Lecture notes for Introduction to Representation Theory

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4.20 Schur polynomials Let ∂ = (∂ 1 , ..., ∂ p ) be a partition of n, and N ⊂ p. Let Define the polynomials N j +N−j j +N−j s(j) i sS N j=1 D (x) = (−1) s x = det(x ). D (x) S (x) := D0 (x) (clearly D 0 (x) is just (x)). It is easy to see that these are indeed polynomials, as D is antisymmetric and therefore must be divisible by . The polynomials S are called the Schur polynomials. Proposition 4.61. m m (x1 + ... + x ) im = ν N (C i )S (x). m :p→N Proof. The identity follows from the Frobenius character formula and the antisymmetry of m m (x) (x 1 + ... + x N ) im . m Certain special values of Schur polynomials are of importance. Namely, we have Proposition 4.62. Therefore, S (1, z, z 2 , ..., z N−1 ) = S (1, ..., 1) = 1→i
Theorem 4.63. (Weyl character formula) The representation L is zero if and only if N < p, where p is the number of parts of ∂. If N ⊂ p, the character of L is the Schur polynomial S (x). Therefore, the dimension of L is given by the formula dim L = 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
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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
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
Page 55 and 56: 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: Lemma 4.56. Let k be a field of cha
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 and 108: Now, in the general case, we prove
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Lecture notes for Introduction to Representation Theory

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