Lecture notes for Introduction to Representation Theory

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2.6 Characters of representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.7 The Jordan-Hölder theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 The Krull-Schmidt theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.10 Representations of tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Representations of finite groups: basic results 33 3.1 Maschke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Character tables, examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.9 Computing tensor product multiplicities using character tables . . . . . . . . . . . . 42 3.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 Representations of finite groups: further results 47 4.1 Frobenius-Schur indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Frobenius determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Algebraic numbers and algebraic integers . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4 Frobenius divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5 Burnside’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 Representations of products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.7 Virtual representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.8 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.9 The Mackey formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.10 Frobenius reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.12 Representations of S n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.13 Proof of Theorem 4.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.14 Induced representations for S n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2
4.15 The Frobenius character formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.16 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.17 The hook length formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.18 Schur-Weyl duality for gl(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.19 Schur-Weyl duality for GL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.20 Schur polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.21 The characters of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.22 Polynomial representations of GL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.23 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.24 Representations of GL 2 (F q ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.24.1 Conjugacy classes in GL 2 (F q ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 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’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.26 Representations of semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Quiver Representations 78 5.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Indecomposable representations of the quivers A 1 , A 2 , A 3 . . . . . . . . . . . . . . . . 81 5.3 Indecomposable representations of the quiver D 4 . . . . . . . . . . . . . . . . . . . . 83 5.4 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.5 Gabriel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6 Reflection Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.7 Coxeter elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.8 Proof of Gabriel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6 Introduction to categories 98 6.1 The definition of a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3 Morphisms of functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.4 Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.5 Representable functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3
Page 1: Introduction to representation theo
Page 5 and 6: 1 Basic notions of representation t
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
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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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