Subgroups of $W(E_8)$ Are there any proper subgroups of the Coxeter group $W(E_8)$ which are also proper overgroups of $W(A_8)$, other than $\text{Aut}(A_8)$?
 A: One approach is to calculate the orbits of $W(A_8)$ on $W(E_8) / W(A_8)$.
I claim these orbits have sizes  $1, 1, 84, 84, 560, 560, 630$.
Given this claim, it's straightforward to check. For any intermediate subgroup $G$, $G / W(A_8)$ must have an order a divisor of $1920$ and must be a union of these orbits, including $1$. We can't write $1920/2= 960$ as a sum of these numbers because $630+1+1+84+84$ is too small but including two of $560$, $560$, $630$ would be too big. We can't write $1920/3=640$ for similar reasons. Because $1920/4 =480<560$, the only remaining possibilities are $1$, $2$, $85$, $86$, $169$, $170$ and none of those is a divisor of $1920$ except $1$ and $2$, which correspond to $W(A_8)$ and its normalizer.
The way I calculated this involves observing that because the inclusion $W(A_8) \subset W(E_8)$ comes from viewing the lattice $A_8$ as an index $3$ sublattice of $E_8$, the kernel of a map $E_8 \to \mathbb Z/3$ arising by dot product with an element of $E_8$ mod $3$, we can represent $W(E_8)/W(A_8)$ as a $W(E_8)$-orbit inside $E_8$ mod $3$ — specifically, the elements with norm congruent to $2$ mod $3$ that aren't roots.
I then found all these elements in a model of $E_8$ on which $W(A_8)=S_9$ acts, i.e. vectors in $((1/3) \mathbb Z)^9$ that sum to zero, and calculated the $S_9$ orbits. They are

*

*$630$ permutations of $(0,1,1,1,1,-1,-1,-1,-1)$

*$1$ permutation of $(1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3)$ (after subtracting $3$ from one entry — mod $3E_8$, it doesn't matter which one)

*$84$ permutations of $(1/3,1/3,1/3, -2/3,-2/3,-2/3,-2/3, -2/3, -2/3)$  (after adding $3$ to one entry)

*$560$ permutations of $(4/3, 4/3, 4/3,-2/3,-2/3,-2/3,1/3,1/3, 1/3) $ (after subtracting $3$ from one entry)

*and the negations of the last three orbits.

