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Introduction to Representation Theory Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina with historical interludes by Slava Gerovitch Contents Chapter 1. Introduction 1 Chapter 2. Basic notions of representation theory 5 x2.1. What is representation theory? 5 x2.2. Algebras 8 x2.3. Representations 10 x2.4. Ideals 15 x2.5. Quotients 15 x2.6. Algebras defined by generators and relations 17 x2.7. Examples of algebras 17 x2.8. Quivers 19 x2.9. Lie algebras 22 x2.10. Historical interlude: Sophus Lie's trials and transformations 26 x2.11. Tensor products 31 x2.12. The tensor algebra 35 x2.13. Hilbert's third problem 36 x2.14. Tensor products and duals of representations of Lie algebras 37 x2.15. Representations of sl(2) 37 iii iv Contents x2.16. Problems on Lie algebras 39 Chapter 3. General results of representation theory 43 x3.1. Subrepresentations in semisimple representations 43 x3.2. The density theorem 45 x3.3. Representations of direct sums of matrix algebras 47 x3.4. Filtrations 49 x3.5. Finite dimensional algebras 49 x3.6. Characters of representations 52 x3.7. The Jordan-H¨oldertheorem 53 x3.8. The Krull-Schmidt theorem 54 x3.9. Problems 56 x3.10. Representations of tensor products 59 Chapter 4. Representations of finite groups: Basic results 61 x4.1. Maschke's theorem 61 x4.2. Characters 63 x4.3. Examples 64 x4.4. Duals and tensor products of representations 67 x4.5. Orthogonality of characters 67 x4.6. Unitary representations. Another proof of Maschke's theorem for complex representations 71 x4.7. Orthogonality of matrix elements 72 x4.8. Character tables, examples 73 x4.9. Computing tensor product multiplicities using character tables 77 x4.10. Frobenius determinant 78 x4.11. Historical interlude: Georg Frobenius's \Principle of Horse Trade" 79 x4.12. Problems 83 x4.13. Historical interlude: William Rowan Hamilton's quaternion of geometry, algebra, metaphysics, and poetry 88 Contents v Chapter 5. Representations of finite groups: Further results 93 x5.1. Frobenius-Schur indicator 93 x5.2. Algebraic numbers and algebraic integers 95 x5.3. Frobenius divisibility 98 x5.4. Burnside's theorem 100 x5.5. Historical interlude: William Burnside and intellectual harmony in mathematics 102 x5.6. Representations of products 107 x5.7. Virtual representations 107 x5.8. Induced representations 107 x5.9. The Frobenius formula for the character of an induced representation 109 x5.10. Frobenius reciprocity 110 x5.11. Examples 112 x5.12. Representations of Sn 112 x5.13. Proof of the classification theorem for representations of Sn 114 x5.14. Induced representations for Sn 116 x5.15. The Frobenius character formula 118 x5.16. Problems 120 x5.17. The hook length formula 121 x5.18. Schur-Weyl duality for gl(V ) 122 x5.19. Schur-Weyl duality for GL(V ) 124 x5.20. Historical interlude: Hermann Weyl at the intersection of limitation and freedom 125 x5.21. Schur polynomials 131 x5.22. The characters of Lλ 132 x5.23. Algebraic representations of GL(V ) 133 x5.24. Problems 135 x5.25. Representations of GL2(Fq) 135 x5.26. Artin's theorem 144 vi Contents x5.27. Representations of semidirect products 145 Chapter 6. Quiver representations 149 x6.1. Problems 149 x6.2. Indecomposable representations of the quivers A1;A2;A3 154 x6.3. Indecomposable representations of the quiver D4 158 x6.4. Roots 164 x6.5. Gabriel's theorem 167 x6.6. Reflection functors 168 x6.7. Coxeter elements 173 x6.8. Proof of Gabriel's theorem 174 x6.9. Problems 177 Chapter 7. Introduction to categories 181 x7.1. The definition of a category 181 x7.2. Functors 183 x7.3. Morphisms of functors 185 x7.4. Equivalence of categories 186 x7.5. Representable functors 187 x7.6. Adjoint functors 188 x7.7. Abelian categories 190 x7.8. Complexes and cohomology 191 x7.9. Exact functors 194 x7.10. Historical interlude: Eilenberg, Mac Lane, and \general abstract nonsense" 196 Chapter 8. Homological algebra 205 x8.1. Projective and injective modules 205 x8.2. Tor and Ext functors 207 Chapter 9. Structure of finite dimensional algebras 213 x9.1. Lifting of idempotents 213 x9.2. Projective covers 214 Contents vii x9.3. The Cartan matrix of a finite dimensional algebra 216 x9.4. Homological dimension 216 x9.5. Blocks 217 x9.6. Finite abelian categories 218 x9.7. Morita equivalence 220 References for historical interludes 221 Mathematical references 227 Chapter 1 Introduction Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to ge- ometry, probability theory, quantum mechanics, and quantum field theory. Representation theory was born in 1896 in the work of the Ger- man mathematician F. G. Frobenius. This work was triggered by a letter to Frobenius by R. Dedekind. In this letter Dedekind made the following observation: take the multiplication table of a finite group G and turn it into a matrix XG by replacing every entry g of this table by a variable xg. Then the determinant of XG factors into a product of irreducible polynomials in fxgg, each of which occurs with multiplicity equal to its degree. Dedekind checked this surprising fact in a few special cases but could not prove it in general. So he gave this problem to Frobenius. In order to find a solution of this problem (which we will explain below), Frobenius created the representation theory of finite groups. The goal of this book is to give a \holistic" introduction to rep- resentation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representa- tion theories of groups, Lie algebras, and quivers as special cases. It is designed as a textbook for advanced undergraduate and beginning 1 2 1. Introduction graduate students and should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract al- gebra. Theoretical material in this book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book covers a number of standard topics in representation theory of groups, associative algebras, Lie algebras, and quivers. For a more detailed treatment of these topics, we refer the reader to the textbooks [S], [FH], and [CR]. We mostly follow [FH], with the exception of the sections discussing quivers, which follow [BGP], and the sections on homological algebra and finite dimensional algebras, for which we recommend [W] and [CR], respectively. The organization of the book is as follows. Chapter 2 is devoted to the basics of representation theory. Here we review the basics of abstract algebra (groups, rings, modules, ideals, tensor products, symmetric and exterior powers, etc.), as well as give the main definitions of representation theory and discuss the objects whose representations we will study (associative algebras, groups, quivers, and Lie algebras). Chapter 3 introduces the main general results about representa- tions of associative algebras (the density theorem, the Jordan-H¨older theorem, the Krull-Schmidt theorem, and the structure theorem for finite dimensional algebras). In Chapter 4 we discuss the basic results about representations of finite groups. Here we prove Maschke's theorem and the orthogonality of characters and matrix elements and compute character tables and tensor product multiplicities for the simplest finite groups. We also discuss the Frobenius determinant, which was a starting point for development of the representation theory of finite groups. We continue to study representations of finite groups in Chapter 5, treating more advanced and special topics, such as the Frobenius- Schur indicator, the Frobenius divisibility theorem, the Burnside the- orem, the Frobenius formula for the character of an induced repre- sentation, representations of the symmetric group and the general 1. Introduction 3 linear group over C, representations of GL2(Fq), representations of semidirect products, etc. In Chapter 6, we give an introduction to the representation theory of quivers (starting with the problem of the classification of configura- tions of n subspaces in a vector space) and present a proof of Gabriel's theorem, which classifies quivers of finite type. In Chapter 7, we give an introduction to category theory, in par- ticular, abelian categories, and explain how such categories arise in representation theory. In Chapter 8, we give a brief introduction to homological algebra and explain how it can be applied to categories of representations. Finally, in Chapter 9 we give a short introduction to the repre- sentation theory of finite dimensional algebras. Besides, the book contains six historical interludes written by Dr. Slava Gerovitch.1 These interludes, written in an accessible and ab- sorbing style, tell about the life and mathematical work of some of the mathematicians who played a major role in the development of modern algebra and representation theory: F. G. Frobenius, S. Lie, W. Burnside, W. R. Hamilton, H. Weyl, S. Mac Lane, and S. Eilen- berg. For more on the history of representation theory, we recommend that the reader consult the references to the historical interludes, in particular the excellent book [Cu]. Acknowledgments. This book arose from the lecture notes of a representation theory course given by the first author to the re- maining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students and its extended version given by the first author to MIT undergrad- uate mathematics students in the fall of 2008. The authors are grateful to the Clay Mathematics Institute for hosting the first version of this course. The first author is very in- debted to Victor Ostrik for helping him prepare this course and thanks 1I wish to thank Prof.
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