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Introduction to Representation Theory by Pavel Etingof, Oleg Golberg

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Introduction to Representation Theory by Pavel Etingof, Oleg Golberg 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 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms 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 9 x2.4. Ideals 15 x2.5. Quotients 15 x2.6. Algebras defined by generators and relations 16 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 30 x2.12. The tensor algebra 35 x2.13. Hilbert's third problem 36 x2.14. Tensor products and duals of representations of Lie algebras 36 x2.15. Representations of sl(2) 37 iii Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms iv Contents x2.16. Problems on Lie algebras 39 Chapter 3. General results of representation theory 41 x3.1. Subrepresentations in semisimple representations 41 x3.2. The density theorem 43 x3.3. Representations of direct sums of matrix algebras 44 x3.4. Filtrations 45 x3.5. Finite dimensional algebras 46 x3.6. Characters of representations 48 x3.7. The Jordan-H¨oldertheorem 50 x3.8. The Krull-Schmidt theorem 51 x3.9. Problems 53 x3.10. Representations of tensor products 56 Chapter 4. Representations of finite groups: Basic results 59 x4.1. Maschke's theorem 59 x4.2. Characters 61 x4.3. Examples 62 x4.4. Duals and tensor products of representations 65 x4.5. Orthogonality of characters 65 x4.6. Unitary representations. Another proof of Maschke's theorem for complex representations 69 x4.7. Orthogonality of matrix elements 70 x4.8. Character tables, examples 71 x4.9. Computing tensor product multiplicities using character tables 75 x4.10. Frobenius determinant 76 x4.11. Historical interlude: Georg Frobenius's \Principle of Horse Trade" 77 x4.12. Problems 81 x4.13. Historical interlude: William Rowan Hamilton's quaternion of geometry, algebra, metaphysics, and poetry 86 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents v Chapter 5. Representations of finite groups: Further results 91 x5.1. Frobenius-Schur indicator 91 x5.2. Algebraic numbers and algebraic integers 93 x5.3. Frobenius divisibility 96 x5.4. Burnside's theorem 98 x5.5. Historical interlude: William Burnside and intellectual harmony in mathematics 100 x5.6. Representations of products 104 x5.7. Virtual representations 105 x5.8. Induced representations 105 x5.9. The Frobenius formula for the character of an induced representation 106 x5.10. Frobenius reciprocity 107 x5.11. Examples 110 x5.12. Representations of Sn 110 x5.13. Proof of the classification theorem for representations of Sn 112 x5.14. Induced representations for Sn 114 x5.15. The Frobenius character formula 115 x5.16. Problems 118 x5.17. The hook length formula 118 x5.18. Schur-Weyl duality for gl(V ) 119 x5.19. Schur-Weyl duality for GL(V ) 122 x5.20. Historical interlude: Hermann Weyl at the intersection of limitation and freedom 122 x5.21. Schur polynomials 128 x5.22. The characters of Lλ 129 x5.23. Algebraic representations of GL(V ) 130 x5.24. Problems 131 x5.25. Representations of GL2(Fq) 132 x5.26. Artin's theorem 141 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms vi Contents x5.27. Representations of semidirect products 142 Chapter 6. Quiver representations 145 x6.1. Problems 145 x6.2. Indecomposable representations of the quivers A1;A2;A3 150 x6.3. Indecomposable representations of the quiver D4 154 x6.4. Roots 160 x6.5. Gabriel's theorem 163 x6.6. Reflection functors 164 x6.7. Coxeter elements 169 x6.8. Proof of Gabriel's theorem 170 x6.9. Problems 173 Chapter 7. Introduction to categories 177 x7.1. The definition of a category 177 x7.2. Functors 179 x7.3. Morphisms of functors 181 x7.4. Equivalence of categories 182 x7.5. Representable functors 183 x7.6. Adjoint functors 184 x7.7. Abelian categories 186 x7.8. Complexes and cohomology 187 x7.9. Exact functors 190 x7.10. Historical interlude: Eilenberg, Mac Lane, and \general abstract nonsense" 192 Chapter 8. Homological algebra 201 x8.1. Projective and injective modules 201 x8.2. Tor and Ext functors 203 Chapter 9. Structure of finite dimensional algebras 209 x9.1. Lifting of idempotents 209 x9.2. Projective covers 210 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents vii x9.3. The Cartan matrix of a finite dimensional algebra 212 x9.4. Homological dimension 212 x9.5. Blocks 213 x9.6. Finite abelian categories 214 x9.7. Morita equivalence 216 References for historical interludes 217 Mathematical references 223 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms 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 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms 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 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms 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.
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