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arXiv · 1709.05532

The Unknown Subgroup of $Aut(E_8)$

Majid Butler·De'janeke Johnson·Tomme Denney·Sandernisha Claiborne·Tianna Robinson
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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 $E_8$.)

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Majid Butler, De'janeke Johnson, Tomme Denney, Sandernisha Claiborne, Tianna Robinson. 2018-02-15. The Unknown Subgroup of $Aut(E_8)$. https://arxiv.org/abs/1709.05532

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