Representation Theory: Using the Schur orthogonality relations, we obtain an orthonormal basis of L^2(G) using matrix coefficients of irreducible representations. This shows the sum of squares of dimensions of irreducibles equals |G|. We also obtain an orthonormal basis of Class(G) using irreducible characters, and from this we see that the number of irreducible classes equals the number of conjugacy classes in G. We also obtain character formulas for multiplicities.

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.