Lecture notes for Introduction to Representation Theory

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[Cu] C. Curtis, Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, AMS, 1999. [CR] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, AMS, 2006. [FH] W. Fulton and J. Harris, Representation Theory, A first course, Springer, New York, 1991. [Fr] Peter J. Freyd, Abelian Categories, an Introduction to the Theory of Functors. Harper and Row (1964). [McL] S. MacLane, Categories for a working Mathematician: 2nd Ed., Graduate Texts in Mathematics 5, Springer, 1998. 108
MIT OpenCourseWare http://ocw.mit.edu 18.712 Introduction to Representation Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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Introduction to representation theo
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4.15 The Frobenius character formul
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1 Basic notions of representation t
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1.2 Algebras Let us now begin a sys
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(ii) If V 2 is irreducible, θ is s
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1.5 Quotients Let A be an algebra a
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Hint. Show that x p and y p are cen
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(b) A = k < x 1 , ..., x m > (the g
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1.10 Tensor products In this subsec
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Similarly, if C is another k-algebr
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(e) Let N v be the smallest N satis
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⇒ ⇒ 2 General results of repres
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y θ(a 1 , . . . , a n ) = a 1 y 1
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Example 2.14. 1. Let A = k[x]/(x n
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V . This number is called the lengt
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Problem 2.24. Let A be an algebra,
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3 Representations of finite groups:
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Exercise. Show that if | G | = 0 in
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If V, W are representations of G, t
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Theorem 3.11. If G is finite, then
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A 4 Id (123) (132) (12)(34) # 1 4 4
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⇒ C C C + 3 C 3 − C 3 − C C
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(a) Derive this classification. Hin
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4 Representations of finite groups:
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Proof. Let V = C[G] be the regular
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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
Page 57 and 58: 1.48). In particular, this understa
Page 59 and 60: For the partition ∂ = (1, ..., 1)
Page 61 and 62: Proof. The proof is obtained easily
Page 63 and 64: Corollary 4.49 easily implies that
Page 65 and 66: Lemma 4.56. Let k be a field of cha
Page 67 and 68: Theorem 4.63. (Weyl character formu
Page 69 and 70: ⇒ ⇒ The goal of this section is
Page 71 and 72: ⇒ 4.24.3 Principal series represe
Page 73 and 74: If ∂ 1 = ⇒ ∂ 2 , let z = xy
Page 75 and 76: Because λ is also a root of unity,
Page 77 and 78: Proof. (i) Let us decompose V = V (
Page 79 and 80: • E 7 : −−−−−−−−
Page 81 and 82: (d) Which groups from Problem 3.24
Page 83 and 84: By identifying V and Y as subspaces
Page 85 and 86: So - considering the first summand
Page 87 and 88: 5.4 Roots From now on, let be a fi
Page 89 and 90: on Z N restricts to the inner produ
Page 91 and 92: 1. Let i Q be a sink. Then either
Page 93 and 94: Proposition 5.31. Let Q be a quiver
Page 95 and 96: Theorem 5.34. There is m N, such t
Page 97 and 98: 3) H n : V = C n with basis v i , W
Page 99 and 100: We also mention that in many exampl
Page 101 and 102: For this reason in category theory,
Page 103 and 104: Dictionary between category theory
Page 105 and 106: Definition 6.21. An abelian categor
Page 107: Now, in the general case, we prove
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