Number Theory: An Introduction to Mathematics

416 X A Character Study

For suppose there existλ(μ)ij ∈Csuch that

∑

i,j,μ

λ(μ)ij αij(μ)(s)=0foreverys∈G.

Multiplying byα(klv)(s−^1 )and summing over alls∈G,weget(g/nv)λ(lkv)=0. Hence

every coefficientλ(lkv)vanishes.

(ii) ∑

s∈G

χμ(s)χv(s−^1 )=gδμv. (6)

This follows from (5) by settingi=j,k=land summing overj,l.

(iii)The irreducible charactersχμare linearly independent.

In fact (iii) follows from (6) in the same way that (i) follows from (5).
Theorthogonality relations(6) for the irreducible characters enable us to decom-
pose a given representationρinto irreducible representations. For ifρ=⊕mμρμis a
direct sum decomposition ofρinto irreducible componentsρμ, where the coefficients
mμare non-negative integers, and ifρhas characterχ,then

χ(s)=

∑

μ

mμχμ(s).

Multiplying byχv(s−^1 ) and summing over alls∈G, we deduce from (6) that

g−^1

∑

s∈G

χ(s)χv(s−^1 )=mv. (7)

Thus the multiplicitiesmvare uniquely determined by the characterχof the represen-
tationρ. It follows thattwo representations are equivalent if and only if they have the
same character.
In the same way we find

g−^1

∑

s∈G

χ(s)χ(s−^1 )=

∑

μ

mμ^2. (8)

Hencea representationρwith characterχis irreducible if and only if

g−^1

∑

s∈G

χ(s)χ(s−^1 )= 1.

The procedure for decomposing a representation into its irreducible components
may be applied, in particular, to the regular representation. Evidently theg×gmatrix
representing an elementshas all its main diagonal elements 0 ifs=eand all its main
diagonal elements 1 ifs=e. Thus the characterχRof the regular representationρR
is given by

χR(e)=g,χR(s)=0ifs=e.