Number Theory: An Introduction to Mathematics

414 X A Character Study

5 Characters of Arbitrary Finite Groups..........................

By definition, thetraceof ann×nmatrixA=(αij)is the sum of its main diagonal
elements:

trA=

∑n

i= 1

αii.

It is easily verified that, for anyn×nmatricesA,Band any scalarsλ,μ,wehave

tr(λA+μB)=λtrA+μtrB,

tr(AB)=tr(BA), tr(A⊗B)=(trA)(trB).

Letρ:s→A(s)be a matrix representation of a groupG.Bythecharacterof the
representationρwe mean the mappingχ:G→Cdefined by

χ(s)=trA(s).

Since tr(T−^1 AT)=tr(AT T−^1 )=trA, equivalent representations have the same char-
acter. The significance of characters stemsfrom the converse, which will be proved
below.
Clearly the characterχof a representationρis aclass function,i.e.

χ(st)=χ(ts) for alls,t∈G.

The degreenof the representationρis determined by its characterχ,sinceA(e)=In
and henceχ(e)=n.
If the representationρis the sum of two representationsρ′andρ′′, the correspond-
ing charactersχ,χ′,χ′′evidently satisfy

χ(s)=χ′(s)+χ′′(s) for everys∈G.

On the other hand, if the representationρis the product of the representationsρ′and
ρ′′,then

χ(s)=χ′(s)χ′′(s) for everys∈G.

Thus the set of all characters of a groupis closed under addition and multiplication.
The character of an irreducible representation will be called simply anirreducible
character.
LetGbe a group andρa representation ofGof degreenwith characterχ.If
sis an element ofGof finite orderm, then by restrictionρdefines a representation
of the cyclic group generated bys. By Proposition 9 and Corollary 11, this represen-
tation is equivalent to a sum of representations of degree 1. Thus ifSis the matrix
representings, there exists an invertible matrixTsuch that

T−^1 ST=diag[ω 1 ,...,ωn]