
Lecture Description
Representation Theory: We present basic concepts about the representation theory of finite groups. Representations are defined, as are notions of invariant subspace, irreducibility and full reducibility. For a general finite group G, we suggest an analogue to the finite abelian case, where every representation may be represented by diagonal matrices with unitary diagonal entries.
Course materials, including problem sets and solutions, available at mathdoctorbob.org/UR-RepTheory.html
Course Index
- RT1: Representation Theory Basics
- RT2: Unitary Representations
- RT3. Equivalence and Examples (Expanded)
- RT4.1. Constructions from Linear Algebra (Expanded)
- RT4.1.1: Complex Conjugate Representations
- RT4.2. Schur's Lemma (Expanded)
- RT5. Mostly Exercises (Expanded)
- RT6. Representations on Function Spaces
- RT7.1: Finite Abelian Groups: Character Orthogonality
- RT7.2. Finite Abelian Groups: Fourier Analysis
- RT7.3. Finite Abelian Groups: Convolution
- RT8.1. Schur Orthogonality Relations
- RT8.2. Finite Groups: Classification of Irreducibles
- RT8.3. Finite Groups: Projection to Irreducibles
- RT9. Basic Tensor Analysis
- RT9.1. Application of Tensors: Normal Modes
- Character Tables for S4 and A4
- Character Tables for S5 and A5
Course Description
Doctor Bob provides 18 short video lectures on the introduction to representation theory. Group representations are where group theory meets linear algebra, and important applications arise in various math subjects (number theory, analysis, algebraic geometry), physics, and chemistry.
We consider the basic representation theory of finite groups. Goals include a look at Fourier series/analysis using groups and elementary character theory.