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the value 1. Thus there is exactly one 1 in each row and column, and 0™s everywhere else. The
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

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