3.16. Lecture 16

3.16.1. Harmonic functions

We can use our understanding of the representation theory of $\mathrm{SO}(3)$ to shed light on the classical theory of spherical harmonics.

We let $\mathcal{P}^{\ell}$ be the subspace of $\mathbb{C}[x,y,z]$ consisting of homogeneous polynomials of degree $\ell$ . This has an action of $\mathrm{SO}(3)$ given by

g\cdot f(\mathbf{x})=f(g^{T}\mathbf{x}),

where $\mathbf{x}=\begin{pmatrix}x\\ y\\ z\end{pmatrix}$ is a vector in $\mathbb{C}^{3}$ . We therefore get a representation of $\mathrm{SO}(3)$ .

Lemma 3.16.1.

The elements $J_{x},J_{y}$ , and $J_{z}$ of $\mathrm{SO}(3)$ act on $\mathcal{P}^{\ell}$ according to the following formulae:

	$\displaystyle J_{x}$	$\displaystyle=z\frac{\partial}{\partial y}-y\frac{\partial}{\partial z}$
	$\displaystyle J_{y}$	$\displaystyle=x\frac{\partial}{\partial z}-z\frac{\partial}{\partial x}$
	$\displaystyle J_{z}$	$\displaystyle=y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}.$

Proof.

Exercise (Problem 3.16.2). ∎

It will be useful in what follows to recall

Lemma 3.16.2.

(Euler’s formula) If $f$ is a homogeneous polynomial of degree $\ell$ , then

x\frac{\partial}{\partial x}{f}+y\frac{\partial}{\partial y}{f}+z\frac{% \partial}{\partial z}{f}=\ell f.

The representation $\mathcal{P}^{\ell}$ is not irreducible. Let $r^{2}\in\mathbb{C}[x,y,z]$ be the polynomial

r^{2}=x^{2}+y^{2}+z^{2}.

Note that $r^{2}$ is clearly invariant under the action of $\mathrm{SO}(3)$ .

Lemma 3.16.3.

The map $\mathcal{P}^{\ell}\to\mathcal{P}^{\ell+2}$ defined by

f\longmapsto r^{2}f

is an injective homomorphism of $\mathrm{SO}(3)$ -representations.

Proof.

We have, for $g\in\mathrm{SO}(3)$ ,

g(r^{2}f)=g(r^{2})g(f)=r^{2}g(f)

as required. ∎

Next, we consider the Laplace operator $\Delta$ given by

f\longmapsto\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y% ^{2}}+\frac{\partial^{2}f}{\partial z^{2}}.

Lemma 3.16.4.

The Laplace operator $\Delta:\mathcal{P}^{\ell}\to\mathcal{P}^{\ell-2}$ is an $\mathrm{SO}(3)$ -homomorphism.

Proof.

We must show that, for $g=(g_{ij})\in\mathrm{SO}(3)$ ,

g\cdot(\Delta f)=\Delta(g\cdot f).

We have

\frac{\partial}{\partial x_{i}}(g\cdot f)({\bf x})=\sum_{j}g_{ij}\frac{% \partial f}{\partial x_{j}}(g^{T}x)

and so

\frac{\partial}{\partial x_{i}}\frac{\partial}{\partial x_{i}}(g\cdot f)({\bf x% })=\sum_{j,k}g_{ij}g_{ik}\frac{\partial}{\partial x_{k}}\frac{\partial}{% \partial x_{j}}(g^{T}x).

We sum over $i$ , for fixed $j$ and $k$ :

\sum_{i}g_{ij}g_{ik}=\delta_{jk}

since $g$ is orthogonal. We therefore obtain

\sum_{i}\frac{\partial^{2}}{\partial x_{i}^{2}}(g\cdot f)({\bf x})=\sum_{j}% \frac{\partial^{2}}{\partial x_{j}^{2}}(f)(g^{T}{\bf x}),

and therefore

\Delta(g\cdot f)=g\cdot\Delta(f).

∎

An element $f\in\mathcal{P}^{\ell}$ is harmonic if $\Delta f=0$ . Since $\dim(\mathcal{P}^{\ell})>\dim(\mathcal{P}^{\ell-2})$ , harmonic polynomials must exist. We write $\mathcal{H}^{\ell}\subseteq\mathcal{P}^{\ell}$ for the space of harmonic polynomials.

Lemma 3.16.5.

On $\mathcal{P}^{\ell}$ , we have

r^{2}\Delta=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}+\ell^{2}+\ell.

Proof.

This is Problem 3.16.2. ∎

It follows that $r^{2}\Delta$ preserves each irreducible subrepresentation of $\mathcal{P}^{\ell}$ (since $J_{x},J_{y},J_{z}$ do). Furthermore, by Schur’s lemma it must act on each irreducible subrepresentation as a scalar. We determine that scalar.

Lemma 3.16.6.

Suppose that $V\subseteq\mathcal{P}^{\ell}$ is an irreducible subrepresentation with highest weight $ik$ . Then, for all $f\in V$ ,

(r^{2}\Delta)(f)=(\ell^{2}+\ell-k^{2}-k)f=(\ell-k)(\ell+k+1)f.

Proof.

Since $r^{2}\Delta$ is a $\mathfrak{so}_{3}$ -homomorphism and $V$ is irreducible, by Schur’s lemma it acts as a scalar on $V$ . It therefore suffices to compute the action on a highest weight vector ${\bf v}\in V$ . So $J_{z}{\bf v}=ik{\bf v}$ and $(J_{x}-iJ_{y}){\bf v}=0$ . It follows that $J_{z}^{2}{\bf v}=-k^{2}{\bf v}$ and, as

	$\displaystyle J_{x}^{2}+J_{y}^{2}$	$\displaystyle=(J_{x}+iJ_{y})(J_{x}-iJ_{y})+i[J_{x},J_{y}]$
		$\displaystyle=(J_{x}+iJ_{y})(J_{x}-iJ_{y})+iJ_{z},$

we have $(J_{x}^{2}+J_{y}^{2}){\bf v}=0{\bf v}-k{\bf v}$ .

Applying the previous lemma gives the result. ∎

Theorem 3.16.7.

For every $\ell\geq 0$ ,

\mathcal{P}^{\ell}=\mathcal{H}^{\ell}\oplus r^{2}\mathcal{P}^{\ell-2}.

The space $\mathcal{H}^{\ell}$ is the irreducible highest weight representation of $\mathrm{SO}(3)$ of dimension $2\ell+1$ , and the space $\mathcal{P}^{\ell}$ , as an $\mathrm{SO}(3)$ -representation, the direct sum of representations of weights $i\ell,i(\ell-2),\ldots$ , each occurring with multiplicity one.

Proof.

We use induction on $\ell$ . The case $\ell=0$ is clear (we just have the trivial representation). For $\ell=1$ we only have degree $1$ polynomials so again $\mathcal{P}^{1}=\mathcal{H}^{1}$ .

Suppose true for $\ell-1$ with $\ell\geq 2$ . By the previous lemma, the space $\mathcal{H}^{\ell}$ is the sum of all the copies inside $\mathcal{P}^{\ell}$ of the irreducible representation with with highest weight $i\ell$ . Since $\mathcal{P}^{\ell-2}$ does not contain this irreducible representation, by the inductive hypothesis, we have

\mathcal{H}^{\ell}\cap r^{2}\mathcal{P}^{\ell-2}=\{0\}.

Since, $\mathcal{H}^{\ell}\neq 0$ as already discussed, its dimension is a positive multiple of $2\ell+1$ . However, its dimension is at most

\dim\mathcal{P}^{\ell}-\dim\mathcal{P}^{\ell-2}=\binom{\ell+2}{2}-\binom{\ell}% {2}=2\ell+1.

It follows that $\mathcal{H}^{\ell}$ is irreducible, and that we have

\mathcal{H}^{\ell}\oplus r^{2}\mathcal{P}^{\ell-2}=\mathcal{P}^{\ell}.

The statement about the decomposition into irreducibles follows. ∎

The proof of this theorem shows that

\mathcal{P}^{\ell}=\mathcal{H}^{\ell}\oplus r^{2}\mathcal{H}^{\ell-2}\oplus r^% {4}\mathcal{H}^{\ell-4}\ldots

so that every polynomial has a unique decomposition as a sum of harmonic polynomials multiplied by powers of $r^{2}$ .

We can go further and give nice bases for the $\mathcal{H}^{\ell}$ by taking weight vectors for $J_{z}$ . First, we have

Lemma 3.16.8.

The function $(x-iy)^{\ell}\in\mathcal{P}^{\ell}$ is a highest weight vector of weight $i\ell$ .

Proof.

This is Problem 3.16.2(a). ∎

We then obtain a weight basis by repeatedly applying the lowering operator

J_{x}+iJ_{y}=(ix-y)\frac{\partial}{\partial z}{}+z(\frac{\partial}{\partial y}% {}-i\frac{\partial}{\partial x}{}).

The functions thus obtained are known as ’spherical harmonics’ (at least, up to normalisation), and give a particularly nice basis for the space of functions on the sphere $S^{2}\subseteq\mathbb{R}^{3}$ . The decomposition of a function into spherical harmonics is analogous to the Fourier decomposition of a function on the unit circle.

Example 3.16.9.

If $\ell=1$ , then $\mathcal{P}^{\ell}=\mathcal{H}^{\ell}$ , and the weight vectors are

x+iy,z,x-iy.

If $\ell=2$ , a basis of $\mathcal{H}^{\ell}$ made up of weight vectors is

(x-iy)^{2},z(x-iy),x^{2}+y^{2}-2z^{2},z(x+iy),(x+iy)^{2}.

3.16.2. Exercises

Problem 41. Prove that the action of $A=(a_{ij})\in\mathfrak{so}_{3}$ is given by

\sum_{i,j}a_{ij}x_{i}\frac{\partial}{\partial x_{j}}

by considering

\frac{d}{dt}f(\exp(tA^{T})\mathbf{x})=\frac{d}{dt}f((\operatorname{Id}+tA^{T})% \mathbf{x})

at $t=0$ (where we rewrite $x,y,z$ as $x_{1},x_{2},x_{3}$ ). Thus prove Lemma 3.16.1.

Problem 42.

(a)

Verify the formula

$r^{2}\Delta=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}+\ell^{2}+\ell$

as operators on $\mathcal{P}^{\ell}$ .
(b)

Find the image of the Casimir element from problem 3.13.3 under our isomorphism $\mathfrak{sl}_{2,\mathbb{C}}\to\mathfrak{so}_{3,\mathbb{C}}$ , and compare to part (a).

Problem 43. Let $\ell\geq 1$ .

(a)

Verify that $(x-iy)^{\ell}$ is a highest weight vector in $\mathcal{H}^{\ell}$ .
(b)

By applying the lowering operator, find weight vectors of weights $i(\ell-1)$ and $i(\ell-2)$ .
(c)

Find a basis of weight vectors in $\mathcal{H}^{\ell}$ when $\ell=1$ and $\ell=2$ (i.e. verify Example 3.16.9).

Problem 44.

(a)

Prove that, for $f\in\mathcal{P}^{\ell}$ ,

$\Delta(r^{2}f)=r^{2}\Delta(f)+2(2\ell+3)f.$
(b)

Find a similar formula for

$\Delta(r^{2k}f)-r^{2k}\Delta(f).$
(c)

Use this to give another proof that

$\mathcal{H}^{\ell}\cap r^{2}\mathcal{P}^{\ell-2}=\{0\}.$

(Hint: if $f$ is in the intersection, let $f=r^{2k}g$ , $g$ not divisible by $r^{2}$ ).

Problem 45.

(a)

Let $V$ be the standard — three-dimensional — representation of $\mathfrak{so}_{3}$ . Find a basis of weight vectors for $\operatorname{Sym}^{2}(V)$ , and decompose it into irreducible subrepresentations.
(b)

Let $\mathcal{H}^{2}$ be the five-dimensional representation of $\mathrm{SO}(3)$ . Decompose $\mathcal{H}^{2}\otimes\mathcal{H}^{2}$ into irreducible representations.