This page describes the results of some lattice computations carried
out by Arie Peterson, specifically in the lattices E_{8} and
E_{7}. We are are interested in the number of roots that can
be orthogonal to lattice vectors of a certain length.

CanonicalOrbits.results gives, for

*d*up to 150, the number of orbits (under the action of the Weyl group) of "canonical" vectors of length 2*d*, 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 E_{8}of length 2*d*.The file

`Roots-a-n.results`reports, for*d*up to*n*, whether there exists a vector in E_{8}of length 2*d*that is orthogonal to exactly*a*roots.Transversal.results contains a list of 135 elements of the Weyl group of E

_{8}, 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.

The file

`E7Roots-a-n.results`reports, for*d*up to*n*, whether there exists a vector in E_{7}of length 2*d*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 E_{7}of length 2*d*.

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 E_{7}. - 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