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A particularly striking application to physics and chemistry is explained in Singer's book Linearity, symmetry, and prediction in the hydrogen atom. The practical problem, in the large, is easy to state: what is the stuff around us made of, and why does it react with other stuff the way it does? More precisely, what explains the structure of the periodic table?

There is no a priori reason that the elements ought to naturally arrange themselves in rows of size $2, 8, 8, 18, 18, ...$ with repeating chemical properties. This periodic structure profoundly shapes the nature of the world around us and so ought to be well worth trying to understand on a deeper level.

Physically, the answer has to do with the way that electrons arrange themselves around a nucleus, one of the classic examples of the breakdown of classical mechanics. The Bohr model posits that electrons are arranged in discrete orbitals $n = 1, 2, 3, ... $ with energy levels proportional to $- \frac{1}{n^2}$ such that the $n^{th}$ energy level admits at most $2n^2$ electrons. This behavior $- \frac{1}{n^2}$ can be empirically deduced by an examination of atomic spectra but the Bohr model still does not provide a conceptual explanation of it.

That explanation comes from full-blown quantum mechanics, which already requires a fair amount of nontrivial mathematics. For our purposes quantum mechanics will be described by a Hilbert space $K = L^2(X)$ where $X$ is the classical phase space (e.g. $\mathbb{R}^3$) and a self-adjoint operator $H : K \to K$, the Hamiltonian, which will describe the evolution of states via the Schrödinger equation.

The simplest case is that of an electron orbiting a single proton, in which case one can explicitly write down the potential. In this case the Schrödinger equation can be solved fairly explicitly and the answer tells you what electron orbitals look like, but it turns out that one can do much better: it is possible to predict the solutions and their properties using representation theory.

To start with, the Coulomb potential has a spherical symmetry, so this endows $K$ with the structure of a unitary representation of $\text{SO}(3)$. By identifying two wave functions together if they lie in the same representation we can hope to have a physical classification of the possible states of an electron; the idea is that physical quantities we care about should be invariant under physical symmetries (e.g. mass, energy, charge). The action of $\text{SO}(3)$ breaks up the space of possible states based on their angular momentum (Noether's theorem). The corresponding representations have dimensions $1, 3, 5, 7, ...$ and indeed we find that we can decompose the number of elements in each row of the periodic table as

$$2 = 1 + 1$$ $$8 = 1 + 1 + 3 + 3$$ $$18 = 1 + 1 + 3 + 3 + 5 + 5$$

corresponding to the possible angular momentum values allowed at each energy level. Of course these symmetry considerations apply to every spherically symmetric system so the $\text{SO}(3)$ symmetry cannot tell us anything more specific.

But it turns out there is even more symmetry to exploit. First of all, remarkably enough the $\text{SO}(3)$ symmetry extends to an $\text{SO}(4)$ symmetry. (I do not really know a conceptual explanation of this, unfortunately; I have a half-baked one which I'm not sure is valid.) The irreducible representations of $\text{SO}(4)$ occurring here are precisely the ones of dimensions $1, 1 + 3, 1 + 3 + 5, ...$ and they break up into irreducible $\text{SO}(3)$ representations in exactly the right way to account for the above pattern up to a factor of $2$. Second of all, the factor of $2$ is accounted for by an additional action of $\text{SU}(2)$ coming from electron spin (the thing that makes MRI machines work).

So representation theory provides a strikingly elegant answer to the question of how the periodic table is arranged (if one accepts that a single proton is a good approximation to a general atomic nucleus). Of course there is much more to say here about the relation between representation theory and physics and chemistry, but I am not the one to ask...