Introduction to representation theory - MIT Mathematics

1 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. What is representation theory ? 5. Algebras 8. Representations 9. Ideals 15. Quotients 15. Algebras defined by generators and relations 16. Examples of algebras 17. Quivers 19. Lie algebras 22. Historical interlude: Sophus Lie's trials and transformations 26. Tensor products 30. The tensor algebra 35. Hilbert's third problem 36. Tensor products and duals of representations of Lie algebras 36. Representations of sl(2) 37. iii iv Contents Problems on Lie algebras 39. Chapter 3. General results of representation theory 41. Subrepresentations in semisimple representations 41. The density theorem 43. Representations of direct sums of matrix algebras 44.

2 Filtrations 45. Finite dimensional algebras 46. Characters of representations 48. The Jordan-Ho lder theorem 50. The Krull-Schmidt theorem 51. Problems 53. Representations of tensor products 56. Chapter 4. Representations of finite groups: Basic results 59. Maschke's theorem 59. Characters 61. Examples 62. Duals and tensor products of representations 65. Orthogonality of characters 65. Unitary representations. Another proof of Maschke's theorem for complex representations 68. Orthogonality of matrix elements 70. Character tables, examples 71. Computing tensor product multiplicities using character tables 74. Frobenius determinant 75. Historical interlude: Georg Frobenius's Principle of Horse Trade 77. Problems 81. Historical interlude: William Rowan Hamilton's quaternion of geometry, algebra, metaphysics, and poetry 86. Contents v Chapter 5. Representations of finite groups: Further results 91.

3 Frobenius-Schur indicator 91. Algebraic numbers and algebraic integers 93. Frobenius divisibility 96. Burnside's theorem 98. Historical interlude: William Burnside and intellectual harmony in Mathematics 100. Representations of products 104. Virtual representations 105. Induced representations 105. The Frobenius formula for the character of an induced representation 106. Frobenius reciprocity 107. Examples 110. Representations of Sn 110. Proof of the classification theorem for representations of Sn 112. Induced representations for Sn 114. The Frobenius character formula 115. Problems 118. The hook length formula 118. Schur-Weyl duality for gl(V ) 119. Schur-Weyl duality for GL(V ) 122. Historical interlude: Hermann Weyl at the intersection of limitation and freedom 122. Schur polynomials 128. The characters of L 129. Algebraic representations of GL(V ) 130. Problems 131. Representations of GL2 (Fq ) 132.

4 Artin's theorem 141. vi Contents Representations of semidirect products 142. Chapter 6. Quiver representations 145. Problems 145. Indecomposable representations of the quivers A1 , A2 , A3 150. Indecomposable representations of the quiver D4 154. Roots 160. Gabriel's theorem 163. Reflection functors 164. Coxeter elements 169. Proof of Gabriel's theorem 170. Problems 173. Chapter 7. Introduction to categories 177. The definition of a category 177. Functors 179. Morphisms of functors 181. Equivalence of categories 182. Representable functors 183. Adjoint functors 184. Abelian categories 186. Complexes and cohomology 187. Exact functors 190. Historical interlude: Eilenberg, Mac Lane, and general abstract nonsense 192. Chapter 8. Homological algebra 201. Projective and injective modules 201. Tor and Ext functors 203. Chapter 9. Structure of finite dimensional algebras 209. Lifting of idempotents 209.

5 Projective covers 210. Contents vii The Cartan matrix of a finite dimensional algebra 211. Homological dimension 212. Blocks 213. Finite abelian categories 214. Morita equivalence 215. References for historical interludes 217. Mathematical references 223. 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.

6 Then the determinant of XG factors into a product of irreducible polynomials in {xg }, 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.

7 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).

8 Chapter 3 introduces the main general results about representa- tions of associative algebras (the density theorem, the Jordan-Ho lder 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.

9 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 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.

10 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. Pavel Etingof and his co-authors for adding technical notes to my historical monograph.