following is an important example.

Let X = n = {1, 2, . . . , n} and G = Sn , the symmetric group of degree n, acting on n in the

usual way. We may take as a basis for k[n], the functions δj (1 j n) given by

1 if k = j,

δj (k) =

0 else.

Relative to this basis, the action of σ ∈ Sn is given by the n — n matrix [σ] whose ij th entry is

1 if i = σ(j),

(6.2) [σ]ij =

0 else.

Taking n = 3, we get

® ® ®

010 001 010

[(132)] = °0 0 1» , [(13)] = °0 1 0» , [(132)(13)] = [(12)] = °1 0 0» .

100 100 001

As expected, we also have

® ® ®

010 001 010

[(132)][(13)] = °0 0 1» °0 1 0» = °1 0 0» = [(132)(13)].

100 100 001

An important fact about permutation representations is the following, which makes their char-

acters easy to calculate.

Proposition 3.22. Let X be a ¬nite G-set, and k[X] the associated permutation represen-

tation. Let g ∈ G and ρg : k[X] ’’ k[X] be the linear transformation induced by g. Then

Tr ρg = |X g | = |{x ∈ X : gx = x}| = number of elements of X ¬xed by g.

Proof. Take the elements of X as a basis for k[X]. Then Tr ρg is the sum of the diagonal

terms in the matrix [ρg ] relative to this basis. Hence

Tr ρg = number of non-zero diagonal terms in [ρg ]

= number of elements of X ¬xed by g

by Equation (6.1).

Our next result which that permutation representations are self-dual.

Proposition 3.23. Let X be a ¬nite G-set, and k[X] the associated permutation represen-

tation. Then there is a G-isomorphism k[X] ∼ k[X]— .

=

7. GENERALIZED PERMUTATION REPRESENTATIONS 33

Proof. Take as a basis of k[X] the elements x ∈ X. Then a basis for the dual space k[X]—

consists of the elements x— . By de¬nition of the action of G on k[X]— = Hom| (k[X], k), we have

(g · x— )(y) = x— (g ’1 x) (g ∈ G, y ∈ X).

A familiar calculation shows that g · x— = (gx)— , and so this basis is also permuted by G. Now

de¬ne a function • : k[X] ’’ k[X]— by

ax x — .

•( ax x) =

x∈X x∈X

This is a k-linear isomorphism also satisfying

ax (gx)— = g · •

•g ax x =• ax (gx) = ax x .

x∈X x∈X x∈X x∈X

Hence • is a G-isomorphism.

7. Generalized permutation representations

It is useful to generalize the notion of permutation representation somewhat. Let V be a

¬nite dimensional k-vector space with a representation of G, ρ : G ’’ GL| (V ); we will usually

write gv = ρg v. We can consider the set of all functions X ’’ V , Map(X, V ), and this is also

a ¬nite dimensional k-vector space with addition and scalar multiplication de¬ned by

(f1 + f2 )(x) = f1 (x) + f2 (x), (tf )(x) = t(f (x)),

for f1 , f2 , f ∈ Map(X, V ), t ∈ k and x ∈ X. There is a representation of G on Map(X, V ) given

by

(g · f )(x) = gf (g ’1 x).

We call this a generalized permutation representation of G.

Proposition 3.24. Let Map(X, V ) be a permutation representation of G, where V has basis

v = {v1 , . . . , vn }. Then the functions δx,j : X ’’ V (x ∈ X, 1 j n) given by

vj if y = x,

δx,j (y) =

0 otherwise,

for y ∈ X, form a basis for Map(X, V ). Hence,

dim| Map(X, V ) = |X| dim| V.

Proof. Let f : X ’’ V . Then for any y ∈ X,

n

f (y) = fj (y)vj ,

j=1

where fj : X ’’ k is a function. It su¬ces now to show that any function h : X ’’ k has a

unique expression as

h= h x δx

x∈X

where hx ∈ k and δx : X ’’ k is given by

1 if y = x,

δx (y) =

0 otherwise.

But for y ∈ X,

h(y) = hx δx (y) ⇐’ h(y) = hy .

x∈X

34 3. REPRESENTATIONS OF FINITE GROUPS

Hence h = h(x)δx is the unique expansion of this form. Combining this with the above

x∈X

we have

n n

f (y) = fj (y)vj = fj (x)δx (y)vj ,

j=1 j=1 x∈X

and so

n

f= fj (x)δxj ,

j=1 x∈X

is the unique such expression, since δxj (y) = δx (y)vj .

Proposition 3.25. If V = V1 • V2 is a direct sum of representations V1 , V2 , then there is a

G-isomorphism

Map(X, V ) ∼ Map(X, V1 ) • Map(X, V2 ).

=

Proof. Recall that every v ∈ V has a unique expression of the form

v = v1 + v2 .

De¬ne a function Map(X, V ) ’’ Map(X, V1 ) • Map(X, V2 ) by

f ’’ f1 + f2

where f1 : X ’’ V1 and f2 : X ’’ V2 satisfy

f (x) = f1 (x) + f2 (x) (x ∈ X).

This is easily seen to be both a linear isomorphism and a G-homomorphism, hence a G-

isomorphism.

Proposition 3.26. Let X = X1 X2 where X1 , X2 ⊆ X are closed under the action of G.

Then there is a G-isomorphism

Map(X, V ) ∼ Map(X1 , V ) • Map(X2 , V ).

=

Proof. Let j1 : X1 ’’ X and j2 : X2 ’’ X be the inclusion maps, which are G-maps.

Then given f : X ’’ V , we have two functions fk : X ’’ V (k = 1, 2) given by

f (x) if x ∈ Xk ,

fk (x) =

0 else.

De¬ne a function Map(X, V ) ∼ Map(X1 , V ) ’’ Map(X2 , V ) by

=

f ’’ f1 + f2 .

This is easily seen to be a linear isomorphism. We have

(g · f )k = g · fk ,

using the fact that Xk is closed under the action of G, hence

g · (f1 + f2 ) = g · f1 + g · f2 .

Thus this map is a G-isomorphism.

These results tell us how to reduce an arbitrary generalized permutation representation to

a direct sum of those induced from a transitive G-set X and an irreducible representation V .

CHAPTER 4

Character theory

1. Characters and class functions on a ¬nite group

Let G be a ¬nite group and ρ : G ’’ GLC (V ) a ¬nite dimensional C-representation of

dimension dimC V = n. For g ∈ G, the linear transformation ρg : V ’’ V will sometimes be

written g· or g. The character of g in the representation ρ is the trace of g on V , i.e.,

χρ (g) = Tr ρg = Tr g.

We can view χρ as a function χρ : G ’’ C, the character of the representation ρ.

Definition 4.1. A function θ : G ’’ C is a class function if for all g, h ∈ G,

θ(hgh’1 ) = θ(g),

i.e., θ is constant on each conjugacy class of G.

Proposition 4.2. For all g, h ∈ G,

χρ (hgh’1 ) = χρ (g).

Hence χρ : G ’’ C is a class function on G.

Proof. We have

ρhgh’1 = ρh —¦ ρg —¦ ρh’1 = ρh —¦ ρg —¦ ρ’1

h

and so

χρ (hgh’1 ) = Tr ρh —¦ ρg —¦ ρ’1 = Tr ρg = χρ (g).

h

Example 4.3. Let G = S3 act on the set 3 = {1, 2, 3} in the usual way. Let V = C[3] be

the associated permutation representation over C, where we take as a basis e = {e1 , e2 , e3 } with

action

σ · ej = eσ(j) .

Let us determine the character of this representation ρ : S3 ’’ GLC (V ).

The elements of S3 written using cycle notation are the following:

1, (12), (23), (13), (123), (132).

The matrices of these elements with respect to e are

® ® ® ® ®

0 10 100 0 01 001 010

I3 , °1 0 0» , °0 0 1» , °0 1 0» , °1 0 0» , °0 0 1» .

0 01 010 1 00 010 100

Taking traces we obtain

χρ (1) = 3, χρ (12) = χρ (23) = χρ (13) = 1, χρ (123) = χρ (132) = 0.

Notice that we have χρ (g) ∈ Z. Indeed, by Proposition 3.22 we have

Proposition 4.4. Let X be a G-set and ρ the associated permutation representation on

C[X]. Then

χρ (g) = |X g | = |{x ∈ X : g · x = x}| = the number of elements of X ¬xed by g.

35

36 4. CHARACTER THEORY

The next result sheds further light on the signi¬cance of the character of a G-representation

over the complex number ¬eld C. This makes use of the linear algebra developed in Section 3

of Chapter 1.

Theorem 4.5. For g ∈ G, there is a basis v = {v1 , . . . , vn } of V consisting of eigenvectors

of the linear transformation g.

Proof. Let d = |g|, the order of g. For v ∈ V ,

(g d ’ IdV )(v) = 0.

Now we can apply Lemma 1.14 with the polynomial f (X) = X d ’ 1, which has d distinct roots

in C.

There may well be a smaller degree polynomial identity satis¬ed by the linear transformation

g on V . However, if a polynomial f (X) satis¬ed by g has deg f (X) d and no repeated linear

factors, then f (X)|(X d ’ 1).

Corollary 4.6. The distinct eigenvalues of the linear transformation g on V are dth roots

of unity. More precisely, if d0 is the smallest natural number such that for all v ∈ V ,

(g d0 ’ IdV )(v) = 0,

then the distinct eigenvalues of g are d0 th roots of unity.

Proof. An eigenvalue » (with eigenvector v» = 0) of g satis¬es

(g d ’ IdV )(v» ) = 0,

hence

(»d ’ 1)v» = 0.

Corollary 4.7. For any g ∈ G we have

n

χρ (g) = »j

j=1

where »1 , . . . , »n are the n eigenvalues of ρg on V , including repetitions.

Corollary 4.8. For g ∈ G we have

χρ (g ’1 ) = χρ (g) = χρ— (g).

Proof. If the eigenvalues of ρg including repetitions are »1 , . . . , »n , then the eigenvalues of

ρg’1 including repetitions are easily seen to be »’1 , . . . , »’1 . But if ζ is a root of unity, then

n

1

’1 = ζ, and so χ (g ’1 ) = χ (g). The second equality follows from Proposition 3.12.

ζ ρ ρ

Now let us return to the idea of functions on a group which are invariant under conjugation.

Denote by Gc the set G and let G act on it by conjugation,

g · x = gxg ’1 .

The set of all functions Gc ’’ C, Map(Gc , C) has an action of G given by

(g · ±)(x) = ±(gxg ’1 )

for ± ∈ Map(Gc , C), g ∈ G and x ∈ Gc . Then the class functions are those which are invari-

ant under conjugation and hence form the set Map(Gc , C)G which is a C-vector subspace of

Map(Gc , C).

2. PROPERTIES OF CHARACTERS 37

Proposition 4.9. The C-vector space Map(Gc , C)G has as a basis the set of functions

∆C : Gc ’’ C for C a conjugacy class in G, determined by

1 if x ∈ C,

∆C (x) =

0 if x ∈ C.

/

Thus dimC Map(Gc , C)G is the number of conjugacy classes in G.

Proof. By the proof of Theorem 3.16, a class function ± : Gc ’’ C function can be

uniquely expressed in the form

±= a x δx

x∈Gc

for suitable ax ∈ Gc . But

g·±= ax (g · δx ) = ax δgxg’1 = agxg’1 δx .

x∈Gc x∈Gc x∈Gc

Hence by uniqueness and the de¬nition of class function, we must have

agxg’1 = ax (g ∈ G, x ∈ Gc ).

Hence,

±= aC δx ,

C x∈C

where for each conjugacy class C we choose any element c0 ∈ C and put aC = ac0 . Here the

outer sum is over all the conjugacy classes C of G. We now ¬nd that

∆C = δx

x∈C