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and the rest of the proof is straightforward.

We will see that the characters of non-isomorphic irreducible representations of G also form
a basis of Map(Gc , C)G . We will set C(G) = Map(Gc , C)G .

2. Properties of characters
In this section we will see some other important properties of characters.

Theorem 4.10. Let G be a ¬nite group with ¬nite dimensional complex representations
ρ : G ’’ GLC (V ) and σ : G ’’ GLC (W ). Then
a) χρ (e) = dimC V and for g ∈ G, |χρ (g)| χρ (e).
b) The tensor product representation ρ — σ has character
χρ—σ = χρ χσ ,
i.e., for g ∈ G,
χρ—σ (g) = χρ (g)χσ (g).
c) Let „ : G ’’ GLC (U ) be a representation which is G-isomorphic to the direct sum of
ρ and σ, so U ∼ V • W . Then
=
χ„ = χρ + χσ ,
i.e., for g ∈ G,
χ„ (g) = χρ (g) + χσ (g).

Proof.
38 4. CHARACTER THEORY

a) The ¬rst statement is immediate from the de¬nition. For the second, using Theo-
rem 4.5, we may choose a basis v = {v1 , . . . , vr } of V for which ρg vk = »k vk , where »k
is a root of unity (hence satis¬es |»k | = 1). Then
r r
|χρ (g)| = | »k | |»k | = r = χρ (e).
k=1 k=1

b) Let g ∈ G. By Theorem 4.5 we can ¬nd bases v = {v1 , . . . , vr } and w = {w1 , . . . , ws }
for V and W consisting of eigenvectors for ρg and σg with corresponding eigenvalues
»1 , . . . , »r and µ1 , . . . , µs . The elements vi — wj form a basis for V — W and by the
formula of Equation (3.1), the action of g on these vectors is given by
(ρ — σ)g · (vi — wj ) = »i µj vi — wj .
Finally Corollary 4.7 implies

Tr(ρ — σ)g = »i µj = χρ (g)χσ (g).
i,j

c) For g ∈ G, choose bases v = {v1 , . . . , vr } and w = {w1 , . . . , ws } for V and W consisting
of eigenvectors for ρg and σg with corresponding eigenvalues »1 , . . . , »r and µ1 , . . . , µs .
Then v ∪ w = {v1 , . . . , vr , w1 , . . . , ws } is a basis for U consisting of eigenvectors for „g
with the above eigenvalues. Then
χ„ (g) = Tr „g = »1 + · · · + »r + µ1 + · · · + µs = χρ (g) + χσ (g).

3. Inner products of characters
In this section we will discuss a way to ˜compare™ characters, using a scalar or inner product
on the vector space of class functions C(G). In particular, we will see that the character of
a representation determines it up to a G-isomorphism. We will again work over the ¬eld of
complex numbers C.
We begin with the notion of scalar or inner product on a ¬nite dimensional C-vector space
V . A function ( | ) : V — V ’’ C is called a hermitian inner or scalar product on V if for
v, v1 , v2 , w ∈ V and z1 , z2 ∈ C,
(LLin) (z1 v1 + z2 v2 |w) = z1 (v1 |w) + z2 (v2 |w),
(RLin) (w|z1 v1 + z2 v2 ) = z1 (w|v1 ) + z2 (w|v2 ),
(Symm) (v|w) = (w|v),
(PoDe) 0 (v|v) ∈ R with equality if and only if v = 0.
A set of vectors {v1 , . . . , vk } is said to be orthonormal if

1 if i = j,
(vi |vj ) = δij =
0 else.

We will de¬ne an inner product ( | )G on C(G) = Map(Gc , C)G , often writing ( | ) when the
group G is clear from the context.

Definition 4.11. For ±, β ∈ C(G), let
1
(±|β)G = ±(g)β(g).
|G|
g∈G

Proposition 4.12. ( | ) = ( | )G is an hermitian inner product on C(G).
3. INNER PRODUCTS OF CHARACTERS 39

Proof. The properties LLin, RLin and Symm are easily checked. We will show that PoDe
holds. We have
1 1
|±(g)|2 0
(±|±) = ±(g)±(g) =
|G| |G|
g∈G g∈G
with equality if and only if ±(g) = 0 for all g ∈ G. Hence (±|±) satis¬es PoDe.
Now let ρ : G ’’ GLC (V ) and θ : G ’’ GLC (W ) be ¬nite dimensional representations
over C. We know how to determine (χρ |χθ )G from the de¬nition. Here is another interpreta-
tion of this quantity. Recall from Proposition 3.13 the representations of G on W — V — and
HomC (V, W ); in fact these are G-isomorphic, W — V — ∼ HomC (V, W ). By Proposition 3.14, the
=
C
— )G and Hom (V, W )G are subrepresentations and are images of
G-invariant subspaces (W — V
C C
— ’’ W — V — and µ : Hom (V, W ) ’’ Hom (V, W ).
G-homomorphisms µ1 : W — V 2

Proposition 4.13. We have
(χθ |χρ )G = Tr µ1 = Tr µ2 .
Proof. Let g ∈ G. By Theorem 4.5 and Corollary 4.7 we can ¬nd bases v = {v1 , . . . , vr }
for V and w = {w1 , . . . , ws } for W consisting of eigenvectors with corresponding eigenvalues
»1 , . . . , »r and µ1 , . . . , µs . The elements wj — vi form a basis for W — V — and moreover g acts


on these by
(θ — ρ— )g (wj — vi ) = µj »i wj — vi ,
— —

using Proposition 3.12. By Corollary 4.7 we have
Tr(θ — ρ— )g = µj »i = ( µj )( »i ) = χθ (g)χρ (g).
i,j j i

By de¬nition of µ1 , we have
1 1
Tr(θ — ρ— )g =
Tr µ1 = χθ (g)χρ (g) = (χθ |χρ ).
|G| |G|
g∈G g∈G

Since µ2 corresponds to µ1 under the G-isomorphism W — V — ∼ HomC (V, W ) we obtain
=
Tr µ1 = Tr µ2 .
Corollary 4.14. For irreducible representations ρ and θ,
1 if ρ and θ are G-equivalent,
(χθ |χρ ) =
0 otherwise.
Proof. By Schur™s Lemma, Theorem 3.7,
1 if ρ and θ are G-equivalent,
dim| Hom| (V, W )G =
0 otherwise.
Since µ2 is the identity on Hom| (V, W )G , we obtain the result.
Thus if we take a collection of non-equivalent irreducible representations {ρ1 , . . . , ρr }, their
characters form an orthonormal set {χρ1 , . . . , χρr } in C(G), i.e.,
(χρi |χρj ) = δij .
By Proposition 4.9 we know that dimC C(G) is equal to the number of conjugacy classes in G.
We will show that the characters of the distinct inequivalent irreducible representations form a
basis for C(G), thus there must be dimC C(G) such distinct inequivalent irreducibles.
Theorem 4.15. The characters of all the distinct inequivalent irreducible representations of
G form an orthonormal basis for C(G).
40 4. CHARACTER THEORY

Proof. Suppose ± ∈ C(G) and for every irreducible ρ we have (±|χρ ) = 0. We will show
that ± = 0.
Suppose that ρ : G ’’ GLC (V ) is any representation of G. Then de¬ne ρ± : V ’’ V by

ρ± (v) = ±(g)ρg v.
g∈G

For any h ∈ G and v ∈ V we have

ρ± (ρh v) = ±(g)ρg (ρh v)
g∈G
« 

= ρh  ±(g)ρh’1 gh v 
g∈G
« 

= ρh  ±(h’1 gh)ρh’1 gh v 
g∈G
« 

= ρh  ±(g)ρg v 
g∈G

= ρh ρ± (v).

Hence ρ± ∈ HomC (V, V )G , i.e., ρ± is G-linear.
Now if we apply this to an irreducible ρ of dimension n say, by Schur™s Lemma, Theorem 3.7,
there must be a » ∈ C for which ρ± = » IdV .
Taking traces, we have Tr ρ± = n». Also

Tr ρ± = ±(g) Tr ρg = ±(g)χρ (g) = |G|(±|χρ— ).
g∈G g∈G

Hence we obtain
|G|
»= (±|χρ— ).
dimC V
If (±|χρ ) = 0 for all irreducible ρ, then as ρ— is irreducible whenever ρ is, we must have ρ± = 0
for every such irreducible ρ.
Since every representation ρ decomposes into a sum of irreducible subrepresentations, it is
easily veri¬ed that for every ρ we also have ρ± = 0 for such an ±.
Now apply this to the regular representation ρ = ρreg on V = C[G]. Taking the basis vector
e ∈ C[G] we have
ρ± (e) = ±(g)ρg e = ±(g)ge = ±(g)g.
g∈G g∈G g∈G

But this must be 0, hence we have
±(g)g = 0
g∈G

in C[G] which can only happen if ±(g) = 0 for every g ∈ G, since the g ∈ G form a basis of
C[G]. Thus ± = 0 as desired.
Now for any ± ∈ C(G), we can form the function
r
± =±’ (±|χρi )χρi ,
i=1
4. CHARACTER TABLES 41

where ρ1 , ρ2 , . . . , ρr is a complete set of non-isomorphic irreducible representation of G. For
each k we have
r
(± |χρk ) = (±|χρk ) ’ (±|χρi )(χρi |χρk )
i=1
r
= (±|χρk ) ’ (±|χρi )δi k
i=1
= (±|χρk ) ’ (±|χρk ) = 0,

hence ± = 0. So the characters χρi span C(G), and orthogonality shows that they are linearly
independent, hence they form a basis.

Recall Theorem 3.10 which says that any representation V can be decomposed into irred-
ucible G-subspaces,
V = V1 • · · · • Vm .

Theorem 4.16. Let V = V1 • · · · • Vm be a decomposition into irreducible subspaces. If
ρk : G ’’ GLC (Vk ) is the representation on Vk and ρ : G ’’ GLC (V ) is the representation on
V , then (χρ |χρk ) = (χρk |χρ ) is equal to the number of the factors Vj G-equivalent to Vk .
More generally, if also W = W1 •· · ·•Wn is a decomposition into irreducible subspaces with
σk : G ’’ GLC (Wk ) the representation on Wk and σ : G ’’ GLC (W ) is the representation on
W , then
(χσ |χρk ) = (χρk |χσ )
is equal to the number of the factors Wj G-equivalent to Vk , and

(χρ |χσ ) = (χσ |χρ )
= (χσ |χρk )
k

= (χσ |χρ ).


4. Character tables
The character table of a ¬nite group G is the array formed as follows. Its columns correspond
to the conjugacy classes of G while its rows correspond to the characters χi of the inequiva-
lent irreducible representations of G. The jth conjugacy class Cj is indicated by displaying a
representative cj ∈ Cj . In the (i, j)th entry we put χi (cj ).
c1 c2 ··· cn
χ1 χ1 (c1 ) χ1 (c2 ) · · · χ1 (cn )
χ2 χ2 (c1 ) χ2 (c2 ) · · · χ2 (cn )
. ..
. .
.
χn χn (c1 ) χn (c2 ) · · · χn (cn )
Conventionally we take c1 = e and χ1 to be the trivial character corresponding to the trivial
1-dimensional representation. Since χ1 (g) = 1 for g ∈ G, the top of the table will always have
the form
e c2 · · · cn
χ1 1 1 · · · 1
Also, the ¬rst column will consist of the dimensions of the irreducibles ρi , χi (e).
For the symmetric group S3 we have
42 4. CHARACTER THEORY

e (1 2) (1 2 3)
χ1 1 1 1
χ2 1 ’1 1
χ3 2 0 ’1
The representations corresponding to the χj will be discussed later. Once we have the character
table of a group G we can decompose an arbitrary representation into its irreducible constituents,
since if the distinct irreducibles have characters χj (1 j r) then a representation ρ on V
has a decomposition
V ∼ n1 V1 • · · · • nr Vr ,
=
where nj Vj ∼ Vj • · · · • Vj means a G-subspace isomorphic to the sum of nj copies of the
=
irreducible representation corresponding to χj . Theorem 4.16 now gives nj = (χρ |χj ). The
non-negative integer nj is called the multiplicity of the irreducible Vj in V . The following

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