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REPRESENTATIONS of FINITE GROUPS 1. Definition And REPRESENTATIONS OF FINITE GROUPS 1. Definition and Introduction 1.1. Definitions. Let V be a vector space over the field C of complex numbers and let GL(V ) be the group of isomorphisms of V onto itself. An element a of GL(V ) is a linear mapping of V into V which has a linear inverse a−1. When V has dimension n and has n a finite basis (ei)i=1, each map a : V ! V is defined by a square matrix (aij) of order n. The coefficients aij are complex numbers. They are obtained by expressing the image a(ej) in terms of the basis (ei): X a(ej) = aijei: i Remark 1.1. Saying that a is an isomorphism is equivalent to saying that the determinant det(a) = det(aij) of a is not zero. The group GL(V ) is thus identified with the group of invertible square matrices of order n. Suppose G is a finite group with identity element 1. A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism ρ : G ! GL(V ) of G to the group of automorphisms of V . We say such a map gives V the structure of a G-module. The dimension V is called the degree of ρ. Remark 1.2. When there is little ambiguity of the map ρ, we sometimes call V itself a representation of G. In this vein we often write gv_ or gv for ρ(g)v. Remark 1.3. Since ρ is a homomorphism, we have equality ρ(st) = ρ(s)ρ(t) for s; t 2 G: In particular, we have ρ(1) = 1; ; ρ(s−1) = ρ(s)−1: A map ' between two representations V and W of G is a vector space map ' : V ! W such that ' V −−−−! W ? ? g? ?g (1.1) y y V −−−−! W ' commutes for all g 2 G. We will call this a G-linear map when we want to distinguish it from an arbitrary linear map between the vector spaces V and W . We can then define Ker ', Im ', and Coker ', which are also G-modules. (Check this.) ∗These are some notes I wrote for the students in the class Representations of Finite Groups SS2013. The materials are from the following two books. J. P. Serre Linear representations of finite groups GTM42 W. Fulton; J. Haris Representation theory Part 1, GTM129 1 2 1.2. Basic examples. 1.2.1. Characters. A representation of degree 1 of a group G is a homomorphism ρ : × × G ! C , where C denotes the multiplicative group of nonzero complex numbers. Since each element of G has finite order, the values ρ(g) of ρ are roots of unity. In particular, jρ(g)j = 1. Degree 1 representations are called characters. If we take ρ(g) = 1 for all g 2 G, we obtain a representation of G which is called the trivial representation or the unit representation. 1.2.2. Regular representation. Let jGj be the order of G, and let V be a vector space of dimension jGj with basis (eg)g2G indexed by the elements g of G. For g 2 G, let ρ(g) be the linear map of V into V which sends eg to egh. This defines a linear representation, which is called the regular representation of G. Its degree is equal to the order of G. Note that eg = ρg(e1). Hence the images of e1 form a basis of V . Conversely, let W be a representation of G containing a vector w such that ρg(w) for g 2 G form a basis of W , then W is isomorphic to the regular representation. Define τ : V ! W by τ(eg) = ρg(w), then τ gives us such an isomorphism. 1.2.3. Permutation representations. Suppose that G acts on a finite set X. This means that, for each g 2 G, there is a permutation x 7! gx of X, satisfying the identities 1x = x; g(hx) = (gh)x if g; h 2 G; x 2 X: Let V be a vector space having a basis (ex)x2X indexed by the elements of X. For g 2 G, let ρg be the linear map of V into V which sends ex to egx. The linear representation of G thus obtained is called the permutation representation associated with X. 1.3. Subrepresentations and irreducible representations. 1.3.1. Some linear algebra. Let V be a vector space, and let W and W 0 be two subspaces of V . The space V is said to be the direct sum of W and W 0 if each x 2 V can be written uniquely in the form x = w + w0, with w 2 W and w0 2 W 0. This amounts to saying that the intersection W \ W 0 = 0 and that dim V = dim W + dim W 0. We then write V = W ⊕ W 0 and say that W 0 is a complement of W in V . The mapping p which sends each x 2 V to its component of w 2 W is called the projection of V onto W associated with the decomposition V = W ⊕ W 0. The image of p is W and p(x) = x for x 2 W . Conversely, if p is a linear map of V into itself satisfying these two properties, one checks that V is the direct sum of W and the kernel W 0 of p. 1.3.2. Subrepresentations. A subrepresentation of a representation V is a vector subspace W of V which is invariant under G. A representation V is called irreducible if there is no proper nonzero invariant subspace W of V . Example 1.4. Take V for example the regular representation of G, and let W be the P subspace of dimension 1 of V generated by the element x = g2G eg. We have ρgx = x for all g 2 G. Therefore, W is a subrepresentation of V , isomorphic to the unit representation. Later we will determine all the subrepresentations of the regular representation. Theorem 1.5. Let ρ : G ! GL(V ) be a linear representation of G in V and let W be a vector subspace of V stable under G. Then there exists a complement W 0 of W in V which stable under G. 3 Proof. We give two proofs of this theorem. (1) Recall a Hermitian product on V is a map ( ; ): V × V ! C such that • (w; z) = (z; w) • (w + w0; z) = (w; z) + (w0; z) • (aw; bz) =ab ¯ (w; z) Let H0 be any Hermitian product on V , we define X H(x; y) = H0(gx; gy): g2G It is easy to check that H is also a Hermitian product on V and H(gx; gy) = H(x; y). Let W 0 be the orthogonal complement of W in V . Then W 0 is stable under G and we have our theorem. (2) Let W 0 be any complement of W in V and let p be the corresponding projection of V onto W . Form the average p0 of the conjugates of p by the elements of G: 1 X p0 = ρ ◦ p ◦ ρ−1: jGj g g g2G 0 Since p maps V into W and ρg preserves W we see that p maps V into W . Note that −1 0 0 ρg x 2 W for x 2 W , so p x = x. Thus p is a projection of V onto W , corresponding to some complement W 0 of W . It is easy to check that 0 −1 0 ρg ◦ p ◦ ρg = p : 0 0 0 0 0 0 Thus ρg ◦ p = p ◦ ρg. If x 2 W and g 2 G, we have p x = 0, so p gx = xp x = 0. That 0 shows that W is stable under G and the theorem follows. 1.3.3. Irreducible representations. Let V be a representation of G. By definition, V is irreducible if V is not the direct sum of two representations except for the trivial decom- position V = 0 ⊕ V . A representation of degree 1 is evidently irreducible. Theorem 1.6. Every representation is a direct sum of irreducible representations. Proof. Let V be a linear representation of G. We proceed by induction on dim(V ). If dim(V ) = 0, there is nothing to prove. Suppose that dim(V ) ≥ 1. If V is irreducible, there is nothing to prove. Otherwise, because of last theorem, V can be decomposed into a direct sum V 0 ⊕ V 00 with dim(V 0) < dim(V ) and dim(V 00) < dim(V ). By the induction hypothesis, V 0 and V 00 are direct sums of irreducible representations, and so the same is true for V . Remark 1.7. Let V be a representation of G, and let V = W1 ⊕· · ·⊕Wk be a decomposition of V into a direct sum of irreducible representations. We can ask if this decomposition is unique. The case where all the ρg are equal to 1 shows that this is not true in general. Nevertheless, we will show later that the number of Wi isomorphic to a given irreducible representation does not depend on the chosen decomposition. 4 1.4. Schur's Lemma. Theorem 1.8. If V and W are irreducible representations of G and ' : V ! W is a G-module homomorphism, then (1) Either ' is an isomorphism or ' = 0. (2) If V = W , then ' = λ · I for some λ 2 C, I the identity map. Proof. The first claim follows from the the fact that Ker ' and Im ' are invariant sub- spaces.
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