Arithmetic groups and the affine E8 Dynkin diagram

Abstract

Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2 to nodes of the affine E8 Dynkin diagram. The purpose of this article is to give an explanation for this latter correspondence using elementary properties of the group PSL2. We also obtain a super analogue of McKay's observation, in which conjugacy classes of the Monster are replaced by conjugacy classes of Conway's group -- the automorphism group of the Leech lattice.