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.

  • Roots-2-500.results
  • Roots-6-300.results
  • Roots-10-300.results
  • Roots-12-200.results
  • Roots-14-300.results
  • Roots-16-61.results
  • Roots-18-61.results
  • Roots-20-61.results
  • Roots-30-200.results

• 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.

  • E7Roots-0-500.results
  • E7Roots-2-300.results
  • E7Roots-4-300.results
  • E7Roots-6-300.results
  • E7Roots-8-300.results
  • E7Roots-10-500.results
  • E7Roots-12-300.results
  • E7Roots-14-1000.results
  • E7Roots-16-588.results
  • E7Roots-18-600.results
  • E7Roots-20-300.results
  • E7Roots-22-300.results
  • E7Roots-24-300.results
  • E7Roots-26-300.results
  • E7Roots-28-500.results

• 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.

• CanonicalOrbits.hs
• E7.hs
• E7Roots.hs
• E7RootType.hs computes, for given d, the root type of d in E₇.
• E8.hs
• Roots.hs
• RootType.hs
• Special.hs
• Weyl.hs is an auxiliary module to compute a certain subgroup transversal (see above).

• vector-0.6.0.2
• data-memocombinators-0.4.0
• numeric-prelude-0.1.3.4
• species-0.3.0.1

G.K. Sankaran