What's the status of the following relationship between Ramanujan's $\tau$ function and the simple Lie algebras?

Question

Qiaochu asked this in the comments to this question. Since this is really his question, not mine, I will make this one Community Wiki. In MR0522147, Dyson mentions the generating function $\tau(n)$ given by $$ \sum_{n=1}^\infty \tau(n)\,x^n = x\prod_{m=1}^\infty (1 - x^m)^{24} = \eta(x)^{24}, $$ which is apparently of interest to the number theorists ($\eta$ is Dedekind's function). He mentions the following formula for $\tau$: $$\tau(n) = \frac{1}{1!\,2!\,3!\,4!} \sum \prod_{1 \leq i < j \leq 5} (x_i - x_j)$$ where the sum ranges over $5$-tuples $(x_1,\dots,x_5)$ of integers satisfying $x_i \equiv i \mod 5$, $\sum x_i = 0$, and $\sum x_i^2 = 10n$. Apparently, the $5$ and $10$ are because this formula comes from some identity of $\eta(x)^{10}$. Dyson mentions that there are similar formulas coming from identities with $\eta(x)^d$ when $d$ is on the list $d = 3, 8, 10, 14,15, 21, 24, 26, 28, 35, 36, \dots$. The list is exactly the dimensions of the simple Lie algebras, except for the number $26$, which doesn't have a good explanation. The explanation of the others is in I. G. Macdonald, Affine root systems and Dedekind's $\eta$-function, Invent. Math. 15 (1972), 91--143, MR0357528, and the reviewer at MathSciNet also mentions that the explanation for $d=26$ is lacking.

So: in the last almost-40 years, has the $d=26$ case explained?

Answer 1

The case of $d=26$ is related to the exceptional Lie algebra $F_4$. Let me quote from the 1980 paper by Monastyrsky which was originally published as a supplement to the Russian translation of the Dyson's paper:

A more careful study of Macdonald's article reveals that the identity for the 26th power of $\eta(x)$ is not really such a mystery. It is related to the exceptional group $F_4$ of dimension 52, where the space of dual roots $F_4^V$ and the space of roots $F_4$ are not the same. Thus, there are two distinct identities associated with $F_4$, one for $\eta^{52} (x)$ and the other for $\eta^{26} (x)$. A similar situation prevails in the case of the algebra $G_2$ of dimension 14, which yields identities for $\eta^{14} (x)$ and $\eta^{7} (x)$. The identities for $\eta^{26} (x)$ and $\eta^{7} (x)$ are considerably more complicated.