Regular Two-Graphs From the Even Unimodular Lattice ...

Abstract

This is a revised version of the first part of [8]. Each two-graph is in one-to-one correspondence with a set of equiangular lines. This implies that a two-graph is represented by a system of vectors of equal odd norm with mutual inner products \Sigma1. This is a special odd system, i.e. a set of vectors with odd inner products. The construction of obtaining odd systems from doubly even lattices introduced in [5], [6] is applied to the even unimodular lattice E 8 \Phi E 8 multiplied by p 2. For the odd system of 456 vectors obtained by this construction, we study odd subsystems of 36 vectors spanning equiangular lines, i.e. subsystems representing regular two-graphs on 36 points. A subsystem of vectors representing a two-graph generates a sublattice of the lattice E 8 \Phi E 8 . These sublattices are distinguished by sets of lattice vectors of norm 2. These sets are root systems. Hence the set of all two-graphs from E 8 \Phi E 8 is partitioned into families of two-graphs with the s..