OFFSET
0,5
COMMENTS
Since Aut(E8) is a reflection group one can compute this using nonnegative combinations of the basis dual to the simple roots, since these are the lattice vectors in a fundamental domain and so include a unique representative of each orbit. - Noam D. Elkies, Apr 07 2008
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..1200 [Computed using Elkies's PARI/GP program]
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Gabriele Nebe and N. J. A. Sloane, Home page for this lattice
Kaiwen Sun and Haowu Wang, Weyl invariant E8 Jacobi forms and E-strings, arXiv:2109.10578 [math.NT], 2021. See Table 5 p. 16.
PROG
(PARI)
M = 2*matid(8); for(i=1, 6, M[i, i+1] = M[i+1, i] = -1); M[3, 8] = M[8, 3] = -1; \\ M is now the Gram matrix for the simple roots of E8
M = 1/M; \\ M is now the Gram matrix for the dual basis to the simple roots; their nonnegative combinations are a fundamental domain for W(E8)
{ orbit_counts(N) =
c = vector(N);
v = vector(8, n, 0);
j = 1;
while(j<9, j = 1; v[1]++; k = v*M*v~; while(k>2*N, v[j]=0; j++; if(j<9, v[j]++; k=v*M*v~, k=0)); if(k, c[k/2]++); );
return(concat(1, c)) }
orbit_counts(100) \\ Noam D. Elkies, Apr 07 2008
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrected by Noam D. Elkies, Apr 07 2008
STATUS
approved