On Edge's correspondence associated with \cdot 222

TL;DR

This paper explicitly describes the correspondence of edges in the Fermat cubic 4-fold over characteristic 2 with certain lattice points in the Leech lattice, and uses this to detail Conway's isomorphism between PSU(6,4) and b7 222.

Contribution

It provides an explicit geometric and lattice-theoretic description of the edge correspondence and details Conway's isomorphism in matrix form.

Findings

Explicit correspondence between Fermat cubic 4-fold edges and Leech lattice points.

Matrix representation of Conway's isomorphism.

Abstract

We describe explicitly the correspondence of Edge between the set of planes contained in the Fermat cubic 4-fold in characteristic 2, and the set of lattice points T of the Leech lattice such that OABT is a regular tetrahedron, where O is the origin of the Leech lattice, and A and B are fixed points of the Leech lattice such that OAB is a regular triangle of edge length 2. Using this description, we present Conway's isomorphism from PSU(6,4) to \cdot 222 in terms of matrices.