THEORY OF LINEAR REPRESENTATIONS OF GROUPS

The following sections are included:

• Linear Representations of a Group

  • Definition of a Linear Representation

  • Group Algebra and the Regular Representation

  • Class Operator and Class Space

• Transformation Operators for a Scalar Function

• Equivalent Representations

• Inequivalent and Irreducible Representations

  • Irreducible Representations

  • Schur Theorem

  • Orthogonal Relation

  • Completeness of Representations

  • Character Tables of Finite Groups

  • The Character Table of the Group T

  • The Character Table of the Group O

  • Self-conjugate Representation

• Subduced and Induced Representations

• Applications in Physics

  • Classification of Static Wave Functions

  • Clebsch–Gordan Series and Coefficients

  • Wigner–Eckart Theorem

  • Normal Degeneracy and Accidental Degeneracy

  • An Example of Application

• Irreducible Bases in Group Algebra

  • Ideal and Idempotent

  • Primitive Idempotent

  • Two-side Ideal

  • Standard Irreducible Basis Vectors

• Exercises