Number Theory: An Introduction to Mathematics

412 X A Character Study

Proof We give a constructive proof due to Schur. Lets→A(s),where

A(s)=

(

P(s) Q(s)

0 R(s)

)

,

be a reducible representation of a groupGof finite orderg. Since the mapping
s→A(s)preserves products, we have

P(st)=P(s)P(t), R(st)=R(s)R(t), Q(st)=P(s)Q(t)+Q(s)R(t). (2)

The non-singular matrix

T=

(

IM

0 I

)

satisfies
(
P(t) Q(t)
0 R(t)

)

T=T

(

P(t) 0

0 R(t)

)

(3)

if and only if

MR(t)=P(t)M+Q(t).

Ta k e

M=g−^1

∑

s∈G

Q(s)R(s−^1 ).

Then, by (2),

P(t)M=g−^1

∑

s∈G

{Q(ts)−Q(t)R(s)}R(s−^1 )

=g−^1

∑

s∈G

Q(ts)R(s−^1 t−^1 )R(t)−Q(t)=MR(t)−Q(t),

and hence (3) holds.
Thus the given reducible representations→A(s)is the sum of two representa-
tionss→P(s)ands→R(s)of lower degree. The result follows by induction on the
degree.

Maschke’s original proof of Proposition 9 depended on showing that every repre-
sentation of a finite group is equivalent to a representation byunitarymatrices. We
briefly sketch the argument. Letρ:s→A(s)be a representation of a finite groupG
by linear transformations of a finite-dimensional vector spaceV. We may supposeV
equipped with a positive definite inner product(u,v). It is easily verified that

(u,v)G=g−^1

∑

t∈G

(A(t)u,A(t)v)