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?