Number Theory: An Introduction to Mathematics

4 Representations of Arbitrary Finite Groups 411

It is easily verified that ifs→A(s)is a matrix representation of degreenof a

groupG,thens→A(s−^1 )t(the transpose ofA(s−^1 )) is a representation of the same

degree, thecontragredient representation.Furthermore,s→detA(s)is a representa-

tion of degree 1.

Again, ifρ:s→A(s)andσ:s→B(s)are matrix representations of a group

G, of degreesmandnrespectively, then the Kronecker product mapping

s→A(s)⊗B(s)

is also a representation ofG,ofdegreemn,since

(A(s)⊗B(s))(A(t)⊗B(t))=A(st)⊗B(st).

We will call this representation simply theproductof the representationsρandσ,and
denote it byρ⊗σ.
The basic problem of representation theory is to determine all possible representa-
tions of a given group. As we will see, all representations may in fact be built up from
certain ‘irreducible’ ones.
Letρbe a representation of a groupGby linear transformations of a vector space
V. If a subspaceUofVisinvariantunderG,i.e.if

ρ(s)U⊆U for everys∈G,

then the restrictions toUof the given linear transformations provide a representation

ρUofGby linear transformations of the vector spaceU. If it happens that there exists

another subspaceWinvariant underGsuch thatVis the direct sum ofUandW,i.e.

V=U+WandU∩W={ 0 }, then the representationρis completely determined by

the representationsρUandρWand will be said simply to be theirsum.

A representationρof a groupGby linear transformations of a vector spaceVis

said to beirreducibleif no nontrivial proper subspace ofVis invariant underG,and

reducibleotherwise. Evidently any representation of degree 1 is irreducible.

A matrix representations→A(s),ofdegreen, of a groupGis reducible if it is

equivalent to a representation in which all matrices have the block form

(

P(s) Q(s)

0 R(s)

)

,

whereP(s)is a square matrix of orderm, 0 <m<n.Thens→P(s)ands→R(s)

are representations ofGof degreesmandn−mrespectively. The given representation

is the sum of these representations if there exists a non-singular matrixTsuch that

T−^1 A(s)T=

(

P(s) 0

0 R(s)

)

for everys∈G.

The following theorem of Maschke (1899) reduces the problem of finding all

representations of afinitegroup to that of finding all irreducible representations.

Proposition 9Every representation of a finite group is (equivalent to) a sum of

irreducible representations.