Number Theory: An Introduction to Mathematics

5 Characters of Arbitrary Finite Groups 415

is a diagonal matrix. Moreover, since

T−^1 SkT=diag[ω 1 k,...,ωkn],

ω 1 ,...,ωnare allm-th roots of unity. Thus

χ(s)=ω 1 +···+ωn

is a sum ofnm-th roots of unity. Since the inverse of a root of unityωis its complex

conjugateω ̄, it follows that

χ(s−^1 )=ω− 11 +···+ω−n^1 =χ(s).

Now letGbe a group of finite orderg,andletρ:s→A(s)andσ:s→B(s)be

irreducible matrix representations ofGof degreesnandmrespectively. For anyn×m

matrixC, form the matrix

T=

∑

s∈G

A(s)CB(s−^1 ).

Sincetsruns through the elements ofGat the same time ass,

A(t)T=TB(t) for everyt∈G.

Therefore, by Schur’s lemma,T=Oifρis not equivalent toσandT=λIifρ=σ.

In particular, takeCto be any one of themnmatrices which have a single entry 1 and

all other entries 0. Then ifA=(αij),B=(βkl),weget

∑

s∈G

αij(s)βkl(s−^1 )=

{

0ifρ,σare inequivalent,

λjkδil ifρ=σ,

whereδil = 1 or 0 according asi =lori = l(‘Kronecker delta’). Since for
(αij)=(βij)the left side is unchanged wheniis interchanged withkandjwith
l,wemusthaveλjk=λδjk. To determineλseti=l,j=kand sum with respect
tok. Since the matrices representingsands−^1 are inverse, we getg 1 =nλ. Thus

∑

s∈G

αij(s)αkl(s−^1 )=

{

g/n ifj=kandi=l,

0otherwise.

Ifμ,vrun through an index set for the inequivalent irreducible representations of

G, then the relations which have been obtained can be rewritten in the form

∑

s∈G

αij(μ)(s)α(klv)(s−^1 )=

{

g/nμ ifμ=v,j=k,i=l,

0otherwise.

(5)

Theorthogonality relations(5) for the irreducible matrix elements have several corol-

laries:

(i)The functionsαij(μ):G→Care linearly independent.