*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*

**Contents**

Chapter 1. General Properties on Linear Representations of Finite Groups 5

*§1.1. Basic Deﬁnitions* 5

*§1.2. Subrepresentations and Irreducible Representations* 5

*§1.3. Schur’s Lemma and Its Applications* 6

*§1.4. Direct Sum and Tensor Product* 7

*§1.5. Complete Reducibility* 7

*§1.6. Characters for Representations* 8

*§1.7. Orthogonality Relations for Characters* 9

*§1.8. The Space of Class Functions on G* 10

*§1.9. Characters of a Group* 12

*§1.10. Restricted Representation* 12

*§1.11. Induced Representations* 13

*§1.12. A Concrete Construction for Induced Representation* 14

*§1.13. Characters of Induced Representations* 15

*§1.14. Restrictions of Induced Representations* 16

*§1.15. Method of Little Group* 17

*§1.16. The Schur Algebra* 18

Chapter 2. *The Group GL(2,*F*q*) and Its Subgroups 21

*§2.1. Notational Conventions* 21

*§2.2. The Subgroups U and P* 22

*§2.3. The Borel Subgroup B* 23

*§2.4. The Group GL(2, F**q*) 24

*§2.5. Inducing Characters from B to G* 26

*§2.6. The Jacquet Module of a Representation of GL(2, F**q*) 27

*§2.7. The conjugacy Classes of GL(2, F**q*) 29

Chapter 3. *The Representations of GL(2,*F*q*) 31

*§3.1. Cuspidal Representations* 31

*§3.2. Characters of F*^{×}* _{q}*2 32

*§3.3. The Small Weil Group* 33

3

*§3.4. Constructing Cuspidal Representations from Non-decomposable Characters* 34

*§3.5. The Correspondence between Cuspidal Representations and Non-decomposable*

Characters 38

*§3.6. Whittakers Models* 40

*§3.7. The Γ-function of a representation of G* 41

*§3.8. Determination of ρ by Γ**ρ* 42

*§3.9. The Bessel Function of a representation* 43

*§3.10. The Computation of Γ**ρ**(υ) for a Non-cuspidal ρ* 44

*§3.11. The Computation of Γ**ρ**(υ) for a Cuspidal ρ* 46

*§3.12. The Characters Table of GL(2, F**q*) 48

Index 51

Bibliography 53

*Chapter 1*

**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 ρ**s* *instead of ρ(s).*

*When V is given, we say that V is a representation space of G, denoted V**ρ* *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***(v**t*)*t**∈G* *indexed by the elements t of G. For s, t∈ G, let ρ**s* *be the linear map of V into V such that*
*ρ**s**(v**t**) = v**st**; 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 ρ**s**|**W* *of ρ*_{s}*to W is then an automorphism of W and*
*we have ρ*_{st}*|**W* *= ρ*_{s}*|**W* *◦ ρ**t**|**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(ρ**s**(v)) = ρ*^{}_{s}*(φ(v))*

5

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

*It is easy to check that for a given φ* *∈ Hom**G**(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∈ Hom**G**(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 φ*1

*, φ*

_{2}

*∈*Hom

_{G}*(V, W ). Suppose that there exist v= 0 in V such that φ*1

*(v) = φ*

_{2}

*(v). Then φ*

_{1}

*= φ*

_{2}

*.*

* Proof. The assumption says that φ*1

*− φ*2

*is not injective. Since φ*1

*− φ*2

*∈ Hom*

*G*

*(V, W ), it implies*

*that φ*_{1}*− φ*2 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 ρ**s**◦ f = f ◦ ρ**s* *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, ρ**s* *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*

*h*^{0}= 1
*g*

*t**∈G*

*(ρ*^{}* _{t}*)

^{−1}*◦ h ◦ ρ*

*t*

*.*

*Then:*

*(1) If ρ and ρ*^{}*are not isomorphic, then we have h*^{0} *= 0.*

*(2) If V = W and ρ = ρ*^{}*, then h*^{0} *is a homothety of ratio (1/n)Tr(h), where n = dim(V ).*

*1.5. Complete Reducibility* 7

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

*t**∈G**Tr((ρ**t*)^{−1}*hρ**t**) = 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 (h**kj**) and likewise h*^{0} *is deﬁned by (h*^{0}_{kj}*). Assume ρ and ρ** ^{}* are given

*in matrix form ρ*

_{t}*= (r*

_{ij}*(t)), 1≤ i, j ≤ n and ρ*

^{}*t*

*= (r*

^{}

_{kl}*(t)), 1≤ k, l ≤ m respectively. We have by the*

*deﬁnition of h*

^{0}:

*h*^{0}* _{kj}* = 1

*g*

*t**∈G,1≤l≤m,1≤i≤n*

*r*^{}_{kl}*(t** ^{−1}*)

*· h*

*li*

*· r*

*ij*

*(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(E**li**) = δ**li* *(δ**ij* *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}*)r**ij**(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 V*_{1} *⊕ V*2 by setting
*(ρ⊕ ρ** ^{}*)

_{s}*(v⊕ w) = ρ*

*s*

*(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) = ρ*

*s*

*(v)⊗ ρ*

^{}*s*

*(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 W*^{0} *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*

*p*^{0}= 1
*g*

*t**∈G*

*(ρ** _{t}*)

^{−1}*◦ p ◦ ρ*

*t*

*,*

*where g is the order of G. Since p maps V into W and ρ**t* *preserves W for all t* *∈ G, we see that*
*p*^{0} *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 p*^{0}*(w) = w for all w* *∈ W . Thus p*^{0} *is a projection of V onto W , corresponding to*
*some complement W*^{0} *= Ker(p*^{0}*) of W . We have moreover ρ*_{s}*◦ p*^{0} *= p*^{0} *◦ ρ**s* *for all s* *∈ G. Hence*
*p*^{0}*◦ ρ**s**(w*^{0}*) = ρ*_{s}*◦ p*^{0}*(w*^{0}*) = 0 for w*^{0} *∈ W*^{0} *and s∈ G, which shows that W*^{0} *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 W*^{0} *of G in V such that V = W* *⊕ W*^{0} *is a direct sum representation of W and W*^{0}.
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 sub-
representations 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 ρ**s* 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 = W*1*⊕ · · · ⊕ W**k* 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 = V*_{1}*⊕ · · · ⊕ V**h* *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** ρ**t**(u), ρ*_{t}*(v)*
property * ρ**s**(u), ρ**s**(v)*

*V such that the matrix form of ρ**s* *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 ρ**s*

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

*χ**ρ**(s) = Tr(ρ**s**) for each s∈ G.*

*1.7. Orthogonality Relations for Characters* 9

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

*We remark that if two representations ρ and ρ*^{}*are isomorphic, then χ**ρ**= 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 ρ**s**is normal, and hence diagonalizable. Thus for s∈ G, a basis*
*(v*1*, . . . , v**n**) of V can be chosen such that ρ**s**(v**i**) = λ**i**v**i**with λ**i* *∈ C*^{∗}*, and so χ**ρ**(s) =*_{n}

*i=1**λ**i*. Also note
*that s∈ G has ﬁnite order, the values λ**i* *are roots of unity; in particular we have λ**i* *= λ*^{−1}* _{i}* . Because

*ρ*

_{s}*−1*

*= ρ*

^{−1}

_{s}*, we have χ*

*ρ*

*(s*

*) =*

^{−1}

_{n}*i=1**λ*^{−1}* _{i}* =

_{n}*i=1**λ**i* *and Tr(ρ*_{tst}*−1**) = Tr(ρ**t**◦ ρ**s**◦ ρ*^{−1}*t* *) = Tr(ρ**s*).

We can summarize what we have shown so far in

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

*(1) χ**ρ**(e) = n.*

*(2) χ*_{ρ}*(s*^{−1}*) = χ*_{ρ}*(s) for s∈ G.*

*(3) χ**ρ**(tst*^{−1}*) = χ**ρ**(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 χ**ρ**· χ**ρ*^{}*.*

**Proof. This is a consequence of followings. Suppose that {v***i**} and {w**j**} are bases of V and W which*
*are eigenvectors of ρ**s* *and ρ*^{}* _{s}* with eigenvalues

*{λ*

*i*

*} and {λ*

^{}*j*

*}, respectively. Then {v*

*i*

*⊕ 0*

*W*

*, 0*

*V*

*⊕ w*

*j*

*}*and

*{v*

*i*

*⊗ w*

*j*

*} are eigenvectors of (ρ ⊕ ρ*

*)*

^{}*s*

*and (ρ*

*⊗ ρ*

*)*

^{}*s*with eigenvalues

*{λ*

*i*

*, λ*

^{}

_{j}*} and {λ*

*i*

*· 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* * χ**ρ**, χ*_{ρ}* = 0.*

*(2) If ρ and ρ*^{}*are isomorphic, then we have* * χ**ρ**, 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 (r**ij**(s)) of ρ**s* *and (r*^{}_{kl}*(s)) of ρ*^{}* _{s}* are unitary

*matrices. Thus (r*

*ij*

*(s))*

^{−1}*= (r*

*ij*

*(s))*

^{T}*and (r*

^{}

_{kl}*(s))*

^{−1}*= (r*

^{}

_{kl}*(s))*

^{T}*. We have then r*

*ij*

*(s*

^{−1}*) = r*

*ji*

*(s) and*

*r*

_{kl}

^{}*(s*

^{−1}*) = r*

_{lk}

^{}*(s). Suppose dim(ρ) = n and dim(ρ*

^{}*) = m. By deﬁnition, χ*

_{ρ}*(s) =*

_{n}*i=1**r*_{ii}*(s) and*
*χ*_{ρ}*(s) =*_{m}

*k=1**r*^{}_{kk}*(s), and hence*
* χ**ρ**, χ*_{ρ}* =*

*m*
*k=1*

*n*
*i=1*

* r**ii**, r*_{kk}^{}* and r**ii**, 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

*r*

*ii*

*, r*

_{kk}

^{}*= 0, and hence χ*

*ρ*

*, χ*

_{ρ}*= 0. If*

*ρ is isomorphic to ρ*

^{}*, then n = m and χ*

*ρ*

*= χ*

*. By Corollary 1.3.4, we have*

_{ρ}*r*

*ii*

*, r*

*kk*

*= δ*

*ik*

*/n, and*hence

*χ*

*ρ*

*, χ*

_{ρ}*= χ*

*ρ*

*, χ*

*ρ*

*=*

*n*

*i,k=1**δ*_{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.*

**Corollary 1.7.2. Let ρ be a representation of G in V with character χ***ρ**and suppose V decomposes into*
*a direct sum of irreducible representations: V = W*1*⊕ · · · ⊕ W**k**. Let θ be an irreducible representation*
*of G in W with character χ*_{θ}*. Then the number of W**i* *which is isomorphic to W is equal to* * χ**ρ**, χ*_{θ}*.*
**Proof. Let χ***i* *be the character of the irreducible representation of G in W**i*. By Proposition 1.6.2, we
*have χ**ρ**= χ*1+*· · · + χ**k*. Thus* χ**ρ**, χ**θ** = χ*1*, χ**θ** + · · · + χ**k**, χ**θ**. By Theorem 1.7.1, χ**i**, χ**θ** is equal*
*to 1 (resp. 0) if W**i* *is (resp. is not) isomorphic to W . The result follows.*
Since * χ**ρ**, 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 W*_{1}*, . . . , W** _{h}* are the distinct non-isomorphic irreducible

*representations occur in W with multiplicities m*

_{1}

*, . . . , m*

_{h}*respectively, and χ*

_{1}

*, . . . , χ*

*denote corre-*

_{h}*sponding characters, then V is isomorphic to m*

_{1}

*W*

_{1}

*⊕ · · · ⊕ m*

*h*

*W*

_{h}*and the character χ*

_{ρ}*of V is equal*

*to m*1

*χ*1+

*· · · + m*

*h*

*χ*

*h*

*with m*

*i*=

*χ*

*ρ*

*, χ*

*i*

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

**Proof. Suppose that χ***ρ**= m*_{1}*χ*_{1}+*· · · + m**h**χ*_{h}*where χ*_{i}*are irreducible characters of G. The orthog-*
*onality relations among the χ** _{i}* imply

*χ*

*ρ*

*, χ*

_{ρ}*=*

_{h}*i=1**m*^{2}* _{i}*. Furthermore,

_{h}*i=1**m*^{2}* _{i}* = 1 if only one of

*the m**i* 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 C*1

*, . . . , C*

*be the distinct conjugacy classes*

_{h}*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 C*

_{1}

*, . . . , 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 of*H 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:*

*ρ**f**(v) =*

*t**∈G*

*f (t)ρ**t**(v), for v∈ V.*

*Because f is a class function on G, we have*
*ρ*^{−1}_{s}*◦ ρ**f* *◦ ρ**s*=

*t**∈G*

*f (t)ρ*_{s}*−1**ts*=

*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*
*ρ**f* =

*t**∈G**f (t)ρ**t* *is a homothety of ratio λ given by:*

*λ =* 1
*n*

*t**∈G*

*f (t)χ(t) =* *g*
*n f, χ.*

*1.8. The Space of Class Functions on G* 11

**Proof. Since ρ***f* *∈ Hom**G**(V, V ) and V is irreducible, by Schur’s lemma (Proposition 1.3.1), ρ**f* *= 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 generate*H.*

**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 χ*1

*, . . . , χ*

_{h}*are the distinct characters of the irreducible representations of G.*

*We know that χ*_{1}*, . . . , χ*_{h}*are also characters of G, and since* * χ**i**, χ*_{i}* = χ**i**, χ*_{i}* = 1, they are also*
irreducible. Therefore, we only have to show that the orthogonal complement of*W =span(χ*1*, . . . , χ** _{h}*)
in

*H is {0}. Let f ∈ W*

^{⊥}*and for any representation ρ of G, put ρ*

*f*=

*t**∈G**f (t)ρ**t*. Since* f, χ**i** = 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*

*, we have*

_{f}*ρ*

_{f}*(v*

*) =*

_{e}*t**∈G**f (t)ρ*_{t}*(v** _{e}*) =

*t**∈G**f (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 χ*1

*, . . . , 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 f***s* *: G→ C be the function on G such that f**s**(t) = 1 if t is conjugate to s and f**s**(t) = 0*
*otherwise. Since f*_{s}*∈ H, by Theorem 1.8.2, it can be written as f**s* =_{h}

*i=1**λ*_{i}*χ*_{i}*. Because χ*_{1}*, . . . , χ** _{h}*
are orthonormal,

*λ**i* =* f**s**, χ**i** =* 1
*g*

*t**∈G*

*f**s**(t)χ**i**(t) =* *c(s)*
*g* *χ**i**(s).*

*We then have for each t∈ G,*

*f**s**(t) =* *c(s)*
*g*

*h*
*i=1*

*χ**i**(s)χ**i**(t).*

*Our proof is complete by evaluating f** _{s}*.

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

* Corollary 1.8.4. Let G be a group of order g. Let χ*1

*, . . . , χ*

_{h}*be all the distinct characters of the*

*irreducible representations of G and let n*

_{1}

*, . . . , n*

_{h}*be the dimensions of their corresponding represen-*

*tations. Then*

_{h}*i=1**n*^{2}_{i}*= g and if s= e then* _{h}

*i=1**n**i**χ**i**(s) = 0.*

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 W*1

*, . . . , W*

_{h}*are distinct irreducible representations of G of dimension n*

_{1}

*, . . . , 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, n*

^{2}

_{1}+

*· · · + n*

^{2}

_{h}*= g. Since G is abelian if and only if h = g, which is equivalent to all*

*the n**i* 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 group*C* ^{∗}* 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∈ V**ρ**such 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 ˆ*A which is isomorphic to A.*

*For arbitrary group, the subgroup of G generated by the set* *{sts*^{−1}*t*^{−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**∈G**χ(s) = 0.*

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

*s**∈G**χ(s) =*

*s**∈G**χ(t)χ(s). Subtracting both side by*

*s**∈G**χ(s), we obtain (χ(t)− 1)*

*s**∈G**χ(s) = 0. Since*

*χ(t)= 1, our proof is complete.*

* Lemma 1.9.2 (Artin’s Lemma). If χ*1

*, . . . , χ*

*n*

*are distinct characters of G, then the only elements*

*a*

_{1}

*, . . . , a*

_{n}*in*

*C such that*

_{n}*i=1**a*_{i}*χ*_{i}*(s) = 0 for all s∈ G are a*1 =*· · · = a**n**= 0.*

**Proof. We prove the result by induction. We may assume that every a***i* *= 0. Since χ*1 *= χ*2, there
*exists t* *∈ G such that χ*1*(t)* *= χ*2*(t). We have* *n*

*i=1**a**i**χ**i**(t)χ**i**(s) = 0 and* *n*

*i=1**a**i**χ*1*(t)χ**i**(s) = 0.*

Subtracting these two relations we obtain _{n}

*i=2**a*_{i}*(χ*_{1}*(t)− χ**i**(1))χ*_{i}*(s) = 0 for all s* *∈ G. Since*
*a*_{2}*(χ*_{1}*(t)− χ*2*(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 ρ**t*, for
*t* *∈ H. Let s ∈ G; the vector space ρ**s**W depends only on the left coset sH of s; indeed, if t* *∈ H,*
*we have ρ*_{st}*(W ) = ρ*_{s}*ρ*_{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*_{s}*W 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 W**τ* are

*1.11. Induced Representations* 13

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

*τ**∈G/H**W**τ* 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 ρ***A**be the restriction to A. Suppose that*
*W* *⊆ V is an irreducible subrepresentation of ρ**A**. By Corollary 1.8.5, we have dim(W ) = 1. Since*
*V** ^{}* =

*τ**∈G/A**W**τ* *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/H**W*_{τ}*). Recall that if τ is a left coset of H in G, W*_{τ}*of V is ρ*_{s}*W for any s* *∈ τ. Therefore, we have dim(V ) =*

*τ**∈G/H**dim(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 (v***t*)*t**∈G* such that
*ρ**s**(v**t**) = v**st**. Let W be the subspace of V with basis (v**t*)_{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/H**W*_{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 W***τ*^{}*= ρ**t**(W** ^{}*)

*⊆ ρ*

*t*

*(W ) = W*

*τ*

*. Since V =*

*τ**∈G/H**W**τ*,
*it implies that V** ^{}* =

*τ**∈G/H**W*_{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*_{t}*w) = ρ*^{}_{t}*f (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 ◦ ρ**s**= 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 (ρ***s**(w)) =*
*ρ*^{}_{s}*(F (w)) = ρ*^{}_{s}*(f (w)). This determines F on ρ**s**(W ) = W**τ* *and hence on V because V =*

*τ**∈G/H**W**τ*.
*This proves the uniqueness of F .*

*For the existence of F ; if v = ρ**s**(w)* *∈ W**τ**, 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 ρ**st**(w*^{}*) = ρ**s**(w) for some t∈ H and w*^{}*∈ W , then we*
*have ρ**t**(w*^{}*) = θ**t**(w*^{}*) = w. Hence, ρ*^{}_{st}*(f (w*^{}*)) = ρ*^{}_{s}*(ρ*^{}_{t}*(f (w*^{}*))) = ρ*^{}_{s}*(f (θ**t**(w*^{}*))) = ρ*^{}_{s}*(f (w)). Again, since*
*V =*

*τ**∈G/H**W*_{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◦ ρ*

*s*

^{}*= ρ*

^{}

_{s}

_{}*◦ F for all s*

^{}*∈ G. In fact, if*

*v = ρ*

_{s}*(w)∈ W*

*τ*

*, then F*

*◦ ρ*

*s*

^{}*(ρ*

_{s}*(w)) = F (ρ*

_{s}*s*

*(w)) = ρ*

^{}

_{s}

_{}

_{s}*(f (w)) = ρ*

^{}

_{s}

_{}*(ρ*

^{}

_{s}*(f (w))) = ρ*

^{}

_{s}

_{}*◦ F (ρ*

*s*

*(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◦ρ*

*s*

*= ρ*

^{}

_{s}*◦F for all s ∈ G. For every ρ*

^{}*s*

*(w)∈ ρ*

^{}*s*

*(W ),*

*we have F (ρ*

*s*

*(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) = θ**t**(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 f**w* *: G→ W deﬁned by*

*f**w**(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**∈R**ρ*_{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/H**W*_{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 V**ρ* 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]}*(V*

*ρ*

*, V*

_{ρ}*). The form (ρ, ρ*

*) is clearly*

^{}*symmetric and bilinear. In fact, decomposing V*

_{ρ}*and V*

*into direct sum of irreducible representations, by Theorem 1.7.1 we have that*

_{ρ}* χ**ρ**, 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 .*

*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**∈I**W**i*

*of vector space permuted transitively by G. Choose i*0 *∈ I and W = W**i*_{0} *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 χ**ρ* *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∈H**r∈G*

*χ*_{θ}*(r*^{−1}*sr).*

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

*r**∈R**ρ**r**(W ). For s∈ G and*
*r* *∈ R, we have that sr = r*^{}*t with r*^{}*∈ R and t ∈ H. We see that ρ**s* *sends ρ**r**(W ) into ρ*_{r}*(W ). We*
*choose a basis of V which is the union of bases of ρ**r**(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*

^{−1}*sr = 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*

^{−1}*sr*

*∈ H. Note*

*that in this case ρ*

*s*

*◦ ρ*

*r*

*= ρ*

*r*

*◦ ρ*

*t*

*= ρ*

*r*

*◦ θ*

*t*

*and ρ*

*r*

*deﬁnes an isomorphism of W into ρ*

*r*

*(W ). Hence*

*the restriction of ρ*

*s*

*on ρ*

*r*

*(W ) is equal to ρ*

*r*

*θ*

*t*

*ρ*

^{−1}

_{r}*and thus its trace is equal to that of θ*

*t*, that is, to

*χ*

_{θ}*(t) = χ*

_{θ}*(r*

^{−1}*sr). Our formula follows from the fact that if r*

^{−1}*sr∈ H , then every element u ∈ rH*

*has the property u*

^{−1}*su∈ H and χ*

*θ*

*(u*

^{−1}*su) = χ*

_{θ}*(r*

^{−1}*sr).*

*Let H be a subgroup of G. For a linear representation of ρ : G*

*→ Aut(V ) with character χ*

*ρ*, we denote by Res

^{G}

_{H}*(χ*

*ρ*

*) the character of the restricted representation ρ*

*H*

*of G on H. For a linear*

*representation of θ : H*

*→ Aut(W ) with character χ*

*θ*, we denote by Ind

^{G}

_{H}*(χ*

*θ*) 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*

* χ**ρ**, Ind*^{G}_{H}*(χ** _{θ}*)

*G*=

*Res*

^{G}*H*

*(χ*

_{ρ}*), χ*

_{θ}*H*

*,*

*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 ρ**^{}*are linear representations of G in V and V*^{}*with characters χ and*
*χ** ^{}*, respectively, then

*χ, χ*

^{}*G*is equal to dim(Hom

*G*

*(V, V*

^{}*)). Lemma 1.11.3 shows that every H-linear*