Introduction to Representation Theory

Introduction to Representation Theory - Etingof
Contents
Chapter 1. Introduction
Chapter 2. Basic notions of representation theory
	2.1. What is representation theory?
	2.2. Algebras
	2.3. Representations
	2.4. Ideals
	2.5. Quotients
	2.6. Algebras defined by generators and relations
	2.7. Examples of algebras
	2.8. Quivers
	2.9. Lie algebras
	2.10. Historical interlude: Sophus Lie’s trials and transformations
	2.11. Tensor products
	2.12. The tensor algebra
	2.13. Hilbert’s third problem
	2.14. Tensor products and duals of representations of Lie algebras
	2.15. Representations of sl(2)
	2.16. Problems on Lie algebras
Chapter 3. General results of representation theory
	3.1. Subrepresentations in semisimple representations
	3.2. The density theorem
	3.3. Representations of direct sums of matrix algebras
	3.4. Filtrations
	3.5. Finite dimensional algebras
	3.6. Characters of representations
	3.7. The Jordan-H¨older theorem
	3.8. The Krull-Schmidt theorem
	3.9. Problems
	3.10. Representations of tensor products
Chapter 4. Representations of finite groups: Basic results
	4.1. Maschke’s theorem
	4.2. Characters
	4.3. Examples
	4.4. Duals and tensor products of representations
	4.5. Orthogonality of characters
	4.6. Unitary representations. Another proof of Maschke’s theorem for complex representations
	4.7. Orthogonality of matrix elements
	4.8. Character tables, examples
	4.9. Computing tensor product multiplicities using character tables
	4.10. Frobenius determinant
	4.11. Historical interlude: Georg Frobenius’s “Principle of Horse Trade”
	4.12. Problems
	4.13. Historical interlude: William Rowan Hamilton’s quaternion of geometry, algebra, metaphysics, and poetry
Chapter 5. Representations of finite groups: Further results
	5.1. Frobenius-Schur indicator
	5.2. Algebraic numbers and algebraic integers
	5.3. Frobenius divisibility
	5.4. Burnside’s theorem
	5.5. Historical interlude: William Burnside and intellectual harmony in mathematics
	5.6. Representations of products
	5.7. Virtual representations
	5.8. Induced representations
	5.9. The Frobenius formula for the character of an induced representation
	5.10. Frobenius reciprocity
	5.11. Examples
	5.12. Representations of Sn
	5.13. Proof of the classi.cation theorem for representations of Sn
	5.14. Induced representations for Sn
	5.15. The Frobenius character formula
	5.16. Problems
	5.17. The hook length formula
	5.18. Schur-Weyl duality for gl(V)
	5.19. Schur-Weyl duality for GL(V)
	5.20. Historical interlude: Hermann Weyl at the intersection of limitation and freedom
	5.21. Schur polynomials
	5.22. The characters of Lλ
	5.23. Algebraic representations of GL(V)
	5.24. Problems
	5.25. Representations of GL2(Fq)
	5.26. Artin’s theorem
	5.27. Representations of semidirect products
Chapter 6. Quiver representations
	6.1. Problems
	6.2. Indecomposable representations of the quivers A1,A2,A3
	6.3. Indecomposable representations of the quiver D4
	6.4. Roots
	6.5. Gabriel’s theorem
	6.6. Re.ection functors
	6.7. Coxeter elements
	6.8. Proof of Gabriel’s theorem
	6.9. Problems
Chapter 7. Introduction to categories
	7.1. The definition of a category
	7.2. Functors
	7.3. Morphisms of functors
	7.4. Equivalence of categories
	7.5. Representable functors
	7.6. Adjoint functors
	7.7. Abelian categories
	7.8. Complexes and cohomology
	7.9. Exact functors
	7.10. Historical interlude: Eilenberg, Mac Lane, and “general abstract nonsense”
Chapter 8. Homological algebra
	8.1. Projective and injective modules
	8.2. Tor and Ext functors
Chapter 9. Structure of finite dimensional algebras
	9.1. Lifting of idempotents
	9.2. Projective covers
	9.3. The Cartan matrix of a finite dimensional algebra
	9.4. Homological dimension
	9.5. Blocks
	9.6. Finite abelian categories
	9.7. Morita equivalence
References for historical interludes
Mathematical references