Spherical Harmonics

The spherical harmonics \(Y^l(x)\) are functions defined on the sphere \(S^2\). They form a basis of the space on function on the sphere:

\[\mathcal{F} = \{ S^2 \longrightarrow \mathbb{R} \}\]

On this space it is natural how the group \(O(3)\) acts, Given \(p_a, p_v\) two scalar representations:

\[[L(g) f](x) = p_v(g) f(p_a(g) R(g)^{-1} x), \quad \forall f \in \mathcal{F}, x \in S^2\]

\(L\) is representation of \(O(3)\). But \(L\) is not irreducible. It can be decomposed via a change of basis into a sum of irreps, In a handwavey notation we can write:

\[Y^T L(g) Y = 0 \oplus 1 \oplus 2 \oplus 3 \oplus \dots\]

where the change of basis are the spherical harmonics! This notation is handwavey because \(x\) is a continuous variable, and therefore the change of basis \(Y\) is not a matrix.

As a consequence, the spherical harmonics are equivariant,

\[Y^l(R(g) x) = D^l(g) Y^l(x)\]
r = s2_grid()

r is a grid on the sphere.