The Unknown Subgroup of $Aut(E_8)$

TL;DR

This paper identifies and describes the subgroup $2A_9$ within the automorphism group of the $E_8$ lattice, providing new perspectives and simple descriptions of its structure and actions.

Contribution

It introduces the subgroup $2A_9$ of $Aut(E_8)$ and offers three novel descriptions of its structure and stabilizing properties from different lattice and vector perspectives.

Findings

$2A_9$ stabilizes a partition of 2160 norm 4 vectors into nine copies of $E_8$

$2A_9$ stabilizes a partition of 135 isotropic points into nine disjoint 4-spaces

The subgroup's structure reveals connections to known isomorphisms like $A_8 \,\cong\, L_4(2)$

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 $E_8$.)