### Lanthanides and the Exceptional Lie Group G_{2}

#### Posted by John Baez

The lanthanides are the 14 elements defined by the fact that their electrons fill up, one by one, the 14 orbitals in the so-called f subshell. Here they are:

lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium.

They are also called ‘rare earths’, but that term is often also applied to 3 more elements. Why? That’s a fascinating puzzle in its own right. But what matters to me now is something else: an apparent connection between the lanthanides and the exceptional Lie group G_{2}!

Alas, this connection remains profoundly mysterious to me, so I’m pleading for your help.

Why are there 14 lanthanides? It’s because

- the electrons in the f subshell have orbital angular momentum $3$,
- the irreducible representation of $SO(3)$ corresponding to angular momentum $j = 3$ has dimension $2j+1 = 7$, and
- each electron can also come in $2$ spin states, for a total of $2 \times 7 = 14$ states.

What does this have to do with the exceptional Lie group $\mathrm{G}_2$? The aforementioned $7$-dimensional representation of $SO(3)$ can also be thought of as the space of imaginary octonions, since — rather amazingly — the key algebraic structure on the imaginary octonions, their cross product, is invariant under this representation of $SO(3)$. Indeed, the 7-dimensional representation of $\mathrm{G}_2$ on the imaginary octonions remains irreducible when restricted to a certain $SO(3)$ subgroup of $\mathrm{G}_2$, sometimes called $SO(3)_{irr}$ — and this gives our friend the $j = 3$ representation of $SO(3)$.

All these facts were noticed *and apparently put to some use* by the mathematician and physicist Giulio Racah, famous for his work on the quantum mechanics of angular momentum. This was recently brought to my attention by Paul Schwahn, who is working to better understand the underlying math.

But Racah’s thoughts remain deeply mysterious, because Schwahn found them in a fragmentary second-hand account, and we haven’t been able to find more details!

I thought me mentioning the f-orbital was just a crackpot idea.

But in the AMS volume

Selected Papers of E. B. Dynkin with Commentary(which also contains Dynkin’s original discovery of SO(3)ᵢᵣᵣ) one finds a short review by Yuval Ne’eman, titled “Dynkin Diagrams in the Physics of Particles, Fields and Strings”. The whole thing is a delight to read, but he writes something particularly interesting about an idea of physicist Giulio Racah:“Racah found ways of applying various simple algebras in classifying higher spectra. His methods, later developed and extended by such as L. Biedenharn and M. Moshinsky, exploited higher rank Lie algebras applied to the representation spaces of $SO(3)$. I recall Racah enjoying (anecdotically) the fact that he had found an application for Cartan’s exceptional $\mathrm{G}(2)$, in studying the f-subshell in atomic spectra. One defines an $SO(7)$ algebra acting on some constructs involving the 7-dimensional f-subshell representation of $SO(3)$ - and the inclusion $G(2) \subset SO(7)$ does it. In these very complicated atomic spectra of the lanthanides, it provides some physical insights.”

I really wonder that these physical insights are…

It’s possible that even if Racah’s thoughts are lost in the dark mist of time, later researchers on group representation theory and the quantum mechanics of atoms have used the Lie group $\mathrm{G}_2$ to understand something about f subshell electrons. For example, Biedenharn’s book may contain some clues. But I haven’t yet turned up any clues yet.

## Re: Lanthanides and the Exceptional Lie Group G2

Here is a hint of $\mathrm{G}_2$ showing up in studies of the f shell:

Journal of Mathematical Chemistry44(1) (2008), 5–19. (paywalled link.)$\mathrm{L}_n(q) = PSL(n,\mathbb{F}_q)$ is the group of invertible $n \times n$ matrices of determinant $1$ over the field with $q$ elements, modulo multiples of the identity matrix. $\mathrm{L}_2(7)$ has 168 elements and it’s famous for lovers of the octonions as the symmetry group of the Fano plane.

I haven’t gotten the actual paper yet. The study of the g shell is deeply unpopular in chemistry since it only applies to elements of atomic number 124 and higher!