Lecture notes for Introduction to Representation Theory

More documents

Recommendations

Info

(where N ⊂ p). In this formula, there are many cancelations. After making some of these cancelations, we obtain the hook length formula. Namely, for a square (i, j) in a Young diagram ∂ (i, j ⊂ 1, i ∗ ∂ j ), define the hook of (i, j) to be the set of all squares (i , j ) in ∂ with i ⊂ i, j = j or i = i, j ⊂ j. Let h(i, j) be the length of the hook of i, j, i.e., the number of squares in it. Theorem 4.53. (The hook length formula) One has dim V = ⎛ i→ j n! h(i, j) . Proof. The formula follows from formula (5). Namely, note that ⎛ l 1 ! = (l 1 − l j ) 1
Lemma 4.56. Let k be a field of characteristic zero. (i) For any finite dimensional vector space U over k, the space S n U is spanned by elements of the form u ... u, u U. (ii) For any algebra A over k, the algebra S n A is generated by elements n (a), a A. Proof. (i) The space S n U is an irreducible representation of GL(U) (Problem 3.19). The subspace spanned by u ... u is a nonzero subrepresentation, so it must be everything. (ii) By the fundamental theorem on symmetric functions, there exists a polynomial P with rational coefficients such that P (H 1 (x), ..., H n (x)) = x 1 ...x n (where x = (x 1 , ..., x n )). Then The rest follows from (i). P ( n (a), n (a 2 ), ..., n (a n )) = a ... a. Now, the algebra A is semisimple by Maschke’s theorem, so the double centralizer theorem applies, and we get the following result, which goes under the name “Schur-Weyl duality”. Theorem 4.57. (i) The image A of C[S n ] and the image B of U(gl(V )) in End(V n ) are centralizers of each other. (ii) Both A and B are semisimple. In particular, V n is a semisimple gl(V )-module. (iii) We have a decomposition of A B-modules V n = V L , where the summation is taken over partitions of n, V are Specht modules for S n , and L are some distinct irreducible representations of gl(V ) or zero. 4.19 Schur-Weyl duality for GL(V ) The Schur-Weyl duality for the Lie algebra gl(V ) implies a similar statement for the group GL(V ). Proposition 4.58. The image of GL(V ) in End(V n ) spans B. Proof. Denote the span of g n , g GL(V ), by B . Let b End V be any element. We claim that B contains b n . Indeed, for all values of t but finitely many, t· Id+b is invertible, so (t · Id + b) n belongs to B . This implies that this is true for all t, in particular for t = 0, since (t · Id + b) n is a polynomial in t. The rest follows from Lemma 4.56. Corollary 4.59. As a representation of S n × GL(V ), V n decomposes as V L , where L = Hom Sn (V , V n ) are distinct irreducible representations of GL(V ) or zero. Example 4.60. If ∂ = (n) then L = S n V , and if ∂ = (1 n ) (n copies of 1) then L = √ n V . It was shown in Problem 3.19 that these representations are indeed irreducible (except that √ n V is zero if n > dim V ). 65
Page 1 and 2:
Introduction to representation theo
Page 3 and 4:
4.15 The Frobenius character formul
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
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: Corollary 4.49 easily implies that
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 and 108: Now, in the general case, we prove
Page 109: MIT OpenCourseWare http://ocw.mit.e
show all

Lecture notes for Introduction to Representation Theory

Create successful ePaper yourself

Delete template?

Save as template?