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 a finite basis (ei)ni=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): Xa(ej) = aijei. iRemark 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 ∈ G. In particular, we have <ul style="display: flex;"><li style="flex:1">ρ(1) = 1, , ρ(s−1) = ρ(s)−1 </li><li style="flex:1">.</li></ul>A map ϕ between two representations V and W of G is a vector space map ϕ : V → W such that ϕV −−−−→ W y y<ul style="display: flex;"><li style="flex:1">g</li><li style="flex:1">g</li></ul>(1.1) V −−−−→ W ϕcommutes for all g ∈ 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 ϕ, Im ϕ, and Coker ϕ, 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 21.2. 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, |ρ(g)| = 1. Degree 1 representations are called characters. If we take ρ(g) = 1 for all g ∈ G, we obtain a representation of G which is called the trivial representation or the unit representation. 1.2.2. Regular representation. Let |G| be the order of G, and let V be a vector space of dimension |G| with basis (eg)g∈G indexed by the elements g of G. For g ∈ 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 ∈ 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 ∈ G, there is a permutation x → gx of X, satisfying the identities 1x = x; g(hx) = (gh)x if g, h ∈ G, x ∈ X. Let V be a vector space having a basis (ex)x∈X indexed by the elements of X. For g ∈ 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 W0 be two subspaces of V . The space V is said to be the direct sum of W and W0 if each x ∈ V can be written uniquely in the form x = w + w0, with w ∈ W and w0 ∈ W0. This amounts to saying that the intersection W ∩ W0 = 0 and that dim V = dim W + dim W0. We then write V = W ⊕ W0 and say that W0 is a complement of W in V . The mapping p which sends each x ∈ V to its component of w ∈ W is called the projection of V onto W associated with the decomposition V = W ⊕ W0. The image of p is W and p(x) = x for x ∈ 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 W0 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 Psubspace of dimension 1 of V generated by the element x = g∈G eg. We have ρgx = x for all g ∈ 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 W0 of W in V which stable under G. 3Proof. 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) = a¯b(w, z) Let H0 be any Hermitian product on V , we define X<ul style="display: flex;"><li style="flex:1">H(x, y) = </li><li style="flex:1">H0(gx, gy). </li></ul>g∈G It is easy to check that H is also a Hermitian product on V and H(gx, gy) = H(x, y). Let W0 be the orthogonal complement of W in V . Then W0 is stable under G and we have our theorem. (2) Let W0 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: X1p0 = <ul style="display: flex;"><li style="flex:1">ρg ◦ p ◦ ρ−g 1 </li><li style="flex:1">.</li></ul>|G| g∈G Since p maps V into W and ρg preserves W we see that p0 maps V into W. Note that ρ−g 1x ∈ W for x ∈ W, so p0x = x. Thus p0 is a projection of V onto W, corresponding to some complement W0 of W. It is easy to check that ρg ◦ p0 ◦ ρ−g 1 = p0. Thus ρg ◦ p0 = p0 ◦ ρg. If x ∈ W0 and g ∈ G, we have p0x = 0, so p0gx = xp0x = 0. That shows that W0 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 decomposition 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. 41.4. Schur’s Lemma. Theorem 1.8. If V and W are irreducible representations of G and ϕ : V → W is a G-module homomorphism, then (1) Either ϕ is an isomorphism or ϕ = 0. (2) If V = W, then ϕ = λ · I for some λ ∈ C, I the identity map. Proof. The first claim follows from the the fact that Ker ϕ and Im ϕ are invariant subspaces. For the second, since C is an algebraic closed field, ϕ must have an eigenvalue λ, i.e., for some λ ∈ C, ϕ − λI has a nonzero kernel. By (1), we must have ϕ − λI = 0 and the claim follows. ꢀProposition 1.9. For any representation V of a finite group G, there is a decomposition V = V1a ⊕ · · · ⊕ Vka 1k,where the Vi are distinct irreducible representations. The decomposition of V into a direct sum of the k factors is unique, as are the Vi that occur and their multiplicities ai. Proof. It follows from Schur’s lemma that if W is another representation of G, with a decomposition W = ⊕jWb and ϕ : V → W is a map of representations, then ϕ must map jj<ul style="display: flex;"><li style="flex:1">the factors Via into the factors Wjb for which Vi </li><li style="flex:1">W . If we apply this to the identity </li></ul>jji∼=map of V to V , the stated uniqueness follows. ꢀ1.5. Construct new representations from old ones. 1.5.1. General construction. If V and W are representations, the direct sum V ⊕ W and the tensor product are also representations, via g(v ⊕ w) = gv ⊕ gw, g(v ⊗ w) = gv ⊗ gw. For a representation V , the nth tensor power V ⊗n is again a representation of G by this Vrule, and the exterior power n(V ) and symmetric power Symmn(V ) are subrepresentations of it. The dual V ∗ = Hom(V, C) of V is also a representation of G via ρ∗(g) = tρ(g−1) : V ∗ → V ∗. Note under this definition, we have hρ∗(g)(v∗), ρ(g)(v)i = hv∗, vi, where h, i is the natural pairing between V and V ∗. 1.5.2. Symmetric square and alternation square. Suppose that the representations V1 and V2 are identical to the same representation V , so that V1 ⊗ V2 = V ⊗ V . If (ei) is a basis of V , let θ be the automorphism of V ⊗ V such that θ(ei ⊗ ej) = ej ⊗ ei, for all (i, j). It follows that θ(x ⊗ y) = y ⊗ x for all x, y ∈ V , and θ is independent of the chosen basis (ei). Moreover, θ2 = 1. The space V ⊗ V then decomposes into a direct sum 2^V ⊗ V = Symm2(V ) ⊕ (V ), 5Vwhere Symm2(V ) is the set of elements z ∈ V ⊗ V such that θ(z) = z and 2(V ) is the set of elements of z ∈ V ⊗ V such that θ(z) = −z. The elements (ei ⊗ ej + ej ⊗ ei)i≤j for a Vbasis of Symm2(V ), and the elements (ei ⊗ ej − ej ⊗ ei)i<j for a basis of 2(V ). We have 2^dim Symm2(V ) = n(n + 1)/2 dim (V ) = n(n − 1)/2 if dim(V ) = n. 1.6. Further remarks. 1.6.1. Complete reducibility. We saw that any representation of a finite group is a direct sum of irreducible representations. This property is called complete reducibility, or semisimplicity. We may show that, for continuous representations, any compact group has this property. Integration over the group with respect to an invariant measure on the group plays the role of averaging in the proof of the theorem. The additive group R, on the other hand, does not have this property. Let V = R2, the representation of <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>1 a 0 1 a → leaves the x axis fixed, but there is no complementary subspace. The complete reducibility may also fail if the vector space V is over a field of finite characteristic. 1.6.2. Representations of abelian groups. If V is a representation of the finite group G, abelian or not, each g ∈ G gives a map ρ(g) : V → V . But this map in general is not a G-module homomorphism: for general h ∈ G, we will have g(h(v)) = h(g(v)). Indeed, ρ(g) : V → V will be a G-linear for every ρ if and only if g is in the center Z(G) of G. In particular, if G is abelian, and V is an irreducible representation, then by Schur’s lemma every element g ∈ G acts on V by a scalar multiple of the identity. Every subspace of V is thus invariant, so that V must be one dimensional. The irreducible representations of an abelian group G are thus simply elements of the dual group, that is, homomorphisms ρ : G → C×. 1.6.3. Representations of S3. We consider the simplest nonabelian group G = S3. To begin with, we have two one-dimensional representations: the trivial representation, which we will denote U, and the alternating representation U0, defined by gv = sgn(g)v, for g ∈ G, v ∈ V . Next, since G comes to us as a permutation group, we have a natural permutation representation, in which G acts on C3 by permuting the coordinates. Explicitly, if (e1, e2, e3) is the standard basis, then g(ei) = eg(i), or equivalently, g(z1, z2, z2) = (zg−1(1), zg−1(2), zg−1(3)). This representation is not irreducible. The line spanned by the sum (1, 1, 1) of the basis vectors is invariant, with complementary subspace V = {(z1, z2, z3) ∈ C3 : z1 + z2 + z3 = 0}. 6This two-dimensional representation V is easily seen to be irreducible, we call it the standard representation of S3. Now we describing an arbitrary representation of S3. (Later we will develop the so called character theory, which is a wonderful tool for describing arbitrary representation of a group.) We have seen that the representation theory of a finite abelian group is virtually trivial, we start our analysis of an arbitrary representation W of S3 by looking at the action of the abelian subgroup U3 = Z/3 ⊂ S3 on W. Let τ be any generator of U3, the space W is spanned by eigenvectors vi for the action of τ, whose eigenvalues are of course all powers of a cube root of unity ω = e2πi/3. Thus W = ⊕Vi where Vi = Cvi and τvi = ωαi vi. Next, we ask how the remaining elements of S3 act on W in terms of this decomposition. Let σ be any transposition, so that τ and σ together generate S3, with the relation στσ = τ2. Let v be an eigenvector of τ with eigenvalue ωi, we have τ(σ(v)) = σ(τ2(v)) = σ(ω2i · v) = ω2i · σ(v). Therefore, σ(v) is again an eigenvector for τ with eigenvalue ω2i. Lemma 1.10. Let σ = (12), τ = (123). Then the standard representation of S3 has a basis α = (ω, 1, ω2), β = (1, ω, ω2), with τα = ωα, τβ = ω2β, σα = β, σβ = α. Suppose we start with an eigenvector v for τ. If the eigenvalue of v is ωi = 1, then σ(v) is an eigenvector with eigenvalue ω2i = ωi, and so is independent of v. Now v and σ(v) together span a two-dimensional subspace V 0 of W invariant under S3. In fact, V 0 is isomorphic to the standard representation, which follows from the lemma. If, on the other hand, the eigenvalue of v is 1, then σ(v) may or may not be independent of v. If it is not, then v spans a one-dimensional subrepresentation of W, isomorphic to the trivial representation if σ(v) = v and to the alternating representation if σ(v) = −v. If σ(v) and v are independent, then v + σ(v) and v − σ(v) span one-dimensional representations of W isomorphic to the trivial representation and alternating representation respectively. Thus, we have the following proposition. Proposition 1.11. For an arbitrary representation W of S3, we can write W = U⊕a ⊕ (U0)⊕b ⊕ V ⊕c Moreover, c is the number of independent eigenvectors for τ with eigenvalue ω, a+c is the .multiplicity of 1 as an eigenvalue of σ, and b+c is the multoplicity of −1 as an eigenvalue of σ. 2. Character theory 2.1. The character of a representation. Let V be a vector space of dimension n having a basis (ei). Let A be a linear map of V into itself with matrix (aij)n×n. By the trace of A we mean the scalar

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