410 X A Character Study
4 Representations of Arbitrary Finite Groups
The problem of extending the character theory of finite abelian groups to arbitrary
finite groups was proposed by Dedekind and solved by Frobenius (1896). Simplifi-
cations were afterwards found by Frobenius himself, Burnside and Schur (1905). We
will follow Schur’s treatment, which is distinguished by its simplicity. It turns out that
for nonabelian groups the concept of ‘representation’ is more fundamental than that of
‘character’.
Arepresentationof a groupGis a mappingρofGinto the set of all linear trans-
formations of a finite-dimensional vector spaceVover the fieldCof complex numbers
which preserves products, i.e.
ρ(st)=ρ(s)ρ(t) for alls,t∈G, (1)and maps the identity element ofGinto the identity transformation ofV :ρ(e)=
I. The dimension of the vector spaceV is called thedegreeof the representation
(although ‘dimension’ would be more natural).
It follows at once from (1) that
ρ(s)ρ(s−^1 )=ρ(s−^1 )ρ(s)=I.Thus, for everys∈G,ρ(s)is an invertible linear transformation ofVandρ(s−^1 )=
ρ(s)−^1. (Hence a representation ofGis ahomomorphismofGinto the groupGL(V)
of all invertible linear transformations ofV.)
Any group has atrivial representationof degree 1 in which every element of the
group is mapped into the scalar 1.
Also, with any groupGof finite orderga representation of degreegmay be de-
fined in the following way. Lets 1 ,...,sgbe an enumeration of the elements ofGand
lete 1 ,...,egbeabasisforag-dimensional vector spaceVoverC. We define a linear
transformationA(si)ofVby its action on the basis elements:
A(si)ej=ek ifsisj=sk.Then, for alls,t∈G,
A(s−^1 )A(s)=I, A(st)=A(s)A(t).Thus the mappingρR:si →A(si)is a representation ofG, known as theregular
representation.
By choosing a basis for the vector space wecan reformulate the preceding defini-
tions in terms of matrices. A representation of a groupGis then a product-preserving
maps→ A(s)ofGinto the group of alln×nnon-singular matrices of complex
numbers. The positive integernis the degree of the representation. However, we must
regard two matrix representationss→ A(s)ands→B(s)asequivalentif one is
obtained from the other simply by changing the basis of the vector space, i.e. if there
exists a non-singular matrixTsuch that
T−^1 A(s)T=B(s) for everys∈G.