Representation Theory: We note how to transfer a group action of a group G on a set X to a group action on F(X), the functions on X. Because F(X) is a vector space, we obtain a representation of the group, and we can apply previous techniques. In particular, the group acts on itself in several ways. We give an overview of features to expect from Fourier analysis on the circle group.

Course materials, including problem sets and solutions, available at mathdoctorbob.org/UR-RepTheory.html

1. RT1: Representation Theory Basics
2. RT2: Unitary Representations
3. RT3. Equivalence and Examples (Expanded)
4. RT4.1. Constructions from Linear Algebra (Expanded)
5. RT4.1.1: Complex Conjugate Representations
6. RT4.2. Schur's Lemma (Expanded)
7. RT5. Mostly Exercises (Expanded)
8. RT6. Representations on Function Spaces
9. RT7.1: Finite Abelian Groups: Character Orthogonality
10. RT7.2. Finite Abelian Groups: Fourier Analysis
11. RT7.3. Finite Abelian Groups: Convolution
12. RT8.1. Schur Orthogonality Relations
13. RT8.2. Finite Groups: Classification of Irreducibles
14. RT8.3. Finite Groups: Projection to Irreducibles
15. RT9. Basic Tensor Analysis
16. RT9.1. Application of Tensors: Normal Modes
17. Character Tables for S4 and A4
18. Character Tables for S5 and A5

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.