Number Theory: An Introduction to Mathematics

438 X A Character Study

Moreover, every irreducible representation ofSU( 2 )is equivalent toρnfor some
n≥0.
To determine the characterχnofρnwe observe that anyg∈Gis conjugate inG
to a diagonal matrix

t=

(

eiθ 0

0 e−iθ

)

,

whereθ∈R.Iffk(z 1 ,z 2 )=zk 1 zn 2 −k( 0 ≤k≤n),then

(Ttfk)(z 1 ,z 2 )=(eiθz 1 )k(e−iθz 2 )n−k=ei(^2 k−n)θfk(z 1 ,z 2 ).

Since the polynomialsf 0 ,...,fnare a basis forVnit follows that

χn(g)=χn(t)=

∑n

k= 0

ei(^2 k−n)θ.

Thusχn(I)=n+1,χn(−I)=(− 1 )n(n+ 1 )and

χn(g)={ei(n+^1 )θ−e−i(n+^1 )θ}/{eiθ−e−iθ}=sin(n+ 1 )θ/sinθifg=I,−I.

From this formula we can easily deduce the decomposition of the product repre-
sentationρm⊗ρninto irreducible components. Since

χm(g)χn(g)=(einθ+ei(n−^2 )θ+···+e−inθ){ei(m+^1 )θ−e−i(m+^1 )θ}/{eiθ−e−iθ}

=χm+n(g)+χm+n− 2 (g)+···+χ|m−n|(g),

we have theClebsch–Gordan formula

ρm⊗ρn=ρm+n+ρm+n− 2 +···+ρ|m−n|.

This formula is the group-theoretical basis for the rule in atomic physics which
determines the possible values of the angular momentum when two systems with given
angular momenta are coupled.
The complex numbersγ,δwith|γ|^2 +|δ|^2 = 1 which specify the matrix
g∈SU( 2 )can be uniquely expressed in the form

γ=ei(ψ+φ)/^2 cosθ/ 2 ,δ=ei(ψ−φ)/^2 sinθ/ 2 ,

where 0≤θ≤π,0≤φ< 2 π,− 2 π≤ψ< 2 π. Then the invariant mean of any
continuous functionf:SU( 2 )→Cis given by

M(f)=( 1 / 16 π^2 )

∫ 2 π

− 2 π

∫ 2 π

0

∫π

0

f(θ,φ,ψ)sinθdθdφdψ.

Another example of a compact group which is neither finite nor abelian is the group
SO( 3 )of all 3×3 real orthogonal matrices with determinant 1. The representations
ofSO( 3 )may actually be obtained from those ofSU( 2 ), since the two groups are