The First Thousand Shells of the E₈ Root Lattice

The E₈ root lattice is the set of all vectors x = ½(x₁,x₂,...,x₈) ∈ Q⁸ such that x₁,x₂,...,x₈ ∈ Z;   x₁ ≡ x₂ ≡ ··· ≡ x₈ mod 2;   and x₁ + x₂ + ··· + x₈ ≡ 0 mod 4. Every such vector has norm x₁² + x₂² + ··· + x₈² equal to an even integer 2m ≥ 0; and the number of lattice vectors of norm 2m is

• 240σ₃(m)   for m≥1, where σ₃(m) is the sum of the cubes of the positive integer divisors of m;
• 1,   for m = 0.

See e.g. J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer, 1990; also M. Viazovska, ‘The sphere packing problem in dimension 8’, Annals of Mathematics (2017) 185 (3): 991–1015. Here we list, for each m=1,2,3,...,1000, all the lattice vectors of norm 2m.

We refer to set of all lattice vectors of norm 2m as the m-th shell of the lattice. This set is a union of orbits of the corresponding Weyl group W(E₈); and each such orbit is in turn a union of orbits under the monomial subgroup W(D₈) < W(E₈). The subgroup W(D₈) is generated by the 8!=40320 coordinate permutations, together with the 2⁷=128 even sign changes (in which the sign is changed for any even number of the eight coordinates). For each m, our text file shell‹m›.txt lists

• the number of W(E₈)-orbits on the m-th shell; and
• for each W(E₈)-orbit, we list
  • the number of suborbits under W(D₈) (i.e. the number of `types' of vectors up to monomial transformations), and finally,
  • for each W(D₈)-orbit, the length of this W(D₈)-orbit, and a representative vector (times 2) of the given type (i.e. from the given W(D₈)-orbit).

For example for m=10,

This file may be retrieved at http://ericmoorhouse/e8/shell10.txt  (alternatively, click on the corresponding entry in the table below)

Substutitute m ∈ {1,2,3,...,100} for 10 to retrieve any of the first 100 shells.  Beyond that point, the file is gzipped; e.g. for the 137th shell, substitute shell137.txt.gz for the gzipped text filename.  The 310th shell is the smallest containing a regular W(E₈)-orbit; i.e. the shortest lattice vectors having trivial stabilizer in W(E₈) have norm 620.  The shortest lattice vectors having a stabilizer of order 2 (orbit of length 348364800) have norm 54; and there is a unique such orbit represented by (7,1,1,1,1,1,0,0,0) in the 27th shell.

Number of orbits per shell

Here is a table of the number of orbits of W(E₈)-orbits on the m-th shell for the first few hundred values of m.  Each table entry is linked to the full shell file (shell‹m›.txt or shell‹m›.txt.gz) described above.

/ revised September, 2019