Lecture notes for Introduction to Representation Theory

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6.6 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.7 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.8 Exact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7 Structure of finite dimensional algebras 106 7.1 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.2 Lifting of idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 Projective covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 X G by replacing every entry g of this table by a variable x g . Then the determinant of X G factors into a product of irreducible polynomials in {x g }, 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 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. 1 For 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 δ : 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 δ(a), a A. Also, if V 1 , V 2 are two representations of A then the direct sum V 1 V 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 indecomposable representations of A. 3. Do 1 and 2 restricting to finite dimensional representations. As mentioned above, the algebra A is often given to us by generators and relations. For example, the universal enveloping algebra U of the Lie algebra sl(2) is generated by h, e, f with defining relations he − eh = 2e, hf − fh = −2f, ef − fe = h. (1) This means that the problem of finding, say, N-dimensional representations of A reduces to solving a bunch of nonlinear algebraic equations with respect to a bunch of unknown N by N matrices, for example system (1) with respect to unknown matrices h, e, f. It is really striking that such, at first glance hopelessly complicated, systems of equations can in fact be solved completely by methods of representation theory! For example, we will prove the following theorem. Theorem 1.1. Let k = C be the field of complex numbers. Then: (i) The algebra U has exactly one irreducible representation V d of each dimension, up to equivalence; this representation is realized in the space of homogeneous polynomials of two variables x, y of degree d − 1, and defined by the formulas δ(h) = x − y , δ(e) = x , δ(f) = y . x y y x (ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite 5
Page 1 and 2: Introduction to representation theo
Page 3: 4.15 The Frobenius character formul
Page 7 and 8: 1.2 Algebras Let us now begin a sys
Page 9 and 10: (ii) If V 2 is irreducible, θ is s
Page 11 and 12: 1.5 Quotients Let A be an algebra a
Page 13 and 14: Hint. Show that x p and y p are cen
Page 15 and 16: (b) A = k < x 1 , ..., x m > (the g
Page 17 and 18: 1.10 Tensor products In this subsec
Page 19 and 20: Similarly, if C is another k-algebr
Page 21 and 22: (e) Let N v be the smallest N satis
Page 23 and 24: ⇒ ⇒ 2 General results of repres
Page 25 and 26: y θ(a 1 , . . . , a n ) = a 1 y 1
Page 27 and 28: Example 2.14. 1. Let A = k[x]/(x n
Page 29 and 30: V . This number is called the lengt
Page 31 and 32: Problem 2.24. Let A be an algebra,
Page 33 and 34: 3 Representations of finite groups:
Page 35 and 36: Exercise. Show that if | G | = 0 in
Page 37 and 38: If V, W are representations of G, t
Page 39 and 40: Theorem 3.11. If G is finite, then
Page 41 and 42: A 4 Id (123) (132) (12)(34) # 1 4 4
Page 43 and 44: ⇒ C C C + 3 C 3 − C 3 − C C
Page 45 and 46: (a) Derive this classification. Hin
Page 47 and 48: 4 Representations of finite groups:
Page 49 and 50: Proof. Let V = C[G] be the regular
Page 51 and 52: Note that any algebraic conjugate o
Page 53 and 54: The proof will be based on the foll
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and the action g(f)(x) = f(xg) ⊕g
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1.48). In particular, this understa
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For the partition ∂ = (1, ..., 1)
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Proof. The proof is obtained easily
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Corollary 4.49 easily implies that
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Lemma 4.56. Let k be a field of cha
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Theorem 4.63. (Weyl character formu
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⇒ ⇒ The goal of this section is
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⇒ 4.24.3 Principal series represe
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If ∂ 1 = ⇒ ∂ 2 , let z = xy
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Because λ is also a root of unity,
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Proof. (i) Let us decompose V = V (
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• E 7 : −−−−−−−−
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(d) Which groups from Problem 3.24
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By identifying V and Y as subspaces
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So - considering the first summand
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5.4 Roots From now on, let be a fi
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on Z N restricts to the inner produ
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1. Let i Q be a sink. Then either
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Proposition 5.31. Let Q be a quiver
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Theorem 5.34. There is m N, such t
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3) H n : V = C n with basis v i , W
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We also mention that in many exampl
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For this reason in category theory,
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Dictionary between category theory
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Definition 6.21. An abelian categor
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Now, in the general case, we prove
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Lecture notes for Introduction to Representation Theory

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