Number Theory: An Introduction to Mathematics

8 Generalizations 437

It may be shown that ifρis a unitary representation of a compact groupGin
a Hilbert spaceH,thenH may be represented as a direct sumH =⊕αHαof
mutually orthogonal finite-dimensional invariant subspacesHαsuch that, for everyα,
the restriction ofρtoHαis irreducible.
In particular, any irreducible unitary representation of a compact group is finite-
dimensional. Consequently it is possible to talk about matrix elements and traces,
i.e. characters, of irreducible unitary representations. The orthogonality relations for
matrix elements and for characters of irreducible representations of finite groups
remain valid for irreducible unitary representations of compact groups if one replaces
g−^1

∑

s∈Gf(s)by the invariant meanM(f).
Furthermore, any function f ∈ C(G)can be uniformly approximated by fi-
nite linear combinations of matrix elements of irreducible unitary representations,
and any class function f ∈ C(G)can be uniformly approximated by finite lin-
ear combinations of characters of irreducibleunitary representations. Finally, in the
direct sum decomposition of the regular representation into finite-dimensional irre-
ducible unitary representations, each irreducible representation occurs as often as its
dimension.
Thus the representation theory of compact groups is completely analogous to that
of finite groups. Indeed we may regard the representation theory of finite groups as
a special case, since any finite group is compact with the discrete topology and any
representation is equivalent to a unitary representation.
An example of a compact group which is neither finite nor abelian is the group
G=SU( 2 )of all 2×2 unitary matrices with determinant 1. The elements ofGhave
the form

g=

[

γδ

−δ ̄ γ ̄

]

,

whereγ,δare complex numbers such that|γ|^2 +|δ|^2 =1. Writingγ =ξ 0 +iξ 3 ,
δ=ξ 1 +iξ 2 , we see that topologicallySU( 2 )is homeomorphic to the sphere

S^3 ={x=(ξ 0 ,ξ 1 ,ξ 2 ,ξ 3 }∈R^4 :ξ 02 +ξ 12 +ξ 22 +ξ 32 = 1 }

and hence is compact andsimply-connected(i.e. it is path-connected and any closed
path can be continuously deformed to a point).
For any integern≥0, letVndenote the vector space of all polynomialsf(z 1 ,z 2 )
with complex coefficients which are homogeneous of degreen. Writingz=(z 1 ,z 2 ),
we have

zg=(γz 1 −δ ̄z 2 ,δz 1 + ̄γz 2 ).

Hence if we define a linear transformationTgofVnby(Tgf)(z) = f(zg),then
ρn:g →Tgis a representation ofSU( 2 )inVn. It may be shown that this repre-
sentation is irreducible and is unitary with respect to the inner product

(∑n

k= 0

αkzk 1 zn 2 −k,

∑n

k= 0

βkz 1 kzn 2 −k

)

=

∑n

k= 0

k!(n−k)!αkβ ̄k.