## Introduction

This page describes the results of some lattice computations carried out by Arie Peterson, specifically in the lattices E8 and E7. We are are interested in the number of roots that can be orthogonal to lattice vectors of a certain length.

## Results

#### E8

• CanonicalOrbits.results gives, for d up to 150, the number of orbits (under the action of the Weyl group) of "canonical" vectors of length 2d, i.e. vectors orthogonal to exactly 14 roots.

• RootType.results gives, for d up to 150, the root type of d. This is a list of the possible numbers of roots orthogonal to a vector in E8 of length 2d.

• The file Roots-a-n.results reports, for d up to n, whether there exists a vector in E8 of length 2d that is orthogonal to exactly a roots.

• Transversal.results contains a list of 135 elements of the Weyl group of E8, which together form a transversal of the subgroup of permutations and even sign changes. This is generated by a Monte Carlo method, so different runs of the program may give different transversals.

#### E7

• The file E7Roots-a-n.results reports, for d up to n, whether there exists a vector in E7 of length 2d that is orthogonal to exactly a roots.

• E7RootType.results gives, for d up to 150, the root type of d. This is a list of the possible numbers of roots orthogonal to a vector in E7 of length 2d.

## Source code

These are the computer programs used to obtain the above results. They are written in haskell, a lazy functional programming language.

Here is a tarball with all these source files. To compile the programs, you need the Haskell platform, and the following haskell packages (these can be installed using the command "cabal install \$package"):
• vector-0.6.0.2
• data-memocombinators-0.4.0
• numeric-prelude-0.1.3.4
• species-0.3.0.1