Lecture notes for Introduction to Representation Theory

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of x := ⎛ x j j in the polynomial Hm (x) im . ⇒ Lemma 4.42. c is proportional to an idempotent. Namely, c 2 n! = dim V c . Proof. Lemma 4.40 implies that c 2 is proportional to c . Also, it is easy to see that the trace of c in the regular representation is n! (as the coefficient of the identity element in c is 1). This implies the statement. Lemma 4.43. Let A be an algebra and e be an idempotent in A. Then for any left A-module M, one has Hom A (Ae, M) ∪ = eM (namely, x eM corresponds to f x : Ae ⊃ M given by f x (a) = ax, a Ae). Proof. Note that 1 − e is also an idempotent in A. Thus the statement immediately follows from the fact that Hom A (A, M) ∪ = M and the decomposition A = Ae A(1 − e). Now we are ready to prove Theorem 4.36. Let ∂ ⊂ µ. Then by Lemmas 4.42, 4.43 Hom Sn (V , V µ ) = Hom Sn (C[S n ]c , C[S n ]c µ ) = c C[S n ]c µ . The latter space is zero for ∂ > µ by Lemma 4.41, and 1-dimensional if ∂ = µ by Lemmas 4.40 and 4.42. Therefore, V are irreducible, and V is not isomorphic to V µ if ∂ = µ. Since the number of partitions equals the number of conjugacy classes in S n , the representations V exhaust all the irreducible representations of S n . The theorem is proved. 4.14 Induced representations for S n Denote by U the representation Ind Sn C. It is easy to see that U can be alternatively defined as P U = C[S n ]a . Proposition 4.44. Hom(U , V µ ) = 0 for µ < ∂, and dim Hom(U , V ) = 1. Thus, U = µ∧ K µ V µ , where K µ are nonnegative integers and K = 1. Definition 4.45. The integers K µ are called the Kostka numbers. Proof. By Lemmas 4.42 and 4.43, Hom(U , V µ ) = Hom(C[S n ]a , C[S n ]a µ b µ ) = a C[S n ]a µ b µ , and the result follows from Lemmas 4.40 and 4.41. Now let us compute the character of U . Let C i be the conjugacy class in S n having i l cycles of length l for all l ⊂ 1 (here i is a shorthand notation for (i 1 , ..., i l , ...)). Also let x 1 , ..., x N be variables, and let m H m (x) = x i i be the power sum polynomials. Theorem 4.46. Let N ⊂ p (where p is the number of parts of ∂). Then ν U (C i ) is the coefficient 6 6 If j > p, we define j to be zero. m∧1 60
Proof. The proof is obtained easily from the Mackey formula. Namely, ν U (C i ) is the number of elements x S n such that xgx −1 P (for a representative g C i ), divided by |P |. The order of P is ⎛ i ∂ i !, and the number of elements x such that xgx −1 P is the number of elements in P conjugate to g (i.e. |C i ∈ P |) times the order of the centralizer Z g of g (which is n!/|C i |). Thus, ν U (C i ) = ⎛ |Z g| j ∂j! |C i ∈ P |. Now, it is easy to see that the centralizer Z g of g is isomorphic to ⎛ m S i m ∼ (Z/mZ) im , so and we get Now, since P = ⎛ j S j , we have ⎛ ν U (C i ) = |Z g | = m im i m !, m m im i m ! |C i ∈ P |. m ⎛ j ∂ j! ∂ j ! |C i ∈ P | = ⎛ , m∧1 mr jm r jm ! where r = (r jm ) runs over all collections of nonnegative integers such that mrjm = ∂ j , r jm = i m . m r j∧1 Indeed, an element of C i that is in P would define an ordered partition of each ∂ j into parts (namely, cycle lengths), with m occuring r jm times, such that the total (over all j) number of times each part m occurs is i m . Thus we get ! ν U (C i ) = i ⎛ m r jm ! But this is exactly the coefficient of x in (x1 + ... + x ) im N m∧1 m (r jm is the number of times we take x j ). m r j m m j 4.15 The Frobenius character formula Let (x) = ⎛ 1→i
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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
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
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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
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Page 45 and 46: (a) Derive this classification. Hin
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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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Page 55 and 56: and the action g(f)(x) = f(xg) ⊕g
Page 57 and 58: 1.48). In particular, this understa
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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 (
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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
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Page 107 and 108: Now, in the general case, we prove
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