Number Theory: An Introduction to Mathematics

6 Induced Representations and Examples 421

Proof Letχdenote the character of the representationρofGandψthe character of
the representationσofH. By (7), the number of times thatρoccurs in the complete
reduction of the induced representationσ ̃is

g−^1

∑

s∈G

ψ( ̃ s)χ(s−^1 )=(gh)−^1

∑

s,u∈G

ψ(u−^1 su)χ(s−^1 ).

If we putu−^1 s−^1 u =t,u−^1 =v,thens−^1 =v−^1 tvand(t,v)runs through all
elements ofG×Gat the same time as(s,u). Therefore

g−^1

∑

s∈G

ψ( ̃ s)χ(s−^1 )=(gh)−^1

∑

t,v∈G

χ(v−^1 tv)ψ(t−^1 )

=h−^1

∑

t∈G

χ(t)ψ(t−^1 )=h−^1

∑

t∈H

χ(t)ψ(t−^1 ),

which is the number of times thatσoccurs in the complete reduction of the restriction
ofρtoH.

Corollary 14Each irreducible representation of a finite group G is contained in a
representation induced by some irreducible representation of a given subgroup H.

A simple, but still significant, application of these results is to the case where the
order of the subgroupHis half that of the whole groupG. The subgroupHis then nec-
essarilynormal(as defined in Chapter I,§7) since, for anyv∈G\H, the elements of
G\Hform both a single left cosetvHand a single right cosetHv. Hence ifs→A(s)
is a representation ofH,thensoalsoiss→A(v−^1 sv), itsconjugate representation.
Sincev^2 ∈H, the conjugate of the conjugate is equivalent to the original representa-
tion. Evidently a representation is irreducible if and only if its conjugate representation
is irreducible.
On the other handGhas a nontrivial characterλof degree 1, defined by

λ(s)=1or−1 according ass∈Hors∈/H.

Ifχis an irreducible character ofG, then the characterχλof the product representa-
tion is also irreducible, since

1 =g−^1

∑

s∈G

χ(s)χ(s−^1 )=

∑

s∈G

χ(s)λ(s)χ(s−^1 )λ(s−^1 ).

Evidentlyχandχλhave the same degree.
Ifψiis the character of an irreducible representation ofH, we will denote byψiv
the character of its conjugate representation. Thus

ψiv(s)=ψi(v−^1 sv).

The representation and its conjugate are equivalent if and only ifψiv(s)=ψi(s)for
everys∈H.