Diagonal lattices and rootless $EE_8$ pairs

TL;DR

This paper explores the properties of diagonal sum lattices, especially for E_8, providing a new model for rootless EE_8 pairs, classifying their isometry types, and embedding them into the Leech lattice.

Contribution

It introduces a natural model for rootless EE_8 lattice pairs, classifies their isometry types, and connects them with conjugacy classes in O(E_8).

Findings

The isometry group contains a wreath product.

A bijection between lattice types and conjugacy classes in O(E_8).

New embeddings into the Leech lattice.

Abstract

Let E be an integral lattice. We first discuss some general properties of an SDC lattice, i.e., a sum of two diagonal copies of E in E \bot E. In particular, we show that its group of isometries contains a wreath product. We then specialize this study to the case of E = E_8 and provide a new and fairly natural model for those rootless lattices which are sums of a pair of EE_8-lattices. This family of lattices was classified in [7]. We prove that this set of isometry types is in bijection with the set of conjugacy classes of rootless elements in the isometry group O(E_8), i.e., those h \in O(E_8) such that the sublattice (h - 1)E_8 contains no roots. Finally, our model gives new embeddings of several of these lattices in the Leech lattice.