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