- Show that the representationπon such polynomials given in section 8.2
(induced from theSU(2) representation onC^2 ) is a unitary representation
with respect to this inner product. - Show that the monomials
z 1 jz 2 k
√
j!k!
are orthonormal with respect to this inner product (hint: break up the
integrals into integrals over the two complex planes, use polar coordinates). - Show that the differential operatorπ′(S 3 ) is self-adjoint. Show thatπ′(S−)
andπ′(S+) are adjoints of each other.
Problem 2:
Using the formulas for theY 1 m(θ,φ) and the inner product of equation 8.3,
show that
- TheY 11 ,Y 10 ,Y 1 −^1 are orthonormal.
- Y 11 is a highest weight vector.
- Y 10 andY 1 −^1 can be found by repeatedly applyingL−to a highest weight
vector.
Problem 3:
Recall that the Casimir operatorL^2 ofso(3) is the operator that in any
representationρis given by
L^2 =L^21 +L^22 +L^23
Show that this operator commutes with theρ′(X) for allX ∈so(3). Use
this to show thatL^2 has the same eigenvalue on all vectors in an irreducible
representation ofso(3).
Problem 4:
For the case of theSU(2) representationπon polynomials onC^2 given in
the notes, find the Casimir operator
L^2 =π′(S 1 )π′(S 1 ) +π′(S 2 )π′(S 2 ) +π′(S 3 )π′(S 3 )
as an explicit differential operator. Show that homogeneous polynomials are
eigenfunctions, and calculate the eigenvalues.
B.5 Chapter
Problem 1:
Consider the action ofSU(2) on the tensor productV^1 ⊗V^1 of two spin
representations. According to the Clebsch-Gordan decomposition, this breaks
up into irreducibles asV^0 ⊕V^2.