It is easily veri¬ed that

µ+ („ · v) = µ+ (v), µ’ („ · v) = ’µ’ (v).

We take V+ = im µ+ and V’ = im µ’ and the direct sum decomposition follows from the identity

v = µ+ (v) + µ’ (v).

2. G-HOMOMORPHISMS AND IRREDUCIBLE REPRESENTATIONS 23

The decomposition in this example corresponds to the two distinct irreducible representa-

tions of Z/2. Later we will see (at least over the complex numbers C) that there is always such a

decomposition of a representation of a ¬nite group G with factors corresponding to the distinct

irreducible representations of G.

Example 3.5. Let D2n be the dihedral group of order 2n described in Section 7.2. This

group is generated by elements ± of order n and β of order 2, subject to the relation

β±β = ±’1 .

We can realise D2n as the symmetry group of the regular n-gon centred at the origin and with

vertices on the unit circle (we take the ¬rst vertex to be (1, 0)). It is easily checked that relative

the standard basis {e1 , e2 } of R2 , we get

cos 2rπ/n ’ sin 2rπ/n cos 2rπ/n ’ sin 2rπ/n

±r = β±r =

sin 2rπ/n cos 2rπ/n ’ sin 2rπ/n ’ cos 2rπ/n

for r = 0, . . . , (n ’ 1).

Thus we have a 2-dimensional representation ρR of D2n over R, where the matrices of ρR (±r )

and ρR (β±r ) are given by the above. We can also view R2 as a subset of C2 and interpret these

matrices as having coe¬cients in C. Thus we obtain a 2-dimensional complex representation

ρC of D2n with the above matrices relative to the C-basis {e1 , e2 }.

2. G-homomorphisms and irreducible representations

Suppose that we have two representations ρ : G ’’ GL| (V ) and σ : G ’’ GL| (W ). Then

a linear transformation f : V ’’ W is called G-equivariant, G-linear or a G-homomorphism

with respect to ρ and σ, if for each g ∈ G the diagram

f

V ’’’ W

’’

¦ ¦

¦ ¦σg

ρg

f

V ’’’ W

’’

commutes, i.e., σg —¦ f = f —¦ ρg or equivalently, σg —¦ f —¦ ρg’1 = f . A G-homomorphism which is

a linear isomorphism is called a G-isomorphism or G-equivalence and we say that the represen-

tations are G-isomorphic or G-equivalent.

We de¬ne an action of G on Hom| (V, W ), the vector space of k-linear transformations

V ’’ W , by

(g · f )(v) = σg f (ρg’1 v) (f ∈ Hom| (V, W )).

This is another G-representation. The G-invariant subspace HomG (V, W ) = Hom| (V, W )G is

then equal to the set of all G-homomorphisms.

If the only G-subspaces of V are {0} and V , ρ is called irreducible or simple.

Given a subrepresentation W ⊆ V , the quotient vector space V /W also admits a linear

action of G, ρW : G ’’ GL| (V /W ), the quotient representation, where

ρW (g)(v + W ) = ρ(g)(v) + W,

which is well de¬ned since whenever v ’ v ∈ W ,

ρ(g)(v ) + W = ρ(g)(v + (v ’ v)) + W = ρ(g)(v) + ρ(g)(v ’ v)) + W = ρ(g)(v) + W.

Proposition 3.6. If f : V ’’ W is a G-homomorphism, then

a) ker f is a G-subspace of V ;

b) im f is a G-subspace of W .

24 3. REPRESENTATIONS OF FINITE GROUPS

Proof. (a) Let v ∈ ker f . Then for g ∈ G,

f (ρg v) = σg f (v) = 0,

so ρg v ∈ ker f . Hence ker f is a G-subspace of V

(b) Let w ∈ im f with w = f (u) for some u ∈ V . Now

σg w = σg f (u) = f (ρg u) ∈ im f,

hence im f is a G-subspace of W .

Theorem 3.7 (Schur™s Lemma). Let ρ : G ’’ GLC (V ) and σ : G ’’ GLC (W ) be irred-

ucible representations of G over the ¬eld C, and let f : V ’’ W be a G-linear map.

a) If f is not the zero map, then f is an isomorphism.

[Remark: Part (a) holds for any ¬eld k.]

b) If V = W and ρ = σ, then f has the form

f (v) = »v (v ∈ V )

form some » ∈ C.

Proof. (a) Proposition 3.6 implies that ker f ⊆ V and im f ⊆ W are G-subspaces. By the

irreducibility of V , either ker f = V (in which case f is the zero map) or ker f = {0} in which

case f is injective. Similarly, irreducibility of W implies that im f = {0} (in which case f is the

zero map) or im f = W in which case f is surjective. Thus if f is not the zero map it must be

an isomorphism.

(b) Let » ∈ C be an eigenvalue of f , with eigenvector v0 = 0. Let f» : V ’’ V be the linear

transformation for which

f» (v) = f (v) ’ »v (v ∈ V ).

For g ∈ G,

ρg f» (v) = ρg f (v) ’ ρg »v

= f (ρg v) ’ »ρg v,

= f» (ρg v),

showing that f» is G-linear. Since f» (v0 ) = 0, Proposition 3.6 shows that ker f» = V . As

dimC V = dimC ker f» + dimC im f» ,

we see that im f» = {0} and so

f» (v) = 0 (v ∈ V ).

A linear transformation f : V ’’ V is sometimes called a homothety if it has the form

f (v) = »v (v ∈ V ).

In this proof, it is essential that we take k = C rather than k = R for example, since we

need the fact that every polynomial over C has a root to guarantee that linear transformations

V ’’ V always have eigenvalues. This theorem can fail to hold for representations over R as

the next example shows.

Example 3.8. Let k = R and V = C considered as a 2-dimensional R-vector space. Let

G = µ4 = {1, ’1, i, ’i}

be the group of all 4th roots of unity with ρ : µ4 ’’ GL| (V ) given by

ρ± z = ±z.

2. G-HOMOMORPHISMS AND IRREDUCIBLE REPRESENTATIONS 25

Then this de¬nes a 2-dimensional representation of G over R. If we use the basis {u = 1, v = i},

then

ρi u = v, ρi v = ’u.

From this we see that any G-subspace of V containing a non-zero element w = au + bv also

contains ’bu + av, and hence it must be all of V (exercise). So V is irreducible.

But the linear transformation • : V ’’ V given by

•(au + bv) = ’bu + av = ρi (au + bv)

is G-linear, but not the same as multiplication by a real number (this is left as an exercise).

Theorem 3.9 (Maschke™s Theorem). Let V be a k-vector space and ρ : G ’’ GL| (V ) a

k-representation. Let W ⊆ V be a G-subspace of V . Then there is a projection onto W which is

G-equivariant. Equivalently, there is a linear complement W of W which is also a G-subspace.

Proof. Let p : V ’’ V be a projection onto W . De¬ne a linear transformation p0 : V ’’

V by

1

ρg —¦ p —¦ ρ’1 (v).

p0 (v) = g

|G|

g∈G

Then for v ∈ V ,

ρg —¦ p —¦ ρ’1 (v) ∈ W

g

since im p = W and W is a G-subspace; hence p0 (v) ∈ W . We also have

1

ρh p(ρ’1 ρg v)

p0 (ρg v) = h

|G|

h∈G

1

ρg ρg’1 h p(ρ’1 h v)

= g ’1

|G|

h∈G

1

ρg’1 h p(ρ’1 h v)

= ρg g ’1

|G|

h∈G

1

= ρg ρh p(ρh’1 v)

|G|

h∈G

= ρg p0 (v),

which shows that p0 is G-equivariant. If w ∈ W ,

1

p0 (w) = ρg p(ρg’1 w)

|G|

g∈G

1

= ρg ρg’1 w

|G|

g∈G

1

= w

|G|

g∈G

1

= (|G|w) = w.

|G|

Hence p0 |W = IdW , showing that p0 has image W .

Now consider W = ker p0 , which is a G-subspace by part (a) of Proposition 3.6. This

is a linear complement for W since given the quotient map q : V ’’ V /W , if v ∈ W then

q(v) = 0 + W implies v ∈ W © W and hence 0 = p0 (v) = v.

26 3. REPRESENTATIONS OF FINITE GROUPS

Theorem 3.10. Let ρ : G ’’ GL| (V ) be a linear representation of a ¬nite group with V

non-trivial. Then there are G-spaces U1 , . . . , Ur ⊆ V , each of which is a non-trivial irreducible

subrepresentation and

V = U1 • · · · • Ur .

Proof. We proceed by Induction on n = dim| V . If n = 1, the result is true with U1 = V .

So assume that the result holds whenever dim| V < n. Now either V is irreducible or there

is a proper G-subspace U1 ⊆ V . By Theorem 3.9, there is a G-complement U1 of U1 in V with

dim| U1 < n. By the Inductive Hypothesis there are irreducible G-subspaces U2 , . . . , Ur ⊆ U1 ⊆

V for which

U1 = U2 • · · · • Ur ,

and so we ¬nd

V = U1 • U2 • · · · • Ur .

We will see later that given any two such collections of non-trivial irreducible subrepresent-

ations U1 , . . . , Ur and W1 , . . . , Ws , we have s = r and for each k, the number of Wj G-isomorphic

to Uk is equal to the number of Uj G-isomorphic to Uk . The proof of this will use characters,

which give further information such as the multiplicity of each irreducible which occurs as a sum-

mand in V . The irreducible representations Uk are called the irreducible factors or summands

of the representation V .

An important example of a G-subspace of any representation ρ on V is the G-invariant

subspace

V G = {v ∈ V : ρg v = v ∀g ∈ G}.

We can construct a projection map V ’’ V G which is G-linear, provided that the characteristic

of k does not divide |G|. In practice, we will be mainly interested in the case where k = C, so

in this section from now on, we will assume that k has characteristic 0 .

Proposition 3.11. Let µ : V ’’ V be the k-linear transformation de¬ned by

1

µ(v) = ρg v.

|G|

g∈G

The following hold.

a) For g ∈ G and v ∈ V , ρg µ(v) = µ(v).

b) µ is G-linear.

c) For v ∈ V G , µ(v) = v. Hence, im µ = V G .

Proof. (a) Let g ∈ G and v ∈ V . Then

1 1 1 1

ρg µ(v) = ρg ρh v = ρg ρh v = ρgh v = ρh v = µ(v).

|G| |G| |G| |G|

h∈G h∈G h∈G h∈G

(b) Similarly, for g ∈ G and v ∈ V ,

1 1 1

µ(ρg v) = ρh (ρg v) = ρhg v = ρk v = µ(v).

|G| |G| |G|

h∈G h∈G k∈G

By (a), this agrees with ρg µ(v). Hence, µ is G-linear.

(c) For v ∈ V G ,

1 1 1

µ(v) = ρg v = v= |G|v = v.

|G| |G| |G|

g∈G g∈G

This also shows that im µ = V G .

3. NEW REPRESENTATIONS FROM OLD 27

3. New representations from old

Let G be a ¬nite group and k a ¬eld. In this section we will see how new representations

can be manufactured out of existing ones. As well as allowing interesting new examples to be

constructed, this sometimes gives ways of understanding representations in terms of familiar

ones. This will be important when we have learnt how to decompose representations in terms

of irreducibles and indeed is sometimes used to construct the latter.

Let V1 , . . . , Vr be k-vector spaces carrying representations ρ1 , . . . , ρr of G. Then for each

g ∈ G and each j, we have the corresponding linear transformation ρj g : Vj ’’ Vj . Using

Proposition 1.18 we obtain a unique linear transformation

ρ1 g — · · · — ρr g : V1 — · · · — Vr ’’ V1 — · · · — Vr .

It is easy to verify that this gives a representation of G on the tensor product V1 —· · ·—Vr , called

the tensor product of the original representations. By Proposition 1.18 we have the formula

(3.1) ρ1 g — · · · — ρr g (v1 — · · · — vr ) = ρ1 g v1 — · · · — ρr g vr

for vj ∈ Vj (j = 1, . . . , r).

Let V, W be k-vector spaces supporting representations ρ : G ’’ GL| (V ) and σ : G ’’

GL| (W ). Recall that the set of all linear transformations V ’’ W is the k-vector space

Hom| (V, W ) with addition and multiplication given by the formul¦

(• + θ)(u) = •(u) + θ(u),

(t•)(u) = t(•(u)) = •(tu)

for •, θ ∈ Hom| (V, W ) and t ∈ k.

We can also de¬ne an action of G on Hom| (V, W ) by

(„g •)(u) = σg •(ρg’1 u).

This turns out to be a linear representation of G on Hom| (V, W ).

As a particular example of this, taking W = k with the trivial action of G (i.e., σg = Id| ),

we obtain an action of G on the dual of V , V — = Hom| (V, k), the contragredient representation

ρ— . Explicitly,

ρ— • = • —¦ ρg’1 .