Number Theory: An Introduction to Mathematics

6 Induced Representations and Examples 423

1 =g−^1

∑

s∈G

ψ ̃i(s)χk(s−^1 )

=g−^1

∑

s∈H

ψ ̃i(s)χk(s−^1 )

=g−^1

∑

s∈H

ψ ̃i(s)χk(s−^1 )λ(s−^1 )

=g−^1

∑

s∈G

ψ ̃i(s)χk(s−^1 )λ(s−^1 ).

Thusχkλalso occurs once in the decomposition ofψ ̃i, and sinceχkλ=χkwe must
haveχkλ=χl.
In the relation

∑

iψi(^1 )

(^2) =h, partition the sum into a sum over pairs of distinct
conjugate characters and a sum over self-conjugate characters:
Σ′{ψi( 1 )^2 +ψiv( 1 )^2 }+Σ′′ψi( 1 )^2 =h.
Then for the corresponding characters ofGwe have
Σ′ψ ̃i( 1 )^2 +Σ′′{χk( 1 )^2 +χl( 1 )^2 }= 2 Σ′{ψi( 1 )^2 +ψiv( 1 )^2 }+ 2 Σ′′ψi( 1 )^2 = 2 h=g.
Since, by Corollary 14, each irreducible character ofGappears in the sum on the left,
it follows from (9) that each occursexactly once. Thus we have proved
Proposition 15Let the finite group G have a subgroup H of half its order. Then each
pair of distinct conjugate characters of H yields by induction a single irreducible char-
acter of G of twice the degree, whereas each self-conjugate character of H yields by
induction two distinct irreducible characters of G of the same degree, which coincide
on H and differ in sign on G\H. The irreducible characters of G thus obtained are all
distinct, and every irreducible character of G is obtained in this way.
We will now use Proposition 15 to determine the irreducible characters of several
groups of mathematical and physical interest. LetSndenote thesymmetricgroup con-
sisting of all permutations of the set{ 1 , 2 ,...,n},Anthealternatinggroup consisting
of all even permutations, andCnthecyclicgroup consisting of all cyclic permutations.
ThusSnhas ordern!,Anhas ordern!/2andCnhas ordern.
The irreducible characters of the abelian groupA 3 =C 3 are all of degree 1 and
can be arranged as a table in the following way, whereωis a primitive cube root of
unity, sayω=e^2 πi/^3 =(− 1 +i

√

3 )/2.

A 3

e (123) (132)

ψ 1 11 1

ψ 2 1 ωω^2

ψ 3 1 ω^2 ω

The groupS 3 containsA 3 as a subgroup of index 2. The elements ofS 3 form three
conjugacy classes:C 1 containing only the identity elemente,C 2 containing the three