to Physics)

Chapter 9: Further Topics (Optional/Advanced)

  • 9.1 Real Representations
  • 9.2 Projective Representations
  • 9.3 Modular Representations (if field characteristic divides group order)
  • 9.4 Connections to Fourier Analysis on Groups
  • 9.5 Applications in Physics or Combinatorics

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Tags: Emmanuel Kowalski, Representation, Theory