Number Theory: An Introduction to Mathematics

6 Induced Representations and Examples 419

by (10). IfρR(C)is the matrix representingCin the regular representation, it fol-
lows that there exists an invertible matrixTsuch thatT−^1 ρR(C)Tis a diagonal ma-
trix, consisting of the matrices(g/ni)^2 Ini, repeatednitimes, for everyi. In partic-
ular,(g/ni)^2 is a root of the characteristic polynomialφ(λ)=det(λIg−ρR(C))
for everyi.ButρR(C)is a matrix with integer entries and hence the polynomial
φ(λ)=λg+a 1 λg−^1 +···+aghas integer coefficientsa 1 ,...,ag. The following
lemma, already proved in Proposition II.16 but reproved for convenience of reference
here, now implies that(g/ni)^2 is an integer and hence thatnidividesg.

Lemma 12Ifφ(λ)=λn+a 1 λn−^1 +···+anis a monic polynomial with integer co-
efficients a 1 ,...,anand r a rational number such thatφ(r)= 0 , then r is an integer.

Proof We can writer=b/c,wherebandcare relatively prime integers andc>0.
Then

bn+a 1 bn−^1 c+···+ancn= 0

and hencecdividesbn.Sincecandbhave no common prime factor, this impliesc=1.

If we apply the preceding argument toCk, rather than toC, we see that there
exists an invertible matrixTksuch thatTk−^1 ρR(Ck)Tkis a diagonal matrix, consisting
of the matrices(hkχik/ni)Inirepeatednitimes, for everyi. Thushkχik/niis a root
of the characteristic polynomialφk(λ)=det(λIg−ρR(Ck)). Since this is a monic
polynomial with integer coefficients, it follows thathkχik/niis an algebraic integer.

6 InducedRepresentationsandExamples

LetHbe a subgroup of finite indexnof a groupG,i.e.Gis the disjoint union of
nleft cosets ofH:

G=s 1 H∪···∪snH.

Also, let there be given a representationσ:t→A(t)ofHby linear transformations
of a vector spaceV. The representationσ ̃:s → A ̃(s)ofG inducedby the given
representationσofHis defined in the following way:
Take the vector spaceV ̃to be the direct sum ofnsubspacesVi,whereViconsists
of all formal productssi·v(v∈V)with the rules of combination

si·(v+v′)=si·v+si·v′, si·(λv)=λ(si·v).

Then we set

A ̃(s)si·v=sj·A(t)v,

wheretandsjare determined fromsandsi by requiring thatt =s−j^1 ssi ∈ H.
The degree of the induced representation ofGis thusntimes the degree of the original
representation ofH.