Lecture notes for Introduction to Representation Theory

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Example 6.15. The category of modules over an algebra A and the category of finite dimensional modules over A are abelian categories. Remark 6.16. The good thing about Definition 6.14 is that it allows us to visualize objects, morphisms, kernels, and cokernels in terms of classical algebra. But the definition also has a big drawback, which is that even if C is the whole category A-mod, the ring A is not determined by C. In particular, two different rings can have equivalent categories of modules (such rings are called Morita equivalent). Actually, it is worse than that: for many important abelian categories there is no natural (or even manageable) ring A at all. This is why people prefer to use the standard definition, which is free from this drawback, even though it is more abstract. We say that an abelian category C is k-linear if the groups Hom C (X, Y ) are equipped with a structure of a vector space over k, and composition maps are k-linear in each argument. In particular, the categories in Example 6.15 are k-linear. 6.8 Exact functors Definition 6.17. A sequence of objects and morphisms X 0 ⊃ X 1 ⊃ ... ⊃ X n+1 in an abelian category is said to be a complex if the composition of any two consecutive arrows is zero. The cohomology of this complex is H i = Ker (d i )/Im(d i−1 ), where d i : X i ⊃ X i+1 (thus the cohomology is defined for 1 ∗ i ∗ n). The complex is said to be exact in the i-th term if H i = 0, and is said to be an exact sequence if it is exact in all terms. A short exact sequence is an exact sequence of the form 0 ⊃ X ⊃ Y ⊃ Z ⊃ 0. Clearly, 0 ⊃ X ⊃ Y ⊃ Z ⊃ 0 is a short exact sequence if and only if X ⊃ Y is injective, Y ⊃ Z is surjective, and the induced map Y/X ⊃ Z is an isomorphism. Definition 6.18. A functor F between two abelian categories is additive if it induces homomorphisms on Hom groups. Also, for k-linear categories one says that F is k-linear if it induces k-linear maps between Hom spaces. It is easy to show that if F is an additive functor, then F (X Y ) is canonically isomorphic to F (X) F (Y ). Example 6.19. The functors Ind G K , ResG K , Hom G(V, ?) in the theory of group representations over a field k are additive and k-linear. Definition 6.20. An additive functor F : C ⊃ D between abelian categories is left exact if for any exact sequence 0 ⊃ X ⊃ Y ⊃ Z, the sequence 0 ⊃ F (X) ⊃ F (Y ) ⊃ F (Z) is exact. F is right exact if for any exact sequence the sequence X ⊃ Y ⊃ Z ⊃ 0, F (X) ⊃ F (Y ) ⊃ F (Z) ⊃ 0 is exact. F is exact if it is both left and right exact. 104
Definition 6.21. An abelian category C is semisimple if any short exact sequence in this category splits, i.e., is isomorphic to a sequence (where the maps are obvious). 0 ⊃ X ⊃ X Y ⊃ Y ⊃ 0 Example 6.22. The category of representations of a finite group G over a field of characteristic not dividing | G | (or 0) is semisimple. Note that in a semisimple category, any additive functor is automatically exact on both sides. Example 6.23. (i) The functors Ind G K , ResG are exact. K (ii) The functor Hom(X, ?) is left exact, but not necessarily right exact. To see that it need not be right exact, it suffices to consider the exact sequence and apply the functor Hom(Z/2Z, ?). 0 ⊃ Z ⊃ Z ⊃ Z/2Z ⊃ 0, (iii) The functor X A for a right A-module X (on the category of left A-modules) is right exact, but not necessarily left exact. To see this, it suffices to tensor multiply the above exact sequence by Z/2Z. Exercise. Show that if (F, G) is a pair of adjoint additive functors between abelian categories, then F is right exact and G is left exact. Exercise. (a) Let Q be a quiver and i Q a source. Let V be a representation of Q, and W a representation of Q i (the quiver obtained from Q by reversing arrows at the vertex i). Prove that there is a natural isomorphism between Hom ⎩ F − i V, W and Hom ⎩ + V, F W i . In other words, the functor F + − i is right adjoint to F i . (b) Deduce that the functor F + is left exact, and F i − is right exact. i 105
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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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Page 55 and 56: and the action g(f)(x) = f(xg) ⊕g
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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
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Page 77 and 78: Proof. (i) Let us decompose V = V (
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Page 85 and 86: So - considering the first summand
Page 87 and 88: 5.4 Roots From now on, let be a fi
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Page 93 and 94: Proposition 5.31. Let Q be a quiver
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Page 107 and 108: Now, in the general case, we prove
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