Spherical Harmonics - Connection With Representation Theory

Connection With Representation Theory

The space H_ℓ of spherical harmonics of degree ℓ is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on H_ℓ by function composition

for ψ a spherical harmonic and ρ a rotation. The representation H_ℓ is an irreducible representation of SO(3).

The elements of H_ℓ arise as the restrictions to the sphere of elements of A_ℓ: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R3. By polarization of ψ ∈ A_ℓ, there are coefficients symmetric on the indices, uniquely determined by the requirement

The condition that ψ be harmonic is equivalent to the assertion that the tensor must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), H_ℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ.

More generally, the analogous statements hold in higher dimensions: the space H_ℓ of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓ-tensors. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner.

The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.