Let G be a group and F be a field. An F-representation of G is a group homomorphism ρ : G → GL ( V

Chapter 3

R EPRESENTATIONS OF FINITE GROUPS

Let G be a group and F be a field. An F-representation of G is a group homomorphism ρ : G → GL ( V

) where V a finite dimensional F-vector space. Denote by Rep

( G ) the category of all F-representations. The goal of this chapter is to develop the general theory of complex representations (F = C) of finite groups G. Some of the results hold for more general F or G. We choose to present those general versions whenever they do not require too much additional efforts.

For general F, the abstract theory of modules and rings developed in the previous two chapters will be helpful (e.g. the Wedderburn–Artin the- orem and the adjoint properties of ⊗ and hom for non-commutative rings).

For F = _{C and} | G | < ∞, the essential tool to study representations is the theory of characters χ

: = tr ρ : G → F which can be regarded as an analogue of the Fourier transform in the discrete non-abelian case. In fact two representations ρ, ρ

are equivalent if and only if χ

=

. Being (class) functions on G, characters are much easier to handle than the actual representations. Hence

Be More Concerned with Your Character than Your Representation!

UCLA basketball coach—John Wooden Based on the character theory, two celebrated results discussed in this chapter are (1) Burnside’s theorem that any finite group with order p

q

, p and q are primes, is solvable; (2) Brauer’s theorem that any character of G is “integrally determined” by linear characters (i.e. dim

V = 1) of certain

“elementary subgroups” H ⊂ G. This result is important in number theory.

1. The basics

Group representations are generalizations of group actions on finite sets. If G acts on S, let V : =

Fs with base S, then ρ : G → GL ( V )

29

defined by ρ ( g ) s : = gs and extending linearly over F:

ρ

( g ) ∑

^a

^{s =} ∑

^a

^{gs =} ∑

^a

^s,

is called the permutation representation.

If S = G, with the action being the group multiplication (on the left), we get the regular representation ρ

on V

= F

. It is clear that F

is the underlying vector space of the group algebra F [ G ] , a fact we will explore in details shortly.

Here are some basic operations on representations. We denote ρ ( g ) ∼ A ∈ M

( F ) if A is the matrix of ρ ( g ) under a chosen basis of V. The dimension m = dim

V = : deg ρ is also called the degree of ρ.

(1)

given ρ

: G → _GL ( V

) _{, i} = 1, 2, we define ρ

⊕

: G → GL ( V

⊕ V

) by (

⊕

)( g ) =

( g ) ⊕

( g ) . If ρ

( g ) ∼ A, ρ

( g ) ∼ B, then

ρ₁

( g ) ⊕

( g ) ∼ ^{A 0} 0 B

! .

(2)

similarly we define ρ

⊗

: G → GL ( V

⊗

V

) by (

⊗

)( g ) =

( g ) ⊗

( g ) : v

⊗ v

7→

( g ) v

⊗

( g ) v

. Here all the tensor products are taken over the field F. If V

=

Fv

, V

=

Fw

, then V

⊗

V

=

Fv

⊗ w

. Under the lexicographic order of the basis, namely v

⊗ w

< v

⊗ w

if i < i

or if i = i

and j < j

, then

ρ₁

( g ) ⊗

( g ) ∼ A ⊗ B =









a

B · · · a

B .. . . .. .. . a

B · · · a

B









∈ M

( F ) ,

where A ∈ M

( F ) , B ∈ M

( F ) .

(3)

given ρ : G → GL ( V ) , G acts on φ ∈ V

= hom

( V, F ) by ( gφ )( v ) : =

( g

v ) . The inverse is inserted to ensure that ( gh )

= g ( hφ ) . This defines the dual representation ρ

.

More precisely, if V =

Fv

, ρ ( g ) ∼ A, then V

=

Fv

where v

is the dual basis such that v

( v

) =

. The usual induced linear transforma- tion on V

has matrix

A. Hence ρ

( g ) ∼

( A

) = (

A )

.

(4)

we say ρ

∼ =

if there is a vector space isomorphism η : V

∼ = V

such that ρ

( g ) =

( g )

for all g ∈ G.

It is clear that the group homomorphism G → GL ( V ) extends lin-

early to an F-algebra homomorphism F [ G ] → End

V. That is, a left F [ G ] -

module structure on V. Conversely, a left F [ G ] -module V leads to a rep-

resentation of G. Thus the notion of sub/quotient/irreducible/completely

1. THE BASICS 31

reducible modules corresponds to the analogous notion of representations.

Whenever there is only one ρ involved, we simply write gv : =

( _g ) _v.

However, the notion of ⊗ and dual defined above for F-representations of G do not correspond directly to the ones for F [ G ] -modules. Their rela- tions will become transparent in later sections.

Here are two basic theorems: Maschke’s theorem on complete reducibil- ity and Clifford’s theorem on restrictions to normal subgroups.

Theorem 3.1 (Maschke). If char F - | ^G | < ∞, then every ρ : G → GL ( V ) is completely reducible.

P

. If there is a G-invariant subspace U ⊂ V, we will show that there exists a G-invariant complemented subspace U

, and then he theorem follows by induction. We give two proofs of it. The first only works for F = R or C. But it gives insights to motivate the second proof.

For F = R or C, there exists a G-invariant inner product on V. Indeed for any inner product ( , )

on V, the “balanced” inner product

( v, w ) : = ∑

( gv, gw )

is clearly G-invariant: ( hv, hw ) = ( v, w ) for all h ∈ G. If U ⊂ V is G- invariant then U

⊂ V is also G-invariant: for v ∈ U

, we have ( gv, u ) = ( v, g

u ) = 0 for all u ∈ U, hence v ∈ U

.

For F with char F - | ^G | < ∞, we start with an arbitrary projection map p

(idempotent) onto U instead. To adjust p

to a G-linear map p, i.e. hp = ph for all h ∈ G, we simply take

p : = ¹

| G | ∑

^g

^p

^g,

and then h

ph = p. Moreover, p is still a projection map onto U. For if u ∈ U, since g ( u ) ∈ U we get p

g ( u ) = g ( u ) and then g

p

g ( u ) = u for all g ∈ G, so p ( u ) = u. Also for any x ∈ V we have p ( x ) ∈ U since p

g ( x ) ∈ U. Thus we have the decomposition V = U ⊕ U

corresponding to 1 = p + ( 1 − p ) where U

: = im ( 1 − p ) is also G-invariant since 1 − p is

an idempotent commuting with the G-action. 

Consequently, there is a unique decomposition up to isomorphisms

= _{∑ m}

, V =

V

where ρ

: G → GL ( V

) are irreducible sub repre-

sentations and ρ

6∼ =

for i 6= j. Thus the study of F-representations with

char F - | ^G | < ∞ is reduced to the study on irreducible ones.

Definition 3.2 (Restrictions and Conjugates).

(1) Let ρ : G → GL ( _V ) , for any subgroup H ⊂ G we define the restriction representation of H on V by

ρH

≡ Res

≡ Res

V : =

|

: H → GL ( V ) .

(2) For H / G, σ : H → _GL ( U ) _{and g} ∈ G, we define the g-conjugate representation of σ by

gσ

: H → GL ( U ) ,

( h ) : =

( ghg

) .

It preserves the lattice of F [ H ] -submodules. Also σ

∼ =

⇒

∼ =

. The next basic result works for any fields F and any group G.

Theorem 3.3 (Clifford). Let ρ : G → GL ( V ) be irreducible. Then H / G implies that ρ

is completely reducible and all irreducible components are conjugated with each other with the same multiplicity.

P

. Let U ⊂ V be an irreducible F [ H ] -submodule, say U = F [ H ] v for some v ∈ V \ { 0 } . Then V = _∑

gU since the sum is G-invariant.

Also each gU is H-invariant: for any y ∈ U, h ∈ H, g

hg ∈ H and hence hgy = g ( g

hg ) y ∈ gU.

Moreover, let σ =

|

acting on U, σ

=

|

acting on gU. Then the above formula means σ

∼ =

and so gU is F [ H ] -irreducible for all g ∈ G.

This implies that ρ

on V is completely reducible. It is also clear that if U

∼ = U

for two irreducible components, the gU

∼ = gU

too (all as F [ H ] - modules). Hence ρ ( g ) permutes homogeneous components.  From now on, let A : = F [ G ] and denote by ρ : A → End

V under the same notation ρ. We will apply results on semi-simple artinian rings in the current setting. As before let

A

: = End

V = C

(

( A )) ⊂ End

V ∼ = M

( F ) , and A

: = End

V ⊃

( A ) . (Recall that A

= A

tautologically.)

If ρ is known to be completely reducible, since dim

V < ∞, the density theorem then implies the double centralizer property A

=

( A ) . This is the case if char F - | ^G | < ∞ by Theorem 3.1.

In general, if | G | < _{∞ then A} = F [ G ] is clearly artinian. In particular

there are only a finite number of irreducible representations up to isomor-

phisms. Much more will be said below!

1. THE BASICS 33

Theorem 3.4. Let | G | < _{∞, then A} = F [ G ] is semi-simple ⇐⇒ char F - | ^G | . P

. If char F - | ^G | < _{∞, then}

A is completely reducible by Theo- rem 3.1. Hence A is semi-simple by the Wedderburn–Artin–Jacobson struc- ture theorem on artinian rings. Conversely, if char F | | G | , then for z : =

∑

g we have gz = z = zg for all g ∈ G, hence Fz ⊂ A is an ideal. But z

= ( _{∑ g} ) z = | G | z = 0 in A, hence Fz is a nilpotent ideal and then A is not semi-simple (again by the structure theorem). 

Now we assume char F - | ^G | < ∞. By the structure theorems, A = F [ G ] = A

⊕ . . . ⊕ A

, A

∼ = M

( _∆

) ,

where ∆

’s are division algebras over F. Let I

be a minimal left ideal of A

, then it is also a minimal left ideal of A. Thus we obtain s equivalence classes of irreducible F-representations ρ

, . . . , ρ

of G. Also

M

( _∆

) =

j=1

M

( _∆

) e

is the decomposition into n

copies of I

as the j-th column spaces. Let d

= dim

∆

then dim

I

= n

d

. This implies

Corollary 3.5. For ρ

which acts on the space

A, we have

=

i=1

n

, | G | = ∑

n

d

. Next we determine the center of A. Clearly

C ( A ) = C ( A

) ⊕ . . . ⊕ C ( A

) , C ( A

) = C ( _∆

) .

On the other hand, let C

, 1 ≤ j ≤ r be the conjugacy classes of G. Then Proposition 3.6. C ( A ) =

Fc

, where c

: = _∑

g.

P

. Let a = _∑

a

g ∈ A, then

h

ah = ∑

^a

^h

^{gh =} ∑

^a

^{g = a}

for all h ∈ G is equivalent to that all the coefficients in the same conjugacy class are the same. That is, a is a linearly combination of c

’s. Also c

’s are clearly linearly independent, hence they form a basis of C ( A ) _.  Corollary 3.7. Let r be the number of conjugacy classes of G and s be the number of irreducible F-representations of G, then

(1) r = dim

C ( A ) = _∑

dim

C ( _∆

) . In particular r = s if and only if

∆

is a central simple algebra over F for all i.

(2) If F = F then ^¯ ∆

= F, r = s and | G | = _∑

n

.

Example 3.8. (1) Cyclic groups: G=Cn= hg|gⁿ =1i, F=Q. Then A=_Q[G] ∼=Q[x]/(xⁿ−1) ∼=^M_d|nQ[x]/`_d(x) ∼=^M_d|nQ(ζ_d)

where`n(x) ∈Z[x]is the d-th cyclotomic polynomial which is irreducible overQ, Q(ζ_d)is the d-th cyclotomic field with ζ_d=e^2π

√−1/d.

In this case A is abelian, hence all ni =1 and∆i = C(∆i) = Q(ζ_di)for some d_i|n. The irreducible representation ρ_dcorresponding to d|n has degree ψ(d).

If we start with F=Q(ζ_n)instead, then

A∼=F[x]/

∏

where Vi ∼=Fe_iis an one dimensional representation with gei =ζⁱ_ne_i. Hence there are r=s=n inequivalent irreducible representations of G.

If we start withQ(ζn) ⊃F⊃Q, the structure of F[G]varies dramatically!

(2) Dihedral groups: G=Dn = hR, S| Rⁿ =1, S²=1, SRS=R⁻¹i. We have

|Dn| = 2n and a set of representatives is given by{R^k, R^kS | 0 ≤ k≤ n−1} ⊃ Cn = hRi. The conjugacy classes are determined by

n r=# conj. classes representatives 2ν+1 ν+2 R⁰, . . . , R^ν, S

2ν ν+3 R⁰, . . . , R^ν, R, RS

Here are a few irreducibleC-representations: let ρ1=1, ρ₂=sgn: R7→ (1), S7→

(−1)be the obvious degree 1 representations on F=C. Since SRS=Rⁿ⁻¹, those degree 1 representations of Cnare generally not representations of Dn.

For n=2ν+1, for each k∈ [1, ν]we define a degree 2 representation

σ_k : R7→ ^w

k 0

0 w^−k

!

, S7→ ^{0 1} 1 0

!

, V=_Ce1⊕Ce2.

Here w=ζn. They are clearly irreducible and inequivalent. We have constructed 2+ν=r irreducibleC-representations hence they are all of them. As a consistency check we compute∑^ri=1n²_i =2×1²+ν×2²=2(2ν+1) =2n= |Dn|.

For n = 2ν, ν ≥ 2, two more degree 1 representations are found: ρ3 : R 7→

(−1), S 7→ (1), ρ₄ := ρ₂⊗ρ₃ : R 7→ (−1), S 7→ (−1). But now we take only σ_k, k ∈ [1, ν−1]since σνis reducible—it contains the invariant subspaceC(e1+e2). This gives all the r=ν+3 irreducibleC-representations. Also ∑^ri=1n²_i =₄×1+ (ν−1) ×2²=4ν=2n= |Dn|as expected.

(3) Quaternion group: G = Q₈ = {±_1,±i,±j,±k} ⊂H^×. Notice that Q8 6∼= D4since every subgroup of Q8is normal which is not the case for D4.

Let F = Q. There are at least two irreducible Q-representations, the trivial one of degree 1 and the natural one of degree 4 acting onH(_Q), the quaternion

1. THE BASICS 35

numbers withQ coordinates. The structure theorem then forces a decomposition Q[Q₈] =_Q⊕_Q⊕_Q⊕_Q⊕_H(_Q).

This is consistent with the fact that there are 5 conjugacy classes of Q₈, namely {1}, {−1}, {i,−i}, {j,−j}, {k,−k}. If we consider F = Q(√

−1) instead, the decomposition becomes

F[Q₈] =F⊕F⊕F⊕F⊕M₂(F)_,

whereH(_Q) ⊗_QF∼= M₂(F). This decomposes the degree 4 irreducible represen- tationH(_Q)into two copies of the degree 2 one V=F^⊕2.

Exercise 3.1. Write down the explicit formulas of the decompositions of F [ G ] in Example 3.8, (2) and (3).

Example 3.8, (1) and (3) suggest the following

Definition 3.9 (Absolute irreducibility and splitting fields).

(1) Let K/F be a field extension, then we define the K-representation

: =

⊗

K by composing ρ with GL ( V

) → GL ( V

⊗

K ) .

(2) A representation ρ is absolutely irreducible if ρ

is irreducible for all extension field K/F. This is equivalent to that ρ

is irreducible.

(3) K is a splitting field of G if all irreducible K-representations of G are absolutely irreducible. In particular, ¯ F is always a splitting field.

In Example 3.8-(1), Q (

) is a splitting field of C

. In Example 3.8-(3), Q ( √

− 1 ) is a splitting field of Q

. These are finite extensions of Q. Accord- ing to the theory of CSA/F, a splitting field can be chosen to be a finite extension of F. More precise statement can be made.

Theorem 3.10. Let char F - | ^G | < _{∞, ρ : G} → GL ( V

) . Then (1) ρ is irreducible ⇐⇒ A

: = End

V is a division F-algebra.

(2) ρ is absolutely irreducible ⇐⇒ A

= F id

.

P

. (1) “ ⇒ ” by Schur’s lemma. For “ ⇐ ”: if ρ is reducible, Maschke’s theorem implies V = U ⊕ U

for two sub representations. The projection p onto U then satisfies p

= p, that is p ( p − 1 ) = 0, but p 6= 0, 1.

(2) “ ⇒ ”: if there is a c ∈ A

\ F id

, then the minimal polynomial m

( x ) ∈ F [ x ] of c is irreducible (since A

is a division F-algebra by (1)). Con- sider the simple extension K = F [ x ] / ( m

( x )) . It is a general fact that the minimal polynomial of a linear transformation is unchanged under field extensions. But m

( x ) factors in K [ x ] , hence 0 = m

( c ) = f ( c ) g ( c ) and A

is not a division F-algebra, this leads to a contradiction by (1). For “ ⇐ ”:

A

= F id

implies A

= K id

. Hence ρ

is irreducible for all K/F. 

Using this result together with knowledge in CSA/F, we may deduce Theorem 3.11. Let char F - | ^G | < ∞. Then F is a splitting field of G if and only if F [ G ] ∼ =

M

( F ) . That is, F splits all the division algebras ∆

appeared in the semi-simple decomposition.

The proof is left to the readers.

2. Complex characters

In this section we work with complex representations of finite groups G, namely F = C unless specified otherwise.

Definition 3.12. Let ρ : G → GL ( V ) be a F-representation. The character of

is the function χ

: G → F defined by χ

( g ) : = tr ρ ( g ) .

At the first sight it seems that characters χ contain less information than the representation ρ. However, for a single matrix A the complete informa- tion of tr A

for all k ∈ N is equivalent to the characteristic polynomial f

( x ) . Hence the trace function over the group ρ ( G ) indeed contain rich informations of ρ. In fact we will show that “χ characterizes ρ for F = _C”!

We start with a few immediate consequences following the definition:

(1) χ

is a class function:

tr ρ ( hgh

) = _{tr ρ} ( h )

( g )

( h )

= _{tr ρ} ( g ) _.

Namely χ

( g ) depend only on the conjugacy class of g. We denote the subspace of class functions by

C ( G, F ) ⊂ F

. (2) If U ⊂ V is ρ ( G ) -invariant, then

χ_ρ

=

+

.

This follow from the observation that for a choice of basis respects V = U ⊕ U

(vector space decomposition) we have

ρ

( g ) ∼

|

( g ) ∗ 0

|

( g )

! .

(3) χ

=

since tr A ⊗ B = tr A tr B, which is clear from

A ⊗ B =









a

B · · · .. . . .. .. .

· · · a

B









, m : = deg ρ

.

2. COMPLEX CHARACTERS 37

(1), (2), (3) work for any F. Now we use the assumption F = _C:

(4) If g

= _{1 then ρ} ( _g )

= _id

_{. Thus m}

( _x ) | ( _x

− 1 ) which implies that all roots w

’s are distinct d-th roots of 1. Then ρ ( g ) is diagonalizable

ρ

( g ) ∼







 w

. ..

w









, m : = deg ρ

In particular χ

( g ) = _∑

w

, which leads to the simple observation:

Corollary 3.13. |

( g )| ≤ deg ρ, with equality holds if and only if ρ ( g ) = w id

where w

= _{1 for d} = _{exp G.}

Moreover, χ

( g ) = deg ρ if and only if ρ ( g ) = id

, i.e. g ∈ ker ρ.

(5) χ

=

since

diag ( w

)

= diag ( w

)

= diag ( w

) = diag ( w

) .

(2) For the regular representation, χreg(1) = |G|and χreg(g) =0 for all g6=1.

(3) For F=C, the number of equivalence classes of irreducible representations s is the same as the number of conjugacy classes r (Corollary 3.7-(2)). A character table is a r×r table to list all possible character values for a finite group G.

For G=Dn, using Example 3.8-(2) we may calculate its character table easily:

for n=2ν+1, it is

1 S R^j

1 1 1 1

sgn 1 −1 1

σ_k 2 0 w^kj+w^−kj

where k, j∈ [1, ν]. Notice that χ characterizes ρ: χρ6=χ_ρ0if ρ6∼=ρ⁰.

The major reason to make the character theory powerful comes from Schur’s orthogonality relations which we describe now. At the beginning we may work with any field F and group G with with char F - | ^G | < _∞.

For ρ : G → GL ( V ) and ρ

: G → GL ( V

) , we have a representation

: G → hom

( V, V

) = V

⊗

V

defined by, for any g ∈ G, e ∈ hom

( V, V

) ,

( g ) e : =

( g ) e ρ ( g )

_.

(Indeed ρ

=

⊗

as defined before.) Now we “symmetrize it”:

Claim 3.15. η ( e ) : = _∑

( g ) e ρ ( g )

∈ hom

( V, V

) .

P

.

ρ⁰

( h )

( e ) = ∑

g

ρ⁰

( hg ) e ρ ( g )

= ∑

g

ρ⁰

( g ) e ρ ( h

g )

= ^ ∑

g

ρ⁰

( g ) e ρ ( g )

^

( h ) =

( e )

( h ) .

This shows that η ( _e ) is a morphism of F [ _G ] _-modules.  If both ρ and ρ

are irreducible, then Schur’s lemma implies that η ( e ) = 0 whenever ρ 6∼ =

.

If ρ

=

( e ) ∈ End

V which is a division F-algebra. If we further assume that F is a splitting field of G, say F = C, or simply that ρ is absolutely irreducible, then we have η ( e ) ∈ F id

by Theorem 3.10.

For F = C, a direct proof is easy: let λ ∈ C be an eigenvalue of η ( e ) , then 0 6= _ker (

( e ) −

) ⊂ V is readily seen to be ρ ( G ) -invariant, hence it equals V since ρ is irreducible, and so η ( e ) =

.

Theorem 3.16 (Schur’s orthogonality relations). Let F be a splitting field of G with char F - | G | < _{∞. ρ}

, . . . , ρ

be the set of irreducible representations with matrices ρ

( g ) ∼ ( T

)( g ) . Then char F - ⁿ

: = deg ρ

for all i and

(i) ∑

^T

^{( g ) T}

^{( g}

^{) = 0 if i 6= j,}

(ii) ∑

^T

^{( g ) T}

^{( g}

^{) =}

^| _n ^G

^{| .}

P

. Let e

be the elementary matrix, then the sum is simply η ( e

)

and (i) follows directly.

For (ii), we have η ( e

)

=

for some λ

∈ F. Since T

( g

) = ( T

( g ))

_,

by summing over k = l ∈ [ _{1, n}

] _{we get 0} 6= | G |

= n

since char F -

| G | . This implies char F - ⁿ

and (ii) follows accordingly.  Remark 3.17. For F = C, we will prove later that n

| | G | . This fails for general F even for cyclic groups, see Example 3.8-(1).

From now on we work only for F = C. A major benefit from it is:

Definition 3.18. For φ, ψ ∈ _C

= { f : G → _C } , we define the (Hermitian) inner product

(

)

: = ¹

| G | ∑

( g )

( g ) .

2. COMPLEX CHARACTERS 39

Corollary 3.19. Let χ

=

, i ∈ [ 1, s ] be the irreducible characters. Then (

, χ

) =

.

P

. Since χ

( g ) = _∑

T

( g ) and

χ_i

( g

) = tr ρ

( g

) = tr ρ

( g )

= tr ρ

( g ) =

( g ) ,

Theorem 3.16-(i) then implies that (

, χ

) = 0 if i 6= j. For i = j, Theorem 3.16-(ii) implies that (

, χ

) = _∑

/n

= 1.  Every complex representation ρ of G can be uniquely decomposed as

ρ

= m

⊕ . . . ⊕ m

, m

∈ _Z

.

Hence χ

= m

+ . . . + m

and then m

= (

, χ

) . This implies Corollary 3.20. For C-representations, ρ ∼ =

if and only if χ

=

.

Also (

, χ

) = _∑

m

, which implies

Corollary 3.21. A C-representation ρ is irreducible if and only if (

, χ

) = 1.

Finally, since “s = r” for F = C, we conclude

Theorem 3.22. The irreducible characters χ

, . . . , χ

form an orthonormal basis of the space of class functions C ( G ) .

S

. The theorem is equivalent to s = r, which is proved via the Wedderburn–Artin structure theorem. Here we give a direct proof using only the character theory. We only need to show

Claim 3.23. If f ∈ _C ( G ) has ( f , χ

) = 0 for all i ∈ [ 1, s ] then f = 0.

For each i ∈ [ 1, s ] , we define

T

: = ∑

f ( g )

( g ) ∈ End

V

. In fact T

is ρ

( G ) -linear: for any h ∈ G we compute

ρ_i

( h ) T

= ∑

^{f ( g )}

^{( hg ) = } ∑

^{f ( g )}

^{( hgh}

^{) }

^{( h )}

= ^ ∑

^{f ( h}

^{gh )}

^{( g ) }

^{( h ) = T}

^{( h )}

since f is a class function. Schur’s lemma implies that T

=

. But tr T

= ( f , χ

) = _{0 hence T}

= 0. In particular this implies

∑

f ( g )

( g ) = 0.

Apply it to the vector 1 we get ∑

f ( g ) g = 0. So f ( g ) = 0 for all g. 

Example 3.24. (1) We had seen that ρreg = _∑^s_i=1n_iρ_i using the structure theorem for F[G] (cf. Corollary 3.5). For F = C this follows from the character theory immediately since the multiplicity m_iof ρ_iin ρregis

mi= (χ_reg, χi) = ¹

|G|

∑

(1)₁ (12)₆ (123)₈ (1234)₆ (12)(34)₃

1 1 1 1 1 1

sgn 1 −1 1 −1 1

ρst 3 1 0 −1 −1

ρst⊗_sgn 3 −1 0 1 −1

W 2 0 −1 0 2

To see it, there are 5 conjugacy classes C_jshown in the top row where the subscript is|Cj|. As a check, we see that∑⁵_j=1|Cj| =24= |S4|.

There are 5 irreducibleC-representations of S4where the first two degree 1 representations ρ1, ρ2are obviously there. From 1+1+n²₃+n²₄+n²₅=24 we see that the remaining 3 must be of degree 3, 3, 2.

To get 3-dimensional representations, the standard way is to make S4acts on C⁴ = ^L4_{i=1C ei} as a permutation representation on the basis. Since v := _∑⁴_i=1e_i spans a S4-invariant line, we get a S4representation on V := C⁴/C v ∼= (C v)^⊥. We call it ρ₃=ρstand it character (written as a vector in the above order) is

χ_st=χ_C4−χ_{C v}= (4, 2, 1, 0, 0) − (1, 1, 1, 1, 1) = (3, 1, 0,−1,−1). ρstis indeed irreducible since(χst, χst) = (3²+6+0+6+3)/24=1.

To get another degree 3 representation ρ4. we tensor ρ3with non-trivial degree 1 representations. It must be irreducible since(χχ⁰, χχ⁰) = (χ, χ)if deg χ⁰=1.

We call the remaining ρ5 of degree 2 by W. χW is easily determined by the others since 0=χreg(g) =_{∑ ni}χ_i(g)for g6=1. The result is χ_W= (2, 0,−1, 0, 2).

We have determined W abstractly. To see it concretely, the idea is to make use of subgroups or quotient groups of S4. For example, we have an exact sequence

1→K4→S4→π S3→1

where K₄ = {1,(12)(34),(13)(24),(14)(23) } /S₄is the Klein 4-group. Then any irreducible representation σ of S3is also irreducible for S4via σ◦π. Since S3∼=D3, we may simply take σ1in Example 3.8-(2) to get W=σ₁◦π. It is readily seen that χσ₁◦π(cf. Example 3.14-(3)) coincides with χWas computed above.

(3) Product groups: if G (resp. G⁰) has irreducibleC-representations ρi(resp. ρ⁰_j), then the irreducibleC-representations of G×G⁰are given precisely by the “outer tensor product” ρi# ρ⁰_j’s where

(ρ# ρ⁰)(g, g⁰):=ρ(g) ⊗ρ⁰(g⁰) ∈GL(V⊗V⁰).

2. COMPLEX CHARACTERS 41

Indeed,

(χ_{ρ# ρ}⁰, χ_{ρ# ρ}⁰) = ¹

|G×G⁰|

∑

g,g⁰

|χ_{ρ# ρ}⁰(g, g⁰)|²

= ¹

|G|

∑

g

|χ_ρ(g)|^{2 1}

|G⁰|

∑

g⁰

|χ_ρ⁰(g⁰)|²= (χ_ρ, χ_ρ)(χ_ρ⁰, χ_ρ⁰).

Hence(χ_{ρ# ρ}⁰, χ_{ρ# ρ}⁰) =1 if and only if(χ_ρ, χ_ρ) =1= (χ_ρ⁰, χ_ρ⁰). This shows that {ρ_ij := ρ_i# ρ⁰_j |i ∈ [1, s], j ∈ [1, s⁰] }gives ss⁰ inequivalent irreducible representa- tions of G×G⁰. To see that they are all of them, we simply notice that

∑

∑

Example 3.24-(3) shows that representation theory for product groups is completely reduced to the study of its factors. In fact, representation the- ory of normal subgroups are sub-theory of the group as shown in Example 3.24-(2). In general there are plenty of subgroup while few of then are nor- mal. Hence it is more practical to study relations of representation theories with subgroups. This will be carried out in later sections.

There is a situation where all good things happen, namely the case of (finite) abelian groups or abelian subgroups.

Proposition 3.25. Let G be a finite group, then G is abelian if and only of all its complex irreducible representations are one-dimensional.

P

. This follows from the structure theorem directly: G is abelian

⇔ G has | G | = r = s conjugacy classes ⇔ all n

= 1 in | G | = _∑

n

.  A direct proof for the ”only if” part is also easy: let ρ : G → GL ( V ) be irreducible. Let g ∈ G and 0 6= ker (

( g ) −

( g ) I

) = : V

for some eigenvalue λ ( g ) ∈ C. Since G is abelian, V

is ρ ( G ) -invariant and hence V = V

. This implies that ρ ( g ) =

( g ) I

for all g ∈ G. But then any C v ⊂ V is ρ ( G ) -invariant hence in fact V is one-dimensional.

Corollary 3.26. Let G be a finite abelian group, then the set of all irreducible C- representations of G forms a group ˆ G : = hom ( G, C

) , the dual group of G, which is isomorphic to G (non-canonically).

P

. Degree 1 representations are necessarily irreducible and equiv-

alent to their characters ρ =

: G → _C

. They form a group under tensor

product, which coincides with multiplication of characters. For G finite

abelian, we get the character group ˆ G as defined.

The fundamental theorem of (finitely generated) abelian groups implies that G =

_G

_{, G}

= h _g

i ∼ = _Z/ ( _e

) _{. Then}

G ˆ = hom (

G

, C

) ∼ = ∏ ^{hom ( G}

^{, C}

^{) ∼ =} ∏

^{∼ = G,}

where ˆ G

∼ =

(the group of e

-th roots of 1) since each ρ ∈ hom ( G

, C

) is

determined by ρ ( g ) ∈

. 

Exercise 3.2. Let A ⊂ G be an abelian subgroup and ρ be an irreducible complex representation of G with degree n

. (1) Show that n

≤ [ G : A ] . (2) For A = C ( G ) , show that n

≤ [ G : A ] _.

Remark 3.27. The orthogonality of characters for abelian groups is essen- tially trivial. The same reasoning as in Corollary 3.26 reduces the problem to the one for cyclic groups, which is a simple exercise in geometric series.

To conclude this section, we emphasize that it is essential, in finite group representations, to construct/analyze invariant subspaces. This is mostly achieved by (i) averaging/symmetrizing a linear transformation or (ii) to work with eigenspaces of an operator lying in the center.

3. Arithmetic properties of characters

Recall that a ∈ C is an algebraic number, i.e. a is algebraic over Q, if there is a monic polynomial f ( x ) ∈ _Q [ x ] such that f ( a ) = 0. Also a is an algebraic integer, i.e. a is integral over Z, if the monic polynomial

f ( x ) ∈ _Z [ x ] . We may always take f ( x ) to be the minimal polynomial.

It is elementary to see that (1) a is algebraic over Q ⇒ ma is integral over Z for some m ∈ Z. (2) If a ∈ Q and integral over Z then a ∈ _Z.

Lemma 3.28. Let a ∈ C, then a is integral over Z ⇐⇒ there is a finitely gener- ated Z-module M ⊂ C such that aM ⊂ M.

P

. If f ( a ) = 0 for f ( x ) = x

+ b

x

+ . . . + b

∈ _Z [ x ] , then M : =

_{Z a}

satisfies aM ⊂ M.

Conversely, given M = _∑

_Zm

⊂ C such that aM ⊂ M, then am

=

∑ a

m

for some a

∈ Z. That is,

∑

^{( aδ}

^{− a}

^{) m}

^{= 0, i ∈ [ 1, n ] .}

Hence f ( a ) = 0 for f ( x ) : = det ( xδ

− a

) ∈ _Z [ x ] , which is monic. 

3. ARITHMETIC PROPERTIES OF CHARACTERS 43

Corollary 3.29. The set R of all algebraic integers is a ring and the set Q of all algebraic numbers is a field. The quotient field of R equals Q.

P

. For a, b ∈ R, we need show that a + b ∈ R and ab ∈ R.

Let f ( x ) , g ( x ) ∈ _Z [ x ] be monic polynomials with f ( a ) = 0, g ( b ) = 0. If deg f ( _x ) = _{m, deg g} ( _x ) = _{n, we set}

M = ∑

i∈[0,m−1],j∈[0,n−1]

Z a

b

.

Then it is clear that

( a + b ) M ⊂ M, abM ⊂ M

since all terms with a degree higher than m − 1 or b degree higher than n − 1 can be reduced using f ( a ) = 0 = g ( b ) .

The proof that Q is a field is entirely the same. We simply replace Z- modules by Q-vector spaces in Lemma 3.28 to get criterion for a being al- gebraic. Then the same proof as above gives the result.

Since R ⊂ Q, its quotient field Q ( R ) ⊂ Q. For the reverse inclusion, a ∈ _Q ⇒ b = ma ∈ R for some m ∈ _{Z, hence a} = b/m ∈ Q ( R ) .  Now we investigate these integral properties for irreducible complex representations ρ

and their characters χ

of a finite group G.

Corollary 3.30. All character values χ ( g ) ’s are algebraic integers.

P

. Roots of unity are algebraic integers. Hence any finite integral combination of them, e.g. χ ( g ) , is too by Corollary 3.29.  Much better/precise results hold through investigation on the “interac- tions” between irreducible representations and conjugacy classes:

Theorem 3.31. Let χ

, . . . , χ

be the irreducible characters of G, n

= deg χ

, C

, . . . , C

the conjugacy classes, and c

= _∑

j

g. Then (1) On V

, ρ

( c

) =

I

is a scalar multiplication with

λ_ij

=

( c

)

n

= | C

|

( g

)

n

, g

∈ C

, i, j ∈ [ _{1, s} ] _.

(2) All these eigenvalues λ

are algebraic integers.

P

. Recall that c

= _∑

j

g, j ∈ [ 1, s ] , is a base of Z = C ( F [ G ]) for any field F (Proposition 3.6). Since c

c

∈ Z, the proof actually implies that

c

c

= ∑

^m

^c

for some m

∈ _Z

. Let F = C and apply ρ

to the above formula.

In doing so we notice that ρ

( c

) =

for an eigenvalue λ ∈ _{C. This} follows from the fact that c

∈ Z and then ker (

( c

) −

) ⊆ V

is a non-trivial ρ

( G ) -invariant subspace, hence equals V

. Taking trace we get n

=

( c

) = | C

|

( g

) for any g

∈ C

, hence the formula for λ

.

Now for each i, ρ

( c

)

( c

) = _{∑ m}

( c

) _gives

= ∑

^m

^.

For M = _∑

_{Z λ}

we get λ

M ⊂ M, hence λ

∈ R as expected.  Corollary 3.32. n

| | G | for all i.

P

. From the orthogonal relation (

, χ

) =

, we compute

= ¹

| G | ∑

∑

( g )

( g ) . For i = j and for any choice of g

∈ C

, we get

Q 3 | G | n

=

∑

χ_i

( g

) ^ | C

|

( g

) n

 ∈ R.

Thus | G | /n

∈ _Z. 

Remark 3.33. In Schur’s orthogonal relation (Theorem 3.16-(ii)), we might have already concluded | G | /n

∈ Z if we know that ρ

( g ) can be repre- sented by matrices T

( g ) over R. Since all the traces are R-valued, this seems to be plausible. It turns out to be a deep question which remains largely open in representation theory of finite groups.

The divisibility in Corollary 3.32 is nice but not optimal, since for abelian groups we indeed have n

= 1 for all i. The following improvement due to Tate takes into account the abelian phenomenon.

Proposition 3.34. Let Z be the center of G, then n

| [ G : Z ] for all i.

P

. Let ρ : G → GL ( V ) be an irreducible complex representation.

Consider the m-th outer tensor product ρ

: G

→ GL ( V

) which is still

irreducible (cf. Example 3.24-(3)).