A Note on Complex Representations of GL(2, Fq

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A Note on Complex Representations of GL(2, F

q

) Hua-Chieh Li

Department of Mathematics, National Tsing Hua University, Hsin Chu, Taiwan, R.O.C.

E-mail address: li@math.nthu.edu.tw

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General Properties on Linear Representations of Finite Groups

All groups we consider in this chapter are ﬁnite group.

1.1. Basic Definitions

Let V be a ﬁnite dimensional vector space over C and let Aut(V ) be the group of automorphisms of V onto itself. A linear representation of a ﬁnite group G on V is a homomorphism ρ : G→ Aut(V ) from G to the group Aut(V ). In this way we have the equalities

ρ(s· t) = ρ(s) ◦ ρ(t) ∀s, t ∈ G, ρ(1) = 1 and ρ(s⁻¹) = ρ(s)⁻¹. We will also frequently write ρs instead of ρ(s).

When V is given, we say that V is a representation space of G, denoted Vρ and also say that G acts on V through ρ. The dimension of V is called the dimension of ρ, denoted dim(ρ). If we have ρ(s) equals to the identity map for all s∈ G, the representation is called the trivial representation.

Example 1.1.1. Let g be the order of G and let V be the vector space of dimension g with a basis (vt)t∈G indexed by the elements t of G. For s, t∈ G, let ρs be the linear map of V into V such that ρs(vt) = vst; this deﬁnes a linear representation, which is called the regular representation of G. Note that if e is the identity of G, the orbit of v_e form a basis of V .

Let ρ and ρ be two representations of the same group G in V and V, respectively. These representations are said to be isomorphic if there exists a linear isomorphism τ : V → V such that τ◦ ρ(s) = ρ(s)◦ τ for all s ∈ G. We shall usually identify isomorphic representations.

1.2. Subrepresentations and Irreducible Representations

Let ρ : G→ Aut(V ) be a linear representation and let W be a subspace of V . Suppose that w ∈ W implies ρ_s(w)∈ W for all s ∈ G. The restriction ρs|W of ρ_s to W is then an automorphism of W and we have ρ_st|W = ρ_s|W ◦ ρt|W. Thus W is stable under the action of G and ρ|W : G → Aut(W ) is a linear representation of G in W ; W is said to be a subrepresentation of V .

There are some important subrepresentations. Let ρ and ρ be representations of G into V and W respectively. A G-linear map from V to W is a linear map φ : V → W such that φ(ρs(v)) = ρ_s(φ(v))

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for all s ∈ G and v ∈ V . We denote the space of all G-linear maps from V to W by HomG(V, W ).

It is easy to check that for a given φ ∈ HomG(V, W ), the space Ker(φ) = {v ∈ V | φ(v) = 0} gives a subrepresentation of G in V and the space Im(φ) = {w ∈ W | w = φ(v) for some v ∈ V } gives a subrepresentation of G in W .

A representation of G in V is called irreducible if there is no proper nonzero subrepresentation of V .

Lemma 1.2.1. Let ρ : G → Aut(V ) and ρ: G→ Aut(W ) be two representations of G. Suppose that φ∈ HomG(V, W ) is not the zero map. Then we have the following:

(1) If V is an irreducible representation of G, then φ is injective.

(2) If W is an irreducible representation of G, then φ is surjective.

In particular, if both V and W are irreducible representations of G, then V and W are isomorphic.

Proof. Since φ is not zero, we have Ker(φ) = V and Im(φ) = {0}. Therefore, V is irreducible implies

Ker(φ) ={0} and W is irreducible implies Im(φ) = W .

Corollary 1.2.2. Let V and W be two representations of G where V is irreducible and let φ1, φ₂ ∈ Hom_G(V, W ). Suppose that there exist v= 0 in V such that φ1(v) = φ₂(v). Then φ₁ = φ₂.

Proof. The assumption says that φ1− φ2 is not injective. Since φ1 − φ2 ∈ HomG(V, W ), it implies

that φ₁− φ2 is the zero mapping by Lemma 1.2.1.

1.3. Schur’s Lemma and Its Applications

For each n× n matrix A, since it is over C which is algebraically closed, there exist eigenvalues of A.

By this, we can derive that there exists a unitary matrix U (i.e. U^T · U = I) such that U^T · A · U is a upper triangular matrix. This is what called Schur’s Theorem in Linear Algebra [1, Section 6.5].

Here, by using similar argument, we have the following:

Proposition 1.3.1 (Schur’s Lemma). Let ρ : G → Aut(V ) be an irreducible representation of G and let f be a linear mapping of V into V such that ρs◦ f = f ◦ ρs for all s∈ G. Then f is a homothety (i.e. f = λI for some λ∈ C where I is the identity map of V ).

Proof. Because f is an endomorphism of V , there exists an eigenvalue λ with eigenvector v ∈ V .

Thus f (v) = λI(v). By Corollary 1.2.2, f is equal to λI.

Let G be a ﬁnite abelian group and let ρ : G → Aut(V ) be a representation of G. It is easy to show that for every s∈ G, ρs is a G-linear mapping of V into V . Hence by Schur’s Lemma, we have the following:

Corollary 1.3.2. Let G be a ﬁnite abelian group and let ρ : G → Aut(V ) be an irreducible represen- tation of G. Then we have that dim(V ) = 1.

We will see latter that there are many applications for Schur’s Lemma. Here we give some impor- tant ones which are very useful for developing character theory.

Corollary 1.3.3. Let ρ : G → Aut(V ) and ρ : G→ Aut(W ) be two irreducible representations of G and let g be the order of G. Let h be a linear mapping of V into W (note: h may not be a G-linear mapping). Put

h⁰= 1 g

t∈G

(ρ_t)⁻¹◦ h ◦ ρt. Then:

(1) If ρ and ρ are not isomorphic, then we have h⁰ = 0.

(2) If V = W and ρ = ρ, then h⁰ is a homothety of ratio (1/n)Tr(h), where n = dim(V ).

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1.5. Complete Reducibility 7

Proof. We have ρsh⁰ = h⁰ρs for all s∈ G. Applying Lemma 1.2.1 and Schur’s Lemma with f = h⁰, we see in case (1) that h⁰ = 0 and in case (2) that h⁰ = λI for some λ ∈ C. For the value of λ, we have nλ = Tr(λI) = (1/g)

t∈GTr((ρt)⁻¹hρt) = Tr(h).

Now we rewrite Corollary 1.3.3 in matrix form. Suppose that dim(W ) = m and the linear mapping h is defined by an m× n matrix (hkj) and likewise h⁰ is defined by (h⁰_kj). Assume ρ and ρ are given in matrix form ρ_t= (r_ij(t)), 1≤ i, j ≤ n and ρt = (r_kl(t)), 1≤ k, l ≤ m respectively. We have by the definition of h⁰:

h⁰_kj = 1 g

t∈G,1≤l≤m,1≤i≤n

r_kl(t⁻¹)· hli· rij(t).

Since h is any linear mapping, we choose h with matrix form E_li, the matrix which is 1 in the (l, i)- place and 0 everywhere else. Notice that Tr(Eli) = δli (δij denotes the Kronecker symbol, equal to 1 if i = j and 0 otherwise). Whence:

Corollary 1.3.4. Keeping the hypothesis and notation of Corollary 1.3.3, we have:

(1) If ρ and ρ are not isomorphic, then 1

g

t∈G

r_kl (t⁻¹)rij(t) = 0, ∀ 1 ≤ k, l ≤ m, 1 ≤ i, j ≤ n.

(2) If V = W and ρ = ρ, then 1 g

t∈G

r_kl(t⁻¹)r_ij(t) =

1/n if i=l and k=j, 0 otherwise.

1.4. Direct Sum and Tensor Product

There are many ways to construct new representations from old ones. Here we introduce direct sum and tensor product. Let ρ : G → Aut(V ) and ρ : G → Aut(W ) be linear representations of G in V and W , respectively. Define a linear representation ρ⊕ ρ of G in V₁ ⊕ V2 by setting (ρ⊕ ρ)_s(v⊕ w) = ρs(v)⊕ ρ_s(w), for all s ∈ G, v ∈ V and w ∈ W . ρ ⊕ ρ is called direct sum representation of the given ρ and ρ. The direct sum of an arbitrary finite number of representations is defined similarly.

The tensor product representation ρ⊗ ρ of G in V ⊗ W of the given representations ρ of G in V and ρ in W is defined by the condition (ρ⊗ ρ)s(v⊗ w) = ρs(v)⊗ ρs(w), for all s ∈ G, v ∈ V and w∈ W . The tensor product of an arbitrary finite number of representations is defined similarly.

We can easily see that

dim(ρ⊕ ρ) = dim(ρ) + dim(ρ) and dim(ρ⊗ ρ) = dim(ρ)· dim(ρ) . 1.5. Complete Reducibility

As in any study, before we begin our attempt to classify the representations of a ﬁnite group in earnest we should try to simplify life by restricting our search somewhat. The key to all this is

Proposition 1.5.1. Let ρ be a linear representation of G in V and let W be a subrepresentation of G in V . Then there exists a complement W⁰ of W in V which is stable under G.

Proof. Choose W an arbitrary complement of W in V , and let p : V → W be the corresponding projection of V onto W (i.e. writing v ∈ V uniquely as v = w + w with w ∈ W and w ∈ W, p(v) = w). Deﬁne

p⁰= 1 g

t∈G

(ρ_t)⁻¹◦ p ◦ ρt,

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where g is the order of G. Since p maps V into W and ρt preserves W for all t ∈ G, we see that p⁰ maps V into W . Furthermore, because p(w) = w and ρ⁻¹_t (w) = ρ_t−1(w) ∈ W for all w ∈ W , it implies that p⁰(w) = w for all w ∈ W . Thus p⁰ is a projection of V onto W , corresponding to some complement W⁰ = Ker(p⁰) of W . We have moreover ρ_s ◦ p⁰ = p⁰ ◦ ρs for all s ∈ G. Hence p⁰◦ ρs(w⁰) = ρ_s◦ p⁰(w⁰) = 0 for w⁰ ∈ W⁰ and s∈ G, which shows that W⁰ is stable under G and

complete the proof.

This proposition says that for any subrepresentation W of G in V , there exists another subrep- resentation W⁰ of G in V such that V = W ⊕ W⁰ is a direct sum representation of W and W⁰. Therefore, an irreducible representation is equivalent to saying that it is not the direct sum of two representations. We have the following complete reducibility property.

Theorem 1.5.2. Every representation is a direct sum of irreducible representations.

Proof. We proceed by induction on the dimension of representation. If the representation is irre- ducible, there is nothing to prove. Otherwise, because of Proposition 1.5.1, it can be decomposed into a direct sum of subrepresentations with smaller dimensions. By the induction hypothesis, these subrepresentations are direct sum of irreducible representations and so is our original representation. Remark . This property is not always true for representations of infinite group or over a field other thanC. For example, the additive group R does not have this property. Note also that the argument of Proposition 1.5.1 would fail if the vector space was over a field of finite characteristic.

We can ask if this decomposition of V is unique. The case where all the ρs are equal to identity shows that this is not true in general (in this case the irreducible representations are lines, and we have an inﬁnity of ways to decompose a vector space into a direct sum of lines). Nevertheless, we have a decomposition of V which is “coarser” than the decomposition into irreducible representations, but which has the advantage of being unique. It is obtained as follows. First decompose V into direct sum of irreducible representations V = W1⊕ · · · ⊕ Wk and then collect together the isomorphic representations. A representation is said to be isotypic if it is a direct sum of isomorphic irreducible representation. Thus, we have V = V₁⊕ · · · ⊕ Vh where every V_i is isotypic. This will be the canonical decomposition we have in mind.

There is another concept for the proof of Proposition 1.5.1 which is very useful.

Let T be a linear mapping of V into V , where V is endowed with an inner product , . Suppose that

T with respect to an orthonormal basis of V . Then U is unitary (i.e. U^T · U = U · U^T = I). We say that an n× n matrix A is normal if A^T · A = A · A^T (so a unitary matrix is normal). Using Schur’s theorem we can prove the spectral theorem which says that if A is normal, then there exists a unitary matrix U such that U^T · A · U is a diagonal matrix. This amounts to saying that A is normal if and only if A possesses a orthonormal basis which are eigenvectors.

Let ρ : G→ Aut(V ) be a linear representation where V is endowed with an inner product , .

Consider the product

t∈G ρt(u), ρ_t(v) property ρs(u), ρs(v)

V such that the matrix form of ρs with respect to this basis is a unitary matrix for every s∈ G. Now, if W is a subrepresentation of G in V , then with respect to the inner product

complement W^⊥ of W in V is stable under G; another proof of Proposition 1.5.1 is thus obtained.

1.6. Characters for Representations

Let ρ : G → Aut(V ) be a linear representation of G in V . Since the trace of the linear mapping ρs

does not depend on the choice of basis of V , we put:

χρ(s) = Tr(ρs) for each s∈ G.

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1.7. Orthogonality Relations for Characters 9

The complex valued function χρ on G thus obtained is called the character of the representation ρ.

We remark that if two representations ρ and ρ are isomorphic, then χρ= χ_ρ.

Suppose that dim(ρ) = n. We have Tr(I) = n, and so χ_ρ(e) = n where e is the identity of G.

Recall that from 1.5, the matrix form of ρsis normal, and hence diagonalizable. Thus for s∈ G, a basis (v1, . . . , vn) of V can be chosen such that ρs(vi) = λiviwith λi ∈ C^∗, and so χρ(s) =_n

i=1λi. Also note that s∈ G has ﬁnite order, the values λi are roots of unity; in particular we have λi = λ⁻¹_i . Because ρ_s−1 = ρ⁻¹_s , we have χρ(s⁻¹) =_n

i=1λ⁻¹_i =_n

i=1λi and Tr(ρ_tst−1) = Tr(ρt◦ ρs◦ ρ⁻¹t ) = Tr(ρs).

We can summarize what we have shown so far in

Proposition 1.6.1. If χρ is the character of a dimension n representation ρ : G→ Aut(V ) of G in V , we have:

(1) χρ(e) = n.

(2) χ_ρ(s⁻¹) = χ_ρ(s) for s∈ G.

(3) χρ(tst⁻¹) = χρ(s) for s, t∈ G.

Proposition 1.6.2. Let ρ and ρ be two linear representations of G in V and W , and let χ_ρ and χ_ρ

be their characters, respectively. Then:

(1) The character of the direct sum representation ρ⊕ ρ is equal to χ_ρ+ χ_ρ. (2) The character of the tensor product representation ρ⊗ ρ is equal to χρ· χρ.

Proof. This is a consequence of followings. Suppose that {vi} and {wj} are bases of V and W which are eigenvectors of ρs and ρ_s with eigenvalues {λi} and {λj}, respectively. Then {vi⊕ 0W, 0V ⊕ wj} and {vi ⊗ wj} are eigenvectors of (ρ ⊕ ρ)s and (ρ ⊗ ρ)s with eigenvalues {λi, λ_j} and {λi · λ_j},

respectively.

1.7. Orthogonality Relations for Characters

Let G be a group of order g. If φ and ψ are two complex valued functions on G, we put:

φ, ψ = 1 g

s∈G

φ(s)ψ(s).

This is an inner product.

Theorem 1.7.1. Let ρ and ρ be two irreducible representations of G with characters χ_ρ and χ_ρ, respectively.

(1) If ρ and ρ are not isomorphic, then we have χρ, χ_ρ = 0.

(2) If ρ and ρ are isomorphic, then we have χρ, χ_ρ = 1.

Proof. Because the character dose not depend on the choices of basis, without lose of generality by suitable choice of basis, we suppose that the matrix form (rij(s)) of ρs and (r_kl(s)) of ρ_s are unitary matrices. Thus (rij(s))⁻¹= (rij(s))^T and (r_kl(s))⁻¹ = (r_kl(s))^T. We have then rij(s⁻¹) = rji(s) and r_kl (s⁻¹) = r_lk (s). Suppose dim(ρ) = n and dim(ρ) = m. By deﬁnition, χ_ρ(s) = _n

i=1r_ii(s) and χ_ρ(s) =_m

k=1r_kk(s), and hence χρ, χ_ρ =

m k=1

n i=1

rii, r_kk and rii, r_kk = 1 g

s∈G

r_ii(s)r_kk (s) = 1 g

s∈G

r_ii(s)r_kk (s⁻¹).

If ρ is not isomorphic to ρ, then by Corollary 1.3.4, we have rii, r_kk = 0, and hence χρ, χ_ρ = 0. If ρ is isomorphic to ρ, then n = m and χρ = χ_ρ. By Corollary 1.3.4, we have rii, rkk = δik/n, and hence χρ, χ_ρ = χρ, χρ =n

i,k=1δ_ik/n = 1.

Theorem 1.7.1 says that in terms of the inner product deﬁned above, the characters of irreducible representations of G are orthonormal. There are many applications of these orthogonality relations.

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Corollary 1.7.2. Let ρ be a representation of G in V with character χρand suppose V decomposes into a direct sum of irreducible representations: V = W1⊕ · · · ⊕ Wk. Let θ be an irreducible representation of G in W with character χ_θ. Then the number of Wi which is isomorphic to W is equal to χρ, χ_θ. Proof. Let χi be the character of the irreducible representation of G in Wi. By Proposition 1.6.2, we have χρ= χ1+· · · + χk. Thus χρ, χθ = χ1, χθ + · · · + χk, χθ. By Theorem 1.7.1, χi, χθ is equal to 1 (resp. 0) if Wi is (resp. is not) isomorphic to W . The result follows. Since χρ, χ_θ does not depend on the decomposition of V , this result says that the number of irreducible representations in any decomposition of V which are isomorphic to W is the same. This shows the fact that the canonical decomposition of V is unique (cf. Section 1.5). This number is called the multiplicity of W occurs in V . If W₁, . . . , W_h are the distinct non-isomorphic irreducible representations occur in W with multiplicities m₁, . . . , m_h respectively, and χ₁, . . . , χ_h denote corre- sponding characters, then V is isomorphic to m₁W₁⊕ · · · ⊕ mhW_h and the character χ_ρof V is equal to m1χ1+· · · + mhχh with mi= χρ, χi. Whence:

Corollary 1.7.3. Two representations have the same character if and only if they are isomorphic.

The above results reduce the study of representations to that of their characters. In particular, we have:

Corollary 1.7.4. If χρ is the character of a representation ρ of G in V , then χρ, χρ is a positive integer. Furthermore, we have χρ, χρ = 1 if and only if V is irreducible.

Proof. Suppose that χρ= m₁χ₁+· · · + mhχ_h where χ_i are irreducible characters of G. The orthog- onality relations among the χ_i imply χρ, χ_ρ = _h

i=1m²_i. Furthermore, _h

i=1m²_i = 1 if only one of

the mi is equal to 1. Our result follows.

1.8. The Space of Class Functions on G

A Complex valued function f on G is called a class function if f (tst⁻¹) = f (s) for all s, t ∈ G. By Proposition 1.6.1, all characters of a representation of G are class functions. Recall that two elements s and s in G are said to be conjugate if there exists t∈ G such that s = tst⁻¹; this is an equivalence relation, which partitions G into conjugacy classes. Let C1, . . . , C_h be the distinct conjugacy classes of G. To say that a function f on G is a class function is equivalent to saying that f is constant on each of C₁, . . . , C_h.

We introduce now the space H of class functions on G. This is an inner product space endowed with the inner product deﬁned in 1.7. The dimension ofH is equal to the number of conjugacy classes of G.

Given a linear representation ρ : G→ Aut(V ) of G in V , for f ∈ H, we deﬁne a linear mapping ρ_f : V → V by:

ρf(v) =

t∈G

f (t)ρt(v), for v∈ V.

Because f is a class function on G, we have ρ⁻¹_s ◦ ρf ◦ ρs=

t∈G

f (t)ρ_s−1ts=

u∈G

f (sus⁻¹)ρ_u=

u∈G

f (u)ρ_u = ρ_f. Hence, ρ_f is a G-linear mapping of V into V .

Lemma 1.8.1. Let G be a group of order g and let f be a class function on G. Suppose that ρ : G → Aut(V ) is an irreducible linear representation of G of dimension n and character χ. Then ρf =

t∈Gf (t)ρt is a homothety of ratio λ given by:

λ = 1 n

t∈G

f (t)χ(t) = g n f, χ.

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1.8. The Space of Class Functions on G 11

Proof. Since ρf ∈ HomG(V, V ) and V is irreducible, by Schur’s lemma (Proposition 1.3.1), ρf = λI.

Because dim(V ) = n, we have

λ n = Tr(λI) = Tr(ρ_f) =

t∈G

f (t)Tr(ρ_t) =

t∈G

f (t)χ(t).

The proof is complete.

Theorem 1.7.1 show that the characters of the irreducible representations of G are orthonormal in H. Therefore, they are linearly independent over C. This amounts to saying that the number of the irreducible representations of G is less than or equal to the number of conjugacy classes of G. In fact, they generateH.

Theorem 1.8.2. The characters of irreducible representations of G form an orthonormal basis of the space of class functions on G.

Proof. Suppose that χ1, . . . , χ_h are the distinct characters of the irreducible representations of G.

We know that χ₁, . . . , χ_h are also characters of G, and since χi, χ_i = χi, χ_i = 1, they are also irreducible. Therefore, we only have to show that the orthogonal complement ofW =span(χ1, . . . , χ_h) inH is {0}. Let f ∈ W^⊥ and for any representation ρ of G, put ρf =

t∈Gf (t)ρt. Since f, χi = 0, Lemma 1.8.1 above shows that ρ_f is the zero mapping so long as ρ is irreducible. However, by Theorem 1.5.2, every representation is a direct sum of irreducible representations. We conclude that for any representation ρ, ρ_f is always the zero mapping.

Now let ρ be the regular representation of G (cf. Example 1.1.1) in the vector space of dimension g with a basis (v_t)_t_∈G. Let e be the identity of G. Computing the image of v_e under ρ_f, we have ρ_f(v_e) =

t∈Gf (t)ρ_t(v_e) =

t∈Gf (t)v_t = 0. Since (v_t)_t_∈G is linearly independent, f (t) = 0 for all

t∈ G and the proof is complete.

This theorem says that the number of irreducible representations of G (up to isomorphic) is equal to the number of conjugacy classes of G. We have another consequence of Theorem 1.8.2:

Proposition 1.8.3. Let χ1, . . . , χ_h be the distinct characters of irreducibles representations of G. Let g be the order of G and for s∈ G, let c(s) be the number of elements in the conjugacy class of s. Then we have:

h i=1

χ_i(s)χ_i(t) =

_g

c(s) if t is conjugate to s , 0 otherwise.

Proof. Let fs : G→ C be the function on G such that fs(t) = 1 if t is conjugate to s and fs(t) = 0 otherwise. Since f_s∈ H, by Theorem 1.8.2, it can be written as fs =_h

i=1λ_iχ_i. Because χ₁, . . . , χ_h are orthonormal,

λi = fs, χi = 1 g

t∈G

fs(t)χi(t) = c(s) g χi(s).

We then have for each t∈ G,

fs(t) = c(s) g

h i=1

χi(s)χi(t).

Our proof is complete by evaluating f_s.

Let e be the identity of G. Then c(e) = 1 and χi(e) equals to the dimension of the corresponding irreducible representation of χi. Hence, we have the following:

Corollary 1.8.4. Let G be a group of order g. Let χ1, . . . , χ_h be all the distinct characters of the irreducible representations of G and let n₁, . . . , n_h be the dimensions of their corresponding represen- tations. Then _h

i=1n²_i = g and if s= e then _h

i=1niχi(s) = 0.

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In Corollary 1.3.2, we know that every irreducible representation of an abelian group has dimension 1. In fact, the converse is also true.

Corollary 1.8.5. G is abelian if and only if all the irreducible representations of G have dimension 1.

Proof. Suppose that W1, . . . , W_h are distinct irreducible representations of G of dimension n₁, . . . , n_h respectively, where h is the number of conjugacy classes of G. Suppose that g is the order of G. By Corollary 1.8.4, n²₁+· · · + n²_h = g. Since G is abelian if and only if h = g, which is equivalent to all

the ni are equal to 1, our claim follows.

1.9. Characters of a Group

A representation of G of dimension 1 is a homomorphism of G into the multiplicative groupC^∗ and is called a character of G. In particular, we call the trivial 1-dimensional representation of G, the unit character of G.

Let ρ be a representation of G. Suppose that µ is a character of G for which there exists a non-zero v∈ Vρsuch that ρ_s(v) = µ(s)v for every s∈ G. Then µ is said to be an eigenvalue of G with respective to ρ and v is said to be an eigenvector of G that belongs to µ.

Let A be a ﬁnite abelian group. Then Proposition 1.8.3 says that the irreducible representation of A are of dimension 1 and that their number is equal to |A|. Hence, in this case, the number of characters of A is equal to the number of A. Furthermore, the set of characters of A forms a multiplicative group ˆA which is isomorphic to A.

For arbitrary group, the subgroup of G generated by the set {sts⁻¹t⁻¹ | s, t ∈ G} is called the commutator subgroup of G and denoted G. G is the smallest normal subgroup of G such that G/G is abelian. We can deduce that, G has [G : G] characters. The following properties for characters are useful.

Lemma 1.9.1 (Orthogonality). If χ is not the unit character of G, then

s∈Gχ(s) = 0.

Proof. Since χ is not the unit character, there exists t ∈ G such that χ(t) = 1. We have

s∈Gχ(s) =

s∈Gχ(t)χ(s). Subtracting both side by

s∈Gχ(s), we obtain (χ(t)− 1)

s∈Gχ(s) = 0. Since

χ(t)= 1, our proof is complete.

Lemma 1.9.2 (Artin’s Lemma). If χ1, . . . , χn are distinct characters of G, then the only elements a₁, . . . , a_n in C such that_n

i=1a_iχ_i(s) = 0 for all s∈ G are a1 =· · · = an= 0.

Proof. We prove the result by induction. We may assume that every ai = 0. Since χ1 = χ2, there exists t ∈ G such that χ1(t) = χ2(t). We have n

i=1aiχi(t)χi(s) = 0 and n

i=1aiχ1(t)χi(s) = 0.

Subtracting these two relations we obtain _n

i=2a_i(χ₁(t)− χi(1))χ_i(s) = 0 for all s ∈ G. Since a₂(χ₁(t)− χ2(t))= 0, this contradicts the validity of the result for n − 1 and complete the proof. Remark . Suppose G is abelian. Then G is canonically isomorphic to the dual ˆG of ˆG. Hence the dual of these two lemmas is also true.

1.10. Restricted Representation

If H ⊆ G is a subgroup, any representation ρ of G in V restricts a representation of H in V , denoted ρ_H (or Res^G_H(V )).

Suppose that W is a subrepresentation of ρ_H, that is, a vector subspace of V stable under ρt, for t ∈ H. Let s ∈ G; the vector space ρsW depends only on the left coset sH of s; indeed, if t ∈ H, we have ρ_st(W ) = ρ_sρ_t(W ) = ρ_s(W ) because ρ_t(W ) = W . Hence, if τ is a left coset of H in G, we can thus deﬁne a subspace W_τ of V to be ρ_sW for any s ∈ τ. Because the set of left cosets of H are permuted among themselves by multiplying an element s∈ G on the left, it is clear that the Wτ are

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1.11. Induced Representations 13

permuted among themselves by the ρs, s∈ G. Their sum

τ∈G/HWτ is thus a subrepresentation of V .

We are interested in the case that G has an abelian subgroup.

Proposition 1.10.1. Let G be a group of order g and let A be an abelian subgroup of G of order a.

Then each irreducible representation of G has dimension≤ g/a.

Proof. Let ρ be an irreducible representation of G in V and ρAbe the restriction to A. Suppose that W ⊆ V is an irreducible subrepresentation of ρA. By Corollary 1.8.5, we have dim(W ) = 1. Since V =

τ∈G/AWτ is thus a subrepresentation of V and V is irreducible, we have that V = V, and

hence dim(V )≤ g/a.

1.11. Induced Representations

Let H be a subgroup of G and let W be a subspace of V which is stable under H. We say that the representation ρ of G in V is induced by the representation θ of H in W , if V is equal to the direct sum of the W_τ, τ ∈ G/H (thus, if V =

τ∈G/HW_τ). Recall that if τ is a left coset of H in G, W_τ of V is ρ_sW for any s ∈ τ. Therefore, we have dim(V ) =

τ∈G/Hdim(W_τ) = [G : H]· dim(W ), where [G : H] is the number of left cosets of H in G, i.e. the index of H in G. Later (Theorem 1.11.4) we will see that given a linear representation θ : H → Aut(W ), there exists a unique (up to isomorphic) representation ρ : G → Aut(V ) such that ρ in V is induced by θ in W . In this case we write V = Ind^G_H(W ) and ρ = Ind^G_H(θ).

From the deﬁnition, it is easy to see that Ind^G_H(W ⊕ W) = Ind^G_H(W )⊕ Ind^G_H(W).

Example 1.11.1. Take for ρ the regular representation of G in V ; V has a basis (vt)t∈G such that ρs(vt) = vst. Let W be the subspace of V with basis (vt)_t_∈H. The representation θ of H in W is the regular representation of H and it is clear that ρ is induced by θ.

Now we show the existence and uniqueness of induced representations.

Lemma 1.11.2. If the representation ρ : G → Aut(V ) is induced by θ : H → Aut(W ), and if W is a subspace of W which is stable under H, then the subspace V=

τ∈G/HW_τ of V is stable under G and the representation of G in V is induced by the representation of H in W.

Proof. Let τ ∈ G/H and t ∈ τ. Then we have Wτ = ρt(W)⊆ ρt(W ) = Wτ. Since V =

τ∈G/HWτ, it implies that V =

τ∈G/HW_τ.

By using the lemma above, we can prove the existence of induced representation of θ : H → Aut(W ). Because Ind^G_H(W ⊕ W) = Ind^G_H(W )⊕ Ind^G_H(W), we may assume the θ is irreducible. In this case, (using Corollary 1.7.2) θ is isomorphic to a subrepresentation of the regular representation of H and the regular representation of H induces the regular representation of G (cf. the example above). Applying Lemma 1.11.2, there exists a subrepresentation of the regular representation of G which is induced by θ.

In next section, we will give a concrete construction for the induced representation.

Lemma 1.11.3. Suppose that the representation ρ : G → Aut(V ) is induced by θ : H → Aut(W ).

Let ρ : G → Aut(V) be a linear representation of G and let f : W → V be a H-linear map (i.e.

f (θ_tw) = ρ_tf (w) for all t∈ H and w ∈ W ). Then there exists a unique linear map F : V → V which extends f (i.e. F (w) = f (w) for all w∈ W ) and satisﬁes F ◦ ρs= ρ_s◦ F for all s ∈ G.

Proof. Let τ ∈ G/H. If F satisﬁes these conditions, for s ∈ τ and w ∈ W , we have F (ρs(w)) = ρ_s(F (w)) = ρ_s(f (w)). This determines F on ρs(W ) = Wτ and hence on V because V =

τ∈G/HWτ. This proves the uniqueness of F .

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For the existence of F ; if v = ρs(w) ∈ Wτ, we deﬁne F (v) = ρ_s(f (w)). This deﬁnition does not depend on the choice of s in τ and w in W . If ρst(w) = ρs(w) for some t∈ H and w ∈ W , then we have ρt(w) = θt(w) = w. Hence, ρ_st(f (w)) = ρ_s(ρ_t(f (w))) = ρ_s(f (θt(w))) = ρ_s(f (w)). Again, since V =

τ∈G/HW_τ, by linearity, there exists a unique linear map F : V → V which extends the partial mappings thus deﬁned on every W_τ. One easily checks that F◦ ρs = ρ_s◦ F for all s ∈ G. In fact, if v = ρ_s(w)∈ Wτ, then F ◦ ρs(ρ_s(w)) = F (ρ_ss(w)) = ρ_s_s(f (w)) = ρ_s(ρ_s(f (w))) = ρ_s ◦ F (ρs(w)). Theorem 1.11.4. Let H be a subgroup of G and let θ : H → Aut(W ) be a linear representation of H in W . Then there exists a unique (up to isomorphic) representation ρ : G→ Aut(V ) such that ρ in V is induced by θ in W .

Proof. Because we have proved the existence, we only have to prove the uniqueness. Suppose that ρ : G → Aut(V ) and ρ : G → Aut(V) are two representations of G induced by θ : H → Aut(W ).

Considering ι : W → V the injection of W into V, by Lemma 1.11.3 there exists a unique linear map F : V → V which is identity on W and satisﬁes F◦ρs = ρ_s◦F for all s ∈ G. For every ρs(w)∈ ρs(W ), we have F (ρs(w)) = ρ_s(F (w)) = ρ_s(w). Hence the image of F contains all the ρ_s(W ) and thus F is onto. Since V and V have the same dimension [G : H] dim(W ), we see that F is an isomorphism

which proves the uniqueness.

1.12. A Concrete Construction for Induced Representation

Let G be a ﬁnite group and let H be a subgroup of G. Let θ : H → Aut(W ) be a linear representation of H. Deﬁne a vector space V to be the set of all functions f : G→ W that satisfy

f (ts) = θt(f (s)) ∀ t ∈ H, s ∈ G.

Thus, an element f ∈ V is uniquely decided by its values on a system of representatives H\G of the right cosets of H in G. Deﬁne an action of G on V by

ρ_s(f )(r) = f (r· s) ∀ r, s ∈ G and f ∈ V.

It is easy to check that ρ gives a linear representation of G with representation space V . We embed W into V by mapping each w∈ W onto the function fw : G→ W deﬁned by

fw(s) =

θ_s(w) if s∈ H, 0 otherwise.

Clearly we have that ρ_t(f_w) = f_θ_t_(w) for all t ∈ H and W is isomorphic onto the subspace of V consisting of functions which vanish oﬀ H.

Let now R be a system of representatives of the left cosets G/H. For every f ∈ V and r ∈ R, we deﬁne a function f_r∈ V by

f_r(s) =

f (s) if s∈ Hr⁻¹, 0 otherwise.

Then f =

r∈Rρ_r(ρ⁻¹_r (f )) and ρ⁻¹_r (f_r) = ρ_r−1(f_r) belongs to W (after identifying W with its image in V ). Thus V =

τ∈G/HW_τ and hence V = Ind^G_H(W ).

There is another point of view of induced representation. Let ρ be a linear representation of G. Then Vρ can be also considered as a module over the group-ring C[G]. Using this form, if ρ is another representation of G, then we write (ρ, ρ) = dim Hom_C[G](Vρ, V_ρ). The form (ρ, ρ) is clearly symmetric and bilinear. In fact, decomposing V_ρand V_ρ into direct sum of irreducible representations, by Theorem 1.7.1 we have that

χρ, χ_ρ = (ρ, ρ) .

From this point of view, for induced representation, we obtain also a canonical isomorphism Ind^G_H(W ) ∼=C[G] ⊗_C[H]W .

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1.13. Characters of Induced Representations 15

This characterization of induced representation makes it obvious that the induced representation exists and is unique. On the other hand, given a C[G]-module V which is a direct sum V = ⊕i∈IWi

of vector space permuted transitively by G. Choose i0 ∈ I and W = Wi₀ and let H be the subgroup H ={ s ∈ G |sW = W }. Then it is clear that the C[G]-module V is induced by the C[H]-module W . This form of induced representation is convenient to prove the following fundamental properties, by using elementary property of tensor product.

Proposition 1.12.1. Let J be a subgroup of H and H be a subgroup of G.

(1) (Lemma 1.11.3) Let W be a C[H]-module and let E be a C[G]-module. Then we have Hom_C[H](W, E) ∼= Hom_C[G](Ind^G_H(W ), E) .

(2) Let U be a C[J]-module. Then

Ind^G_J(U ) = Ind^G_H(Ind^H_J(U )) . 1.13. Characters of Induced Representations

Let ρ : G→ Aut(V ) be a linear representation of G which is induced by the representation θ : H → Aut(W ) and let χ_ρ and χ_θ be the corresponding characters. Since by the uniqueness, θ determines ρ up to isomorphic, we ought to be able to compute χ_ρ from χ_θ.

Theorem 1.13.1. Let ρ : G → Aut(V ) be a linear representation of G which is induced by the representation θ : H→ Aut(W ) and let χρ and χ_θ be the corresponding characters. Let h be the order of H. For each s∈ G, we have

χ_ρ(s) = 1 h

r−1sr∈Hr∈G

χ_θ(r⁻¹sr).

Proof. Choose R being a system of representatives of G/H, so V =

r∈Rρr(W ). For s∈ G and r ∈ R, we have that sr = rt with r ∈ R and t ∈ H. We see that ρs sends ρr(W ) into ρ_r(W ). We choose a basis of V which is the union of bases of ρr(W ), r ∈ R. The indices r such that r = r give zero diagonal terms, and for the indices r such that r = r, ρ_r(W ) is stable under ρ_s (because W is stable under ρ_t= θ_t, for t∈ H). Observe that r = r if and only if r⁻¹sr = t∈ H. We thus only have to compute the trace of the restriction of ρ_s on ρ_r(W ) for those r ∈ R such that r⁻¹sr ∈ H. Note that in this case ρs◦ ρr = ρr◦ ρt = ρr◦ θt and ρr deﬁnes an isomorphism of W into ρr(W ). Hence the restriction of ρs on ρr(W ) is equal to ρrθtρ⁻¹_r and thus its trace is equal to that of θt, that is, to χ_θ(t) = χ_θ(r⁻¹sr). Our formula follows from the fact that if r⁻¹sr∈ H , then every element u ∈ rH has the property u⁻¹su∈ H and χθ(u⁻¹su) = χ_θ(r⁻¹sr). Let H be a subgroup of G. For a linear representation of ρ : G → Aut(V ) with character χρ, we denote by Res^G_H(χρ) the character of the restricted representation ρH of G on H. For a linear representation of θ : H → Aut(W ) with character χθ, we denote by Ind^G_H(χθ) the character of the representation of G induced by θ.

Theorem 1.13.2 (Frobenius Reciprocity). Let H be a subgroup of G. Let ρ : G → Aut(V ) be a linear representation of G with character χ_ρ and let θ : H → Aut(W ) be a linear representation of H with character χ_θ. Then we have

χρ, Ind^G_H(χ_θ)G= Res^GH(χ_ρ), χ_θH,

where , G and , H denote the inner products of the spaces of class functions on G and H deﬁned in 1.7.

Proof. Observe ﬁrst that if ρ and ρare linear representations of G in V and V with characters χ and χ, respectively, then χ, χG is equal to dim(HomG(V, V)). Lemma 1.11.3 shows that every H-linear

A Note on Complex Representations of GL(2, Fq