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Introduction to Representation Theory

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Introduction to Representation Theory Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina February 2, 2011 Contents 1 Basic notions of representation theory 5 1.1 What is representation theory? . ......... 5 1.2 Algebras........................................ ... 7 1.3 Representations................................. ...... 7 1.4 Ideals .......................................... 10 1.5 Quotients ....................................... 11 1.6 Algebras defined by generators and relations . ............. 11 1.7 Examplesofalgebras.............................. ...... 11 1.8 Quivers ......................................... 13 1.9 Liealgebras..................................... 15 1.10 Tensorproducts................................. ...... 17 1.11 Thetensoralgebra ............................... ...... 19 1.12 Hilbert’sthirdproblem. ......... 19 arXiv:0901.0827v5 [math.RT] 1 Feb 2011 1.13 Tensor products and duals of representations of Lie algebras.............. 20 1.14 Representations of sl(2) .................................. 20 1.15 ProblemsonLiealgebras . ....... 21 2 General results of representation theory 23 2.1 Subrepresentations in semisimple representations . ................. 23 2.2 Thedensitytheorem ............................... ..... 24 2.3 Representations of direct sums of matrix algebras . ............... 24 2.4 Filtrations..................................... ..... 25 2.5 Finitedimensionalalgebras . ......... 26 1 2.6 Characters of representations . .......... 27 2.7 TheJordan-H¨oldertheorem . ........ 28 2.8 TheKrull-Schmidttheorem . ....... 29 2.9 Problems ........................................ 30 2.10 Representations of tensor products . ............ 31 3 Representations of finite groups: basic results 33 3.1 Maschke’sTheorem................................ ..... 33 3.2 Characters...................................... 34 3.3 Examples ........................................ 35 3.4 Duals and tensor products of representations . ............. 36 3.5 Orthogonality of characters . ......... 37 3.6 Unitary representations. Another proof of Maschke’s theorem for complex represen- tations............................................ 38 3.7 Orthogonality of matrix elements . .......... 39 3.8 Charactertables,examples . ........ 40 3.9 Computing tensor product multiplicities using charactertables ............ 42 3.10Problems ....................................... 43 4 Representations of finite groups: further results 47 4.1 Frobenius-Schurindicator . ......... 47 4.2 Frobeniusdeterminant . ....... 48 4.3 Algebraic numbers and algebraic integers . ............ 49 4.4 Frobeniusdivisibility . ......... 51 4.5 Burnside’sTheorem ............................... ..... 52 4.6 Representationsofproducts . ......... 54 4.7 Virtualrepresentations. ......... 54 4.8 InducedRepresentations . ........ 54 4.9 TheMackeyformula ................................ 55 4.10 Frobeniusreciprocity. ......... 56 4.11Examples ....................................... 57 4.12 Representations of Sn ................................... 58 4.13 ProofofTheorem4.36 .. .. .. .. .. .. .. .. .. ...... 59 4.14 Induced representations for Sn .............................. 60 2 4.15 TheFrobeniuscharacterformula . .......... 61 4.16Problems ....................................... 63 4.17 Thehooklengthformula. ....... 63 4.18 Schur-Weyl duality for gl(V ) ............................... 64 4.19 Schur-Weyl duality for GL(V )............................... 65 4.20 Schurpolynomials ............................... ...... 66 4.21 The characters of Lλ .................................... 66 4.22 Polynomial representations of GL(V )........................... 67 4.23Problems ....................................... 68 4.24 Representations of GL2(Fq)................................ 68 4.24.1 Conjugacy classes in GL2(Fq)........................... 68 4.24.2 1-dimensional representations . .......... 70 4.24.3 Principal series representations . ........... 71 4.24.4 Complementary series representations . ........... 73 4.25 Artin’stheorem................................. ...... 75 4.26 Representations of semidirect products . .............. 76 5 Quiver Representations 78 5.1 Problems ........................................ 78 5.2 Indecomposable representations of the quivers A1, A2, A3 ................ 81 5.3 Indecomposable representations of the quiver D4 .................... 83 5.4 Roots ........................................... 87 5.5 Gabriel’stheorem................................ ...... 89 5.6 ReflectionFunctors............................... ...... 90 5.7 Coxeterelements ................................. ..... 93 5.8 ProofofGabriel’stheorem. ........ 94 5.9 Problems ........................................ 96 6 Introduction to categories 98 6.1 Thedefinitionofacategory . ....... 98 6.2 Functors........................................ 99 6.3 Morphismsoffunctors ............................. 100 6.4 Equivalence of categories . .........100 6.5 Representablefunctors . ........101 3 6.6 Adjointfunctors ................................. 102 6.7 Abeliancategories ............................... 103 6.8 Exactfunctors ................................... 104 7 Structure of finite dimensional algebras 106 7.1 Projectivemodules ............................... 106 7.2 Liftingofidempotents .. .. .. .. .. .. .. .. .......106 7.3 Projectivecovers ................................ 107 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 geometry, probability theory, quantum mechanics and quantum field theory. Representation theory was born in 1896 in the work of the German 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 x , each of which occurs with multiplicity { g} 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 representation theory of finite groups. 1 The present lecture notes arose from a representation theory course given by the first author to the remaining 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 undergraduate math students in the Fall of 2008. The lectures are supplemented by many problems and exercises, which contain a lot of additional material; the more difficult exercises are provided with hints. The notes cover a number of standard topics in representation theory of groups, Lie algebras, and quivers. We mostly follow [FH], with the exception of the sections discussing quivers, which follow [BGP]. We also recommend the comprehensive textbook [CR]. The notes should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra. Acknowledgements. The authors are grateful to the Clay Mathematics Institute for hosting the first version of this course. The first author is very indebted to Victor Ostrik for helping him prepare this course, and thanks Josh Nichols-Barrer and Thomas Lam for helping run the course in 2004 and for useful comments. He is also very grateful to Darij Grinberg for very careful reading of the text, for many useful comments and corrections, and for suggesting the Exercises in Sections 1.10, 2.3, 3.5, 4.9, 4.26, and 6.8. 1For more on the history of representation theory, see [Cu]. 4 1 Basic notions of representation theory 1.1 What is representation theory? In technical terms, representation theory studies representations of associative algebras. Its general content can be very briefly summarized as follows. An associative algebra over a field k is a vector space A over k equipped with an associative bilinear multiplication a, b ab, a, b A. We will always consider associative algebras with unit, → ∈ i.e., with an element 1 such that 1 a = a 1= a for all a A. A basic example of an associative · · ∈ algebra is the algebra EndV of linear operators from a vector space V to itself. Other important examples include algebras defined by generators and relations, such as group algebras and universal enveloping algebras of Lie algebras. A representation of an associative algebra A (also called a left A-module) is a vector space V equipped with a homomorphism ρ : A EndV , i.e., a linear map preserving the multiplication → and unit. A subrepresentation of a representation V is a subspace U V which is invariant under all ⊂ operators ρ(a), a A. Also, if V ,V are two representations of A then the direct sum V V ∈ 1 2 1 ⊕ 2 has an obvious structure of a representation of A. A nonzero representation V of A is said to be irreducible if its only subrepresentations are 0 and V itself, and indecomposable if it cannot be written as a direct sum of two nonzero subrepresentations. Obviously, irreducible implies indecomposable, but not vice versa. Typical problems of representation theory are as follows: 1. Classify irreducible representations of a given algebra A. 2. Classify
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