The Unknown Subgroup of $Aut(E_8)$

Abstract

The $E_8$ lattice has been thoroughly studied for more than a century and nearly all the maximal subgroups of $W(E_8)$ have been described-all except $2A_9$. We will show that $2A_9$ has simple descriptions from three different perspectives: looking at $E_8/2E_8$; looking at the lattice's norm 2 vectors; and looking at its norm 4 vectors. Two of the three descriptions are especially simple: $2A_9$ stabilizes a partition of the 2160 norm 4 vectors into nine scale copies of $E_8$; and $2A_9$ stabilizes a partition of the 135 isotropic points of $E_8/2E_8$ into nine disjoint isotropic 4-spaces of 15 points each (so the well-known known isomorphism $A_8$ $\cong$ $L_4$(2) is visible within $E_8$.)