At fixed $l$, the $2l+1$ spherical harmonics:

$$ Y_l^m(\theta, \phi) \ \ \ m \in [-l, -l+1, \cdots, l-1, l] $$

are a $2l+1$ dimensional irreducible$^1$ representation of $SO(3)$, with Casimir invariant $l(l+1)$. [1] means they are closed under rotations, so for some arbitrary rotation, $R_{{\bf r} \rightarrow {\bf r'}}$:

$$ Y_l^m(\theta', \phi')= \sum_{m'=-l}^l \big[D^{(l)}_{mm'}(R)\big]^*Y_l^{m'}(\theta, \phi) $$

That is: dipoles rotate into other dipoles, quadrupoles rotate into other quadrupoles, and so on.

I'm not sure how it relates to representation theory, but one thing that makes the $Y_l^m(\theta,\phi)$ useful in physics is that they are eigenfunctions under $z$-rotations, with eigenvalue $\exp(im\phi)$:

$$ R_{\phi}\big[Y_l^m(\theta, \phi)\big] = e^{im\phi} Y_l^m(\theta, \phi)$$

and also something that is no obvious (to me) from pictures of atomic orbitals:

$$ \sum_{m=-l}^l ||Y_l^m(\theta, \phi)||^2 \propto 1 $$

that is, a closed shell is isotropic, it has spherical symmetry.

At fixed $l$, the $2l+1$ spherical harmonics:

$$ Y_l^m(\theta, \phi) \ \ \ m \in [-l, -l+1, \cdots, l-1, l] $$

are a $2l+1$ dimensional irreducible$^1$ representation of $SO(3)$, with Casimir invariant $l(l+1)$. [1] means they are closed under rotations, so for some arbitrary rotation, $R_{{\bf r} \rightarrow {\bf r'}}$:

$$ Y_l^m(\theta', \phi')= \sum_{m'=-l}^l \big[D^{(l)}_{mm'}(R)\big]^*Y_l^{m'}(\theta, \phi) $$

That is: dipoles rotate into other dipoles, quadrupoles rotate into other quadrupoles, and so on.

I'm not sure how it relates to representation theory, but one thing that makes the $Y_l^m(\theta,\phi)$ useful in physics is that they are eigenfunctions under $z$-rotations, with eigenvalue $\exp(im\phi)$:

$$ R_{\phi}\big[Y_l^m(\theta, \phi)\big] = e^{im\phi} Y_l^m(\theta, \phi)$$

and also something that is not obvious (to me) from pictures of atomic orbitals:

$$ \sum_{m=-l}^l ||Y_l^m(\theta, \phi)||^2 \propto 1 $$

that is, a closed shell is isotropic, it has spherical symmetry.

At fixed $l$, the $2l+1$ spherical harmonics:

$$ Y_l^m(\theta, \phi) \ \ \ m \in [-l, -l+1, \cdots, l-1, l] $$

are a $2l+1$ dimensional representation of $SO(3)$, with Casimir invariant $l(l+1)$. That means they are closed under rotations, so for some arbitrary rotation, $R_{{\bf r} \rightarrow {\bf r'}}$:

$$ Y_l^m(\theta', \phi')= \sum_{m'=-l}^l \big[D^{(l)}_{mm'}(R)\big]^*Y_l^{m'}(\theta, \phi) $$

That is: dipoles rotate into other dipoles, quadrupoles rotate into other quadrupoles, and so on.

I'm not sure how it relates to representation theory, but one thing that makes the $Y_l^m(\theta,\phi)$ useful in physics is that they are eigenfunctions under $z$-rotations, with eigenvalue $\exp(im\phi)$:

$$ R_{\phi}\big[Y_l^m(\theta, \phi)\big] = e^{im\phi} Y_l^m(\theta, \phi)$$

and also something that is no obvious (to me) from pictures of atomic orbitals:

$$ \sum_{m=-l}^l ||Y_l^m(\theta, \phi)||^2 \propto 1 $$

that is, a closed shell is isotropic, it has spherical symmetry.

At fixed $l$, the $2l+1$ spherical harmonics:

$$ Y_l^m(\theta, \phi) \ \ \ m \in [-l, -l+1, \cdots, l-1, l] $$

are a $2l+1$ dimensional irreducible$^1$ representation of $SO(3)$, with Casimir invariant $l(l+1)$. [1] means they are closed under rotations, so for some arbitrary rotation, $R_{{\bf r} \rightarrow {\bf r'}}$:

$$ Y_l^m(\theta', \phi')= \sum_{m'=-l}^l \big[D^{(l)}_{mm'}(R)\big]^*Y_l^{m'}(\theta, \phi) $$

That is: dipoles rotate into other dipoles, quadrupoles rotate into other quadrupoles, and so on.

I'm not sure how it relates to representation theory, but one thing that makes the $Y_l^m(\theta,\phi)$ useful in physics is that they are eigenfunctions under $z$-rotations, with eigenvalue $\exp(im\phi)$:

$$ R_{\phi}\big[Y_l^m(\theta, \phi)\big] = e^{im\phi} Y_l^m(\theta, \phi)$$

and also something that is no obvious (to me) from pictures of atomic orbitals:

$$ \sum_{m=-l}^l ||Y_l^m(\theta, \phi)||^2 \propto 1 $$

that is, a closed shell is isotropic, it has spherical symmetry.