REPRESENTATION OF FINITE GROUPS · representations can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Repre-sentations

REPRESENTATION OF FINITE GROUPS

A Project Report Submitted

in Partial Fulfilment of the Requirements

for the Degree of

MASTER OF SCIENCE

in

Mathematics and Computing

by

SHUBHAM GUPTA

(Roll No. 132123032)

to the

DEPARTMENT OF MATHEMATICS

INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI

GUWAHATI - 781039, INDIA

April 2015

CERTIFICATE

This is to certify that the work contained in this report entitled “REP-

RESENTATION OF FINITE GROUPS” submitted by SHUBHAM

GUPTA (Roll No: 132123032) to Department of Mathematics, Indian

Institute of Technology Guwahati towards the requirement of the course

MA699 Project has been carried out by him under my supervision.

Guwahati - 781039 (Dr. SHYAMASHREE UPADHYAY)

April 2015 Project Supervisor

ii

ABSTRACT

In mathematics, representation theory is a technique for analyzing abstract

groups in terms of groups of linear transformations. In particular, group

representations can be used to represent group elements as matrices so that

the group operation can be represented by matrix multiplication. Repre-

sentations of groups are important because they allow many group-theoretic

problems to be reduced to problems in linear algebra, which is well under-

stood. They are also important in physics because, for example, they describe

how the symmetry group of a physical system affects the solutions of equa-

tions describing that system. In general, representation of groups is a vast

topic that has been extensively studied in mathematics over many years. In

this thesis, we confine our attention to the study of representations of finite

groups only.

iv

Contents

1 Complete Reducibility and Schur’s Lemma 1

1.1 Definition of representation . . . . . . . . . . . . . . . . . . . 1

1.2 Constructing more representation from the given ones . . . . 3

1.3 Permutation Representation . . . . . . . . . . . . . . . . . . . 6

1.4 Complete Reducibility . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Applications of Schur’s Lemma . . . . . . . . . . . . . . . . . 9

1.6 Examples of Irreducible Representation . . . . . . . . . . . . . 10

2 Characters 15

2.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The First Projection Formula and Its Consequences . . . . . . 17

Bibliography 24

vi

Chapter 1

Complete Reducibility and

Schur’s Lemma

In this thesis, all vector spaces that we will consider will be over the alge-

braically closed field C of complex numbers.

1.1 Definition of representation

Definition 1.1.1. A representation of a finite group G on a finite dimen-

sional complex vector space V is a homomorphism ρ : G → GL(V ) of G to

the group of automorphism of V. i.e. representation is a map ρ : G→ GL(V )

such that

ρ(g1g2) = ρ(g1)ρ(g2) ∀g1, g2 ∈ G

Generally we call V itself a representation of the group G. The dimension of

the vector space V is called the degree of ρ.

Exapmles: (i) Given any group G, take V=F(the 1-dimensional space)

1

and ρ : G → GL(V ), g 7→ (id : F → F )). This is known as trivial/principal

representation. So degρ = 1.

(ii) Let G = C4 be the cyclic group of order 4. Let x be a generator of G. If

ρ : G→ GL(V ) (where V = C4) is a representation of G, then we must have

ρ(x) = X where X is a 4× 4 complex matrix such that X4 = id.

⇒ we can take X to be a diagonal matrix with entries in the set {1,−1, i,−i}.

So, ρ : G→ GL(V ) given by ρ(x) = X where

X =

1 0−1

i

0 −i

is a representation of G.

Definition 1.1.2. Let (V, ρ) and (W, ρ′) are two representations of the

group G. A map φ between representation V and W is a vector space

map φ : V → W such that

W

V

V

W

φ

φ

ρ(g)ρ(g)

Figure 1.1: A map φ between representation

the diagram commutes ∀g ∈ G. We call this as G-linear map. A map

2

φ : V → W is G-linear if φ is linear and the above diagram commutes ∀g ∈ G

i.e.

ρ′(g) ◦ φ(v)=φ ◦ ρ(g)(v) ∀g ∈ G, v ∈ V .

1.2 Constructing more representation from

the given ones

For this section, let us fix (V, ρ) and (W, ρ′) to be two representations of the

finite group G and let φ : V → W be a G-linear map.

(i) Define Kerφ = {v ∈ V |φ(v) = 0} ⊆ V . It is easy to check that Kerφ is

a subsapce of V. Now we will show that Kerφ is also a representation of G.

For this we must have that ρ(g)(v) ∈ Kerφ ∀v ∈ Kerφ and for all g ∈ G.

Observe that φ(ρ(g)(v)) = [φ ◦ ρ(g)](v) = [ρ′(g) ◦ φ](v) = ρ′(g)(φ(v)) =

ρ′(g)(0) = 0 as ρ′(g) ∈ GL(V )

=⇒ ρ(g)(v) ∈ Kerφ

Definition 1.2.1. Let (V, ρ) be a representation of the group G. Then we say

that ρ is faithful if the group homomorphism ρ : G→ GL(V ) is injective.

(ii) Define Imφ = {φ(v) ∈ W |v ∈ V } ⊆W. Now we will show that Im φ is

also representation of G. For this we must have that ρ′(g)(w) ∈ Imφ for all

w ∈ Imφ. Observe that ρ′(g)(w) = ρ′(g)(φ(v)) for some v ∈ V . Since φ is a

G-linear map, therefore ρ′(g)(φ(v)) = φ ◦ ρ(g)(v) = φ(ρ(g)v) ∈ Imφ.

Definition 1.2.2. Let ρ : G → GL(V ) be a representation of G. We say

that W ⊆ V is a G-subspace if it is a subspace and is ρ(G)−invariant,

i.e. ρg(W ) ⊆ W for all g ∈ G. e.g. {0} and V. In this definition if W

3

is a subspace then the corresponding map G → GL(W ), g 7→ ρ(g)|W is a

representation of G, a subrepresentation of ρ.

(iii) Define CoKerφ = W/Imφ , then in a similar way CoKerφ is also a

representation.

(iv) Direct sum: Let (V, ρ) and (W, ρ′) are two representation of G, then the

direct sum V⊕W is also a representation. Define ρo : G→ GL(V ⊕W ) by

ρo(g)(v, w) = (ρ(g)(v), ρ′(g)(w)).

(v) Tensor Product: V⊗

W have basis of the form {v⊗

w|v ∈ V,w ∈ W}

. So any arbitrary element of V⊗

W is of the form∑n

i=1 ai(vi⊗

wi) where

ai ∈ C, vi ∈ V,wi ∈ W .

V⊗

W is also a representation of G, given by the homomorphism π : G →

GL(V⊗

W ) which is defined as

π(g)(v⊗

w) = ρ(g)v⊗

ρ′(g)w ∀v ∈ V,w ∈ W .

(vi) The Dual V ∗ = Hom(V,C) of V is also a representation of G. If ρ :

G→ GL(V ) is a representation and ρ∗ : G→ GL(V ∗) is the dual, we should

have

< ρ∗(g)(v∗), ρ(g)(v) >=< v∗, v > for all g ∈ G, v ∈ V, v∗ ∈ V ∗. This in

turn forces us to define the dual representation by ρ∗(g) = ρ(g−1)t : V ∗ →

V ∗ ∀g ∈ G.

Because < ρ∗(g)(v∗), ρ(g)(v) >=< ρ(g)tρ∗(g)(v∗), v > which should be equal

to < v∗, v > ∀v ∈ V & ∀v∗ ∈ V ∗ means we must have ρ(g)tρ∗(g) = Id, that

is ρ∗(g) = [ρ(g)t]−1 = [ρ(g−1]t.

(vii) Hom(V,W): Given two representations (V, ρ) and (W, ρ′) of G, Hom(V,W)

is also a representation of G because Hom(V,W ) ∼= V ∗⊗

W where the

isomorphism is given by T : V ∗⊗

W → Hom(V,W ) as T (v∗⊗

w)(v) =

4

v∗(v)w ∀v ∈ V,w ∈ W, v∗ ∈ V ∗.

Define π : G→ GL(Hom(V,W )) by

π(g)(Q) := ρ′(g) ◦Q ◦ ρ(g−1) for any Q ∈ Hom(V,W ), since Hom(V,W ) ∼=

V ∗⊗

W we can take Q to be of the formn∑i=1

ai(v∗i

⊗wi). Observe that for

any v∗⊗

w ∈ V ∗⊗

W ,

π(g)(T (v∗⊗

w)) = T · π(g)(v∗⊗

w) [ since T is G-linear ]

= T [ρ∗(g)v∗⊗

ρ′(g)w

= T [ρ(g−1)]tv∗⊗

ρ′(g)w

⇒ π(g)(T (v∗⊗

w))(v) = T [[ρ(g−1)]tv∗⊗

ρ′(g)w](v)

= (ρ(g−1)tv∗)(v) · ρ′(g)w

= ρ′(g)[(ρ(g−1)tv∗(v)w]

= ρ′(g)[< ρ(g−1)tv∗, v > w]

= ρ′(g)[< v∗, ρ(g−1)v > w]

= ρ′(g)[v∗(ρ(g−1)v)w]

= ρ′(g) · T (v∗⊗

w)(ρ(g−1)v)

= [ρ′(g) ◦ T (v∗⊗

w) ◦ ρ(g−1)](v) ∀v ∈ V.

Hence π(g)(T (v∗⊗

w))(v) = [ρ′(g) ◦ T (v∗⊗

w) ◦ ρ(g−1)](v)

⇒ π(g)(T (v∗⊗

w)) = ρ′(g) ◦ T (v∗⊗

w) ◦ ρ(g−1)

∴ for any Q ∈ Hom(V,W ),we have

π(g)(Q) = ρ′(g) ◦Q ◦ ρ(g−1)

5

1.3 Permutation Representation

Definition 1.3.1. Let X be any finite set and suppose that G acts on the

left on X , i.e. G → Aut(X) is a homomorphism to the permutation group

of X, there is an associated permutation representation of G, which is

given by ρ : G→ GL(V )

ρ(g)(∑axex) =

∑axegx where V is a vector space having basis {ex : x ∈ X}.

Example: The regular representation (denoted by RG or R) is the per-

mutation representation of G corresponds to the left action of G on itself.

Alternatively let R be the space of complex valued function on G, where an

element g of the group G acts on the function α ∈ R by (gα)(h) = α(g−1h).

1.4 Complete Reducibility

Definition 1.4.1. A representation (V, ρ) is called irreducible if there is

no proper nonzero G subspace W of V.

Proposition 1.4.2. If W is a subrepresentation of a representation V of a

finite group G, then there is a complementary invariant subspace W ′ of V,

so that V = W⊕

W ′

Proof. Choose an arbitrary subspace U complementary to W, letπo : V → W

be a projection given by the direct sum decomposition V = W⊕

U , and

average the map πo over G:that is take

π(v) =∑

g∈G g(πo(g−1v)).

This will then be a G-linear map from V onto W, which is multiplication by

6

|G| on W;its kernel will, therefore,be a subspace of V invariant under G and

complementary to W.

Corollary 1.4.3. Any representation of a finite group is a direct sum of

irreducible representations.

The above corollary is also known as Complete Reducibility Theo-

rem.

Definition 1.4.4. The representation ρ : G → GL(V ) is complete re-

ducible or semisimple, if it is a direct sum of irreducible representations.

Evidently, irreducible ⇒ complete reducible, but not conversely.

Example:

G =

1 n

0 1

: n ∈ Z

, V = C2, natural action (gv is matrix multiplication).

V is not completely reducible.(Note that G not finite.)

Schur’s Lemma: If V and W are irreducible representations of G and

φ : V → W is a G-module homomorphism, then

(i) Either φ is an isomorphism, or φ = 0.

(ii) If V=W,the φ = λ · I for some λ ∈ C, I the identity.

Proof. First claim is trivial as Kerφ and Imφ are invariant subspaces. For

the second claim since C is algebraically closed φ must have an eigenvalue λ

i.e. for some λ ∈ C, φ− λI has a nonzero kernel. then by (i) φ− λI = 0, so

φ = λI.

7

Proposition 1.4.5. For any representation V of a finite group G, there is a

decomposition

V = V ⊕a11

⊕...⊕

V ⊕akk ,

where the Vi are distinct irreducible representations. The decomposition of V

into a direct sum of the k factors is unique.

Proof. Such a decomposition exists, follows from corollary 1.4.3. More pre-

cisely, if V has another decomposition, say

V = W⊕b11

⊕...⊕

W⊕bk′k′ (where W1, . . . ,Wk′ are distinct irreducible repre-

sentations), then we need to show that k = k′, and after some rearrangement

each Vi ∼= Wi&ai = bi.

Let p ∈ {1, 2, ..., ai}&q ∈ {1, 2, ..., bj} be arbitrary. Consider the com-

position of the following maps, Vili,p−−−−−→

inclusionV = V ⊕a11

⊕...⊕

V ⊕akk

V−→Id

=

W⊕b11

⊕...⊕

W⊕bk′k′

Prj.q(Projection)−−−−−−−−−−→ Wj.

This composition is a G-module homomorphism. So, by Schur’s lemma, this

composed map is either an isomorphism or a zero map. This itself implies

that each Vi is isomorphic to some Wj and hence k = k′.

Let HomG(Vi, V ) denote the space of all G-homomorphisms from Vi to V .

Similarly, HomG(Wj, V ). Using Schur lemma we can say dim(HomG(Vi, V )) =

ai and dim(HomG(Wj, V )) = bj. But since Vi is isomorphic to Wj as G-

representation, we get ai = bj. Hence after some rearrangement, we have

Vi ∼= Wi & ai = bi.

Corollary 1.4.6. Let HomG(V,W ) denote the vector space of all G-homomorphisms

V → W . If V,W are irreducible complex G-spaces, then

8

dimCHomG(V,W ) =

1 if V,W are G-isomorphic

0 otherwise

Proof. If V,W are not isomorphic then the only G-homomorphisms V → W is

0 by Schur’s lemma. Assume V ∼=G W and Θ1,Θ2 ∈ HomG(V,W ), both 6= 0.

Then Θ2 is an isomorphism by Schur’s lemma andΘ−12 Θ1 ∈ HomG(V, V ). So

Θ−12 Θ1 = λid for some λ ∈ C. Then Θ1 = λΘ2.

1.5 Applications of Schur’s Lemma

1. If G has a faithful complex irreducible representation then Z(G) is cyclic.

Proof. Let ρ : G → GL(V ) be faithful irreducible complex representation.

Let z ∈ Z(G),so zg=gz for all g ∈ G. Consider the map φz : v 7→ zv for

v ∈ V . This is a G-endomorphism on V, hence is multiplication by a scalar

µz, say(by Schur). Then the map Z(G) → C×, z 7→ µz, is a representation

of Z and is faithful(since ρ is). Thus Z(G) is isomorphic to a finite subgroup

of C×. hence is cyclic.

2. The irreducible complex representations of a finite abelian group G

are all 1-dimensional.

Proof. Let G be a finite abelian group and let ρ : G → GL(V ) be an irre-

ducible complex representation. For any fixed g ∈ G, the map ρ(g) : V → V

is a G-linear map (because G is abelian). Now since V is irreducible, we have

by Schur’s lemma that ρ(g) = λg.Id for some λg ∈ C. Since ρ(g) = λgId,

therefore every subspace of V is invariant under G. But V is irreducible,

hence it must be 1 dimensional.

9

1.6 Examples of Irreducible Representation

If G is a abelian group 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.

Thus every subspace of V is thus invariant; so that V must be one dimen-

sional. The irreducible representations of an abelian group G are thus simply

elements of the dual group, thet is , homomorphisms

ρ : G→ C∗

Now we consider the simplest nonabelian group G = S3 ,here we have two

one dimensional representations:first is trivial representation (denoted by U)

and the second one is alternating representation (denoted by U ′)defined by

ρ(g)(v) = sgn(g)(v) for g ∈ G, v ∈ C.

Since G is a permutation group, we have a natural permutation represen-

tation in which G acts on C by permuting the coordinates. Explicitly, if

{e1, e2, e3} is the standard basis, then g · e1 = eg(1) or we can write

g · (z1, z2, z3) = (zg−1(1), zg−1(2), zg−1(3)).

This representation is not irreducible. Consider the subspace W=span(1,1,1).Thus

the complementary subspace W⊥ = {(z1, z2, z3) :< (z1, z2, z3).(1, 1, 1) >=

0 i.e. z1 + z2 + z3 = 0}

W⊥ is an invariant under S3. We can easily prove that W⊥ is an irreducible

representation of S3. The proof is as follows:

Suppose there exist a non zero proper subrepresentation W ′ of W⊥. Clearly

since W⊥ has dimension 2, we must have dim(W ′) = 1. Let 0 6= wo ∈ W ′

wo = (w1, w2, w3) (say w2 6= 0)

10

let g=(1 3),then ρ(13)(w1, w2, w3) = (w3, w2, w1)

We have ρ(g) : W ′ → W ′. Since W ′ is 1-dimensional, let 0 6= wo =

(w1, w2, w3),then we can assume that {wo} is a basis of W ′. Now ρ(g)wo ∈

W ′. Since {wo} is a basis of W ′

∴ ρ(g)wo = λwo, where λ is a scalar.

(w3, w2, w1) = λ(w1, w2, w3) for some λ ∈ C

⇒ w3 = λw1, w2 = λw2, w1 = λw3

Since w2 6= 0, then λ = 1

⇒ w3 = w1 (1.1)

Since wo = (w1, w2, w3) ∈ W⊥ ∴, w1+w2+w3 = 0⇒ w2 = −w1−w3 = −2w1

Now w2 6= 0⇒ w1 6= 0

Now take g′ =(2 3)

then ρ(g′)(w1, w2, w3) = (w1, w3, w2)

by the same logic

(w1, w3, w2) = λ′(w1, w2, w3) for some λ′ ∈ C

⇒ w1 = λ′w1, w3 = λ′w2, w2 = λ′w3

Since w1 6= 0⇒ λ′ = 1

⇒ w2 = w3 (1.2)

by (1.1) and (1.2) we get w1 = w2 = w3 and w2 6= 0

wo = (w2, w2, w2) where w2 6= 0

But wo ∈ W ′ ⊂ W⊥ So we must have

w2 + w2 + w2 = 0

⇒ 3w2 = 0⇒ w2 = 0, which is contradiction.

So, it is not possible for W⊥ to have any nonzero proper subrepresentation.

11

⇒ W⊥ is an irreducible representation of S3. We call this 2-dimensional

representation is standard representation of S3

Now we look for any other arbitrary representation of S3. Let W be an

arbitrary representation of G = S3. S3 has an abelian subgroup of order 3,

say u3. Take τ to be generator of u3, the space W is spanned by eigenvectors

vi for the action of τ , whose eigenvalues are powers of a cube root of unity

ω = e2πi/3. Thus

W =⊕

Vi

where Vi = Cvi and τvi = ωαivi. Now the question arises that how the

remaining elements of S3 acts on W in terms of this decomposition. For this

, let σ be any transposition, so that τ and σ together generates S3, with the

relation στσ = τ 2. Now we see that where σ sends an eigenvector v for the

action of τ , say with the eigenvalue ωi, for this we see that how τ acts on

σ(v)

τ(σ(v)) = σ(τ 2(v)) = σ(ω2i · v) = ω2i · σ(v)

This means that if v is an eigenvector for τ with eigenvalue ωi, then σ(v) is

again an eigenvector for τ , with eigenvalue ω2i.

we can easily verify that with σ=(1 2), τ=(1 2 3), the standard representation

has a basis α=(ω, 1, ω2), β = (1, ω, ω2), with

ρ(τ)α = ωα, ρ(τ)β = ω2β, ρ(σ)α = β, ρ(σ)β = α (1.3)

Suppose now we start with such an eigenvector v for τ .

Case I: If the eigenvalue of v is ωi 6= 1, then σ(v) is an eigenvector with

eigenvalue ω2i 6= ωi, and so is independent of v, and v and σ(v) together

span a two-dimensional subspace V ′ of W invariant under S3. In fact, V ′ is

12

isomorphic to the standard representation by (1.3).

Case II: If the eigenvalue of v is1, 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 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 representation of W iso-

morphic to the trivial and alternating representation respectively.

Thus we see that the only three irreducible representations of S3 are the

trivial, alternating and standard representations U,U ′ and V. Moreover for

an arbitrary representation W of S3 we can write

W = U⊕a⊕

U ′⊕b⊕

V ⊕c; and we have a way to determine the multiplicities

a,b and c: c, for example is the number of independent eigenvectors for τ

with eigenvalue ω whereas a+c is the multiplicity of 1 as eigenvalue of σ,

and b+c is the multiplicity of -1 as an eigenvalue of σ. Suppose W is any

arbitrary representation of S3. Then by comeplete reducibility theorem, we

know that W = U⊕a ⊕ U ′⊕b ⊕ V ⊕c where ⊕ denotes direct sum, U denotes

the trivial representation, U ′ denotes the alternating representation and V

denotes the standard representation. Now take any v in W. Then we know

that v is an eigen vector of τ with eigen value either ω, or ω2, or 1. If v is an

eigen vector of τ with eigen value ω, then the coordinates of v corresponding

to U⊕a and U ′⊕b are all 0, because U and U’ are the trivial and the alternat-

ing representations respectively. So we can think of v actually as an element

of V ⊕c. Now how many linearly independent vectors are there in V ⊕c having

eigen value ω (for τ action)? For each copy of V inside V ⊕c, there is one such

vector. And there are c copies of V in V ⊕c. So the total number of linearly

13

independent eigen vectors of τ (inside V ⊕c) having eigen value ω is c. Since

any eigen vector of τ (inside W) is the same as an eigen vector of τ(inside

V ⊕c), it follows that the total number of linearly independent eigen vectors

of τ (inside W) is equal to c.

Now the proof that a+c=the multiplicity of 1 as an eigen value of σ is as

follows: For each of the a-many copies of the trivial representation U inside

W, we will get one vector (take this vector to be any generator of the one-

dimensional vector space U) which will be an eigen vector for σ having eigen

value 1 (This is simply because U is the trivial representation). So we will get

total a many linearly independent vectors (inside U⊕a) this way, which are

eigen vectors of σ having eigen value 1. For each of the b-many copies of U ′,

we will get no eigen vector of σ having eigen value 1 (This is simply because

U ′ is the alternating representation, σ acts on any vector in U ′ by taking it

to its negative). Lastly, for each of the c many copies of V, we will get one

vector v + σ(v) which will be an eigen vector for σ with eigen value 1 (This

is because σ(v+ σ(v)) = (v+ σ(v)), where v and σ(v) denote the generators

of V. This way, we will get total c many linearly independent eigen vectors

of σ (inside V ⊕c) having eigen value 1. So multiplicity of 1 as an eigen value

of σ=the total number of linearly independent eigen vectots of σ (inside W)

having eigen value 1=a+c. [a many come from U⊕a and c many come from

V ⊕c]. The proof that b+c=the multiplicity of -1 as an eigen value of σ is

similar to the proof of a+c, just replace U by U ′ and v+σ(v) by v − σ(v).

14

Chapter 2

Characters

2.1 Characters

Definition 2.1.1. If (V, ρ) is a representation of G, its charater χV is the

complex-valued function on the group defined by

χV (g) = Tr(ρ(g))|V

the trace of ρ(g) on V.

In particular, we have χV (hgh−1) = χV (g).

so that χV is constant on the conjugacy class of G, such a function is called

a class function. Note that χV (1) = dim V.

Proposition 2.1.2. Let (V, ρ) and (W, ρ′) are representations of G. Then

(i) χV ⊕W = χV + χW

(ii) χV ⊗W = χV · χW

(iii) χV ∗ = χ̄V .

15

Proof. We compute the values of these characters on a fixed element g ∈ G.

For the action of g, V has eigenvalues {λi} and W has eigenvalues {µi}. Then

λi + µj and {λi ·µj} are eigenvalues for V⊕

W and V⊗

W , from which first

two formulas follow. Similarly since ρ∗(g) = [(ρ(g))−1]t, therefore the eigen

values for g on V ∗ are λ−1i . If n is the order of g, then all the eigen values

λi are n-th roots of unity, therefore they have all modulus 1, which implies

that λ−1i = λ̄i. Hence {λ−1i = λ̄i} are the eigenvalues for g on V ∗, from this

the third equality follows.

The original fixed-point formula : If (V, ρ) is the permutation repre-

sentation associated to the action of a group G on a finite set X, Then χV (g)

is the number of elements of X fixed by G.

Proof: If (V, ρ) is the permutation representation associated to the action

of a finite group G on a finite set X. Then V=C-span of {ex|x ∈ X} and

ρ : G→ GL(V ) is given by

ρ(g)(∑

x∈X axex) =∑

x∈X axegx.

clearly, for any fixed g ∈ G the matrix of the invertible linear map

ρ(g) : V → V with respect to the basis {ex|x ∈ X} of V will have the

following property :-

In the i-th column, (where i ∈ X) only the entry corresponding to the

position g · i equals 1, all other entries are zero.

Hence the only non-zero elements on the diagonal of the matrix of ρ(g) will

correspond to those x ∈ X for which gx=x, and these non-zero matrix entries

are all equal to 1.

this implies that Tr(ρ(g)|V ) = #{x ∈ X|gx = x}= No. of elements of X

16

fixed by g.

Examples: We compute the character table of S3. to begin with, the trivial

representation takes values (1,1,1) on the three conjugacy classes [1],[(1 2)],

and [(1 2 3)], whereas the alternating representation has values (1,-1,1). To

see the character of the standard representation note that the permutation

representation decomposes: C3=U⊕

V ; since the character of the permu-

tation representation has, (by the original fixed point theorem) the values

(3,1,0), we have χV = χC3 − χU = (3, 1, 0) − (1, 1, 1) = (2, 0,−1). The

charater table of S3.

S3 1 3 2

1 (12) (123)

trivial 1 1 1

alternating U ′ 1 -1 1

standard V 2 0 -1

2.2 The First Projection Formula and Its Con-

sequences

Let (V, ρ) be a representation of of G. We know that V = V ⊕a11

⊕V ⊕a22

⊕...⊕

V ⊕ann ,

where V1, V2, ... are irreducible representations. Out of these V1, V2, ..., Vn one

has to be the trivial representation, call it Vk. Now we will find out that how

many copies (ak) of Vk appear in the above decomposition of V. Set

V G = {v ∈ V : gv = v ∀g ∈ G}, is a vector subspace of V and V G is also

a subrepresentation of (V, ρ). Clearly V G = V ⊕akk and since each Vk is one

17

dimensional, we have dim(V G) = ak.

We will construct a linear map φ : V → V G which will be the projection

onto V G.

Define φ ∈ End(V ) as

φ =1

|G|Σg∈Gρ(g).

Observe that φ is G-linear since Σg∈Gρ(g) = Σg∈Gρ(h)ρ(g)ρ(h−1) for every

h ∈ G.

Proposition 2.2.1. The map φ is a projection of V onto V G.

Proof. First suppose that v = φ(w) = 1|G|

∑ρ(g)w. Then, for any h ∈ G,

ρ(h)v = 1|G|

∑ρ(hg)w = 1

|G|∑ρ(g)w. So the image of φ is contained in

V G. Conversely, if v ∈ V G, then φ(v) = 1|G|

∑v = v, so V G ⊂ Im(φ), and

φ ◦ φ = φ.

Note: Since φ : V → V is the projection onto V G , therefore Trace(φ|V ) =

Trace(φ|V G). But since φ is a projection onto V G, therefore φ|V G = id|V G .

So Trace(φ|V G) = dim(V G).

Now we have a way of finding explicitly the direct sum of the trivial

subrepresentations of a given representation, although the formula can be

hard to use if it does not simplify. If we just know the number ak of copies

of the trivial representation Vk appearing in the decomposition of V , we can

do this numerically, since this number will be just the trace of the projection

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φ. We have

ak = dim(V G) = Trace(φ|V G) = Trace(φ|V ) =1

|G|∑g∈G

Trace(ρg) =1

|G|∑g∈G

χV (g).

(2.1)

In particular, we observe that for an irreducible representation V other than

the trivial one , the sum over all g ∈ G of the values of the character χV is

zero.

We know that if (V, ρ) and (W, ρ′) are representation of G, then Hom(V,W )

is also a representation of G.

Hom(V,W )G={G-module homomorphisms from V to W}.

If V is irreducible then by Schur’s lemma dim Hom(V,W )G is the multi-

plicity of V in W . Similarly, if W is irreducible, then by Schur s lemma

dim(Hom(V,W )G) is the multiplicity of W in V , and in the case where both

V and W are irreducible, we have

dimHom(V,W )G =

1 if V ∼= W

0 if V � W.

But now the character χHom(V,W ) of the representation Hom(V,W ) ≈

V ∗⊗

W is given by

χHom(V,W )(g) = χV (g) · χW (g).

Now we can apply formula (2.1) to obtain

1

|G|∑g∈G

χV (g) · χW (g) =

1 if V ∼= W

0 if V � W.(2.2)

To express this, let

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Cclass(G) = {class functions on G}

and define an Hermitian inner product on Cclass(G) by

(α, β) =1

|G|∑g∈G

α(g)β(g). (2.3)

The following theorem now follows from 2.2:

Theorem 2.2.2. In terms of an inner product, the characters of the irre-

ducible representations of G are orthonormal.

Corollary 2.2.3. The number of irreducible representations of G is less than

or equal to the number of conjugacy classes.

Proof. Since the indicator functions of the conjugacy classes of G form a

basis for Cclass(G), therefore a maximal linearly independent set in Cclass(G)

has N many elements, where N denotes the number of distinct conjugacy

classes of G. It follows from the above theorem that characters of the distinct

irreducible representations of G form an orthonormal (hence linearly inde-

pendent) subset of Cclass(G). Therefore the total number of such (distinct

irreducible) representations must be less than or equal to the cardinality of

a maximal linearly independent set.

Theorem 2.2.4. The number of irreducible representations of G is in fact

equal to the number of conjugacy classes.

Proof. We have already shown that the χV i are orthonormal where Viis are

irreducible representation, so so it remains to show that they span Cclass(G).

To do this, suppose that µ ∈ Cclass(G) is orthogonal to each χV i that is

(µ, χV i) = 0 or all i with 1 ≤ i ≤ n. We want to show that µ must be the

20

zero function.

Given a representation (W, ρ′) of G, let

ρ′µ =∑

g∈G µ(g)ρ′(g).

Now suppose that (W, ρ′) is an irreducible representation (that is ρ′ = ρi for

some i). We show that ρ′µ = 0 Because each ρ′ can be written as a direct

sum of irreducible representations, any ρ (irreducible or not) must be the

zero map. To show that ρ′ being irreducible implies ρ′µ = 0 we show that ρ′µ

is a G-linear map and then apply Schur’s lemma. For each h ∈ G we have

ρ′µρ′(h) =

∑g∈G µ(g)ρ′(g)ρ′(h) =

∑g∈G µ(g)ρ′(gh).

Because µ is a class function µ(xy) = µ(yx). Therefore ,∑g∈G µ(g)ρ′(gh) =

∑r∈G µ(rh−1)ρ′(r) =

∑r∈G µ(h−1r)ρ′(r) =

∑u∈G µ(u)ρ′(hu).

=∑

u∈G µ(u)ρ′(h)ρ′(h) = ρ′(h)ρµ

for r=gh and u = h−1r. Therefore ρ′µ is a G-linear function. By Schur’s

lemma ρ′µ = λ · I for some eigenvalue λ. Now, Trace(λ · I) = kλ and

Trace(ρ′µ) =∑

g∈G µ(g)Trace(ρ′(g)). Therefore,

λ = 1k

∑g∈G µ(g)χW (g) = |G|

k(µ, χW ) = 0

So ρ′µ = 0 as we know that ρ′µ = 0 for each representation (W, ρ′).

Specifically if (W, ρ′) is the regular representation and {eg|g ∈ G} is a basis

for W, then

0 = 0 · e1 =∑

g∈G µ(g)ρ′(g)e1 =∑

g∈G µ(g)eg.

Since the eg are linearly independent,µ(g) = 0 for all g ∈ G.

Therefore {χVi |1 ≤ i ≤ n} is an orthonormal basis for Cclass(G).

Corollary 2.2.5. Any representation is determined by its character.

Proof. As we know that V ∼= V ⊕a11

⊕...⊕

V ⊕akk , where Vi are irreducible

subrepresentation, then χV =∑aiχVi and the χVi are linearly independent.

21

Therefore the ai are determined by the coefficients (when the character of any

representation V is written as a linear combination of the basis {χVi}).

Corollary 2.2.6. A representation V is irreducible if and only if (χV , χV ) =

1

Proof. Since we can write V ∼= V ⊕a11

⊕...⊕

V ⊕akk , then (χV , χV ) =∑a2i .

But since V is irreducible, we must have that exactly one of the ai s is one

and all others are zero. Therefore (χV , χV ) = 1.

Corollary 2.2.7. The multiplicity ai of Vi in V is the inner product of χV

with χVi, i.e. (χV , χVi).

Proof. As we know that the characters of irreducible representations of G

are orthonormal. Therefore by the previous corollary it is clear that ai =

(χV , χVi).

Let R be the regular representation of G. Then by the original fixed-point

formula χR(g) = the number of elements of G fixed by g.

χR(g) =

0 if g 6= e

|G| if g = e.

by equation 2.3 we have

(χR, χR) = 1|G|

∑g∈G χR(g)χR(g)

= 0 + 1|G|χR(e)χR(e) = |G|

therefore by corollary 2.2.6 R is irreducible iff |G| = 1 i.e.,iff G = {e}.

22

Thus if we set R =⊕

V ⊕aii where Vi are all the distinct irreducible repre-

sentation of G, then

ai = (χVi , χR) = 1|G|χVi(e) · |G| = dimVi.

Therefore we conclude that

Corollary 2.2.8. Any irreducible representation V of G appears in the reg-

ular representation dim V times.

Since the regular representation R of a finite group G is finite dimen-

sional, we get that given a finite group G, the set of all distinct irreducible

representation of G is a finite set. Any irreducible representation Vi appears

in R (Regular representation) dim(Vi) times.

Since R =⊕n

i=1 V⊕dimVii and dim(R) = |G|, we have |G| = dim(R) =

n∑i=1

dim(Vi)2

Now since R =⊕n

i=1 V⊕dimVii

χR =n∑i=1

(dimVi)χVi

now evaluate this at some (6= e)g ∈ G

0 = χR(g) =n∑i=1

(dimVi)χVi(g).

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Bibliography

[1] Representation Theory A First Course, William Fulton and Joe Harris,

Graduate texts in Mathematics, Springer,1991.

[2] Algebra, Second edition, Michael Artin, PHI learning Private limited,

2011.

[3] Lecture notes on Representation theory, S.Martin, Lent term

2009,2010,2011.

[4] Some elementary results in representation theory, Isaac Ottoni Wilhelm.

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