Lecture Notes

Students are assigned readings in these lecture notes each week. The present lecture notes arose from a representation theory course given by Prof. Etingof in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students. The students in that course — Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Elena Yudovina, and Dmitry Vaintrob — co-authored the lecture notes which are published here with their permission.

Complete Lecture Notes (PDF - 1.3MB)

Introduction (PDF)

Chapter 1: Basic Notions of Representation Theory (PDF)

1.1 What is representation theory?
1.2 Algebras
1.3 Representations
1.4 Ideals
1.5 Quotients
1.6 Algebras defined by generators and relations
1.7 Examples of algebras
1.8 Quivers
1.9 Lie algebras
1.10 Tensor products
1.11 The tensor algebra
1.12 Hilbert's third problem
1.13 Tensor products and duals of representations of Lie algebras
1.14 Representations of sl(2)
1.15 Problems on Lie algebras

Chapter 2: General Results of Representation Theory (PDF)

2.1 Subrepresentations in semisimple representations
2.2 The density theorem
2.3 Representations of direct sums of matrix algebras
2.4 Filtrations
2.5 Finite dimensional algebras
2.6 Characters of representations
2.7 The Jordan-Hölder theorem
2.8 The Krull-Schmidt theorem
2.9 Problems
2.10 Representations of tensor products

Chapter 3: Representations of Finite Groups: Basic Results (PDF)

3.1 Maschke's Theorem
3.2 Characters
3.3 Examples
3.4 Duals and tensor products of representations
3.5 Orthogonality of characters
3.6 Unitary representations. Another proof of Maschke's theorem for complex representations
3.7 Orthogonality of matrix elements
3.8 Character tables, examples
3.9 Computing tensor product multiplicities using character tables
3.10 Problems

Chapter 4: Representations of Finite Groups: Further Results (PDF)

4.1 Frobenius-Schur indicator
4.2 Frobenius determinant
4.3 Algebraic numbers and algebraic integers
4.4 Frobenius divisibility
4.5 Burnside's theorem
4.6 Representations of products
4.7 Virtual representations
4.8 Induced representations
4.9 The Mackey formula
4.10 Frobenius reciprocity
4.11 Examples
4.12 Representations of Sn
4.13 Proof of theorem 4.36
4.14 Induced representations for Sn
4.15 The Frobenius character formula
4.16 Problems
4.17 The hook length formula
4.18 Schur-Weyl duality
4.19 Schur-Weyl duality for GL(V)
4.20 Schur polynomials
4.21 The characters of Lλ
4.22 Polynomial representations of GL(V)
4.23 Problems
4.24 Representations of GL2(Fq)

4.24.1 Conjugacy classes in GL2(Fq)
4.24.2 1-dimensional representations
4.24.3 Principal series representations
4.24.4 Complementary series representations

4.25 Artin's theorem
4.26 Representations of semidirect products

Chapter 5: Quiver Representations (PDF)

5.1 Problems
5.2 Indecomposable representations of the quivers A1,A2,A3
5.3 Indecomposable representations of the quiver D4
5.4 Roots
5.5 Gabriel's theorem
5.6 Reflection functors
5.7 Coxeter elements
5.8 Proof of Gabriel's theorem
5.9 Problems

Chapter 6: Introduction to Categories (PDF)

6.1 The definition of a category
6.2 Functors
6.3 Morphisms of functors
6.4 Equivalence of categories
6.5 Representable functors
6.6 Adjoint functors
6.7 Abelian categories
6.8 Exact functors

Chapter 7: Structure of Finite Dimensional Algebras (PDF)

7.1 Projective modules
7.2 Lifting of idempotents
7.3 Projective covers