# 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.