Barnes–Wall lattice
In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter–Todd lattice.
The automorphism group of the Barnes–Wall lattice has order 89181388800 = 221 35 52 7 and has structure 21+8 PSO8+(F2). There are 4320 vectors of norm 4 in the Barnes–Wall lattice (the shortest nonzero vectors in this lattice).
The genus of the Barnes–Wall lattice was described by Scharlau & Venkov (1994) and contains 24 lattices; all the elements other than the Barnes–Wall lattice have root system of maximal rank 16.
The Barnes–Wall lattice is described in detail in (Conway & Sloane 1999, section 4.10).
References
- Barnes, E. S.; Wall, G. E. (1959), "Some extreme forms defined in terms of Abelian groups", J. Austral. Math. Soc., 1 (1): 47–63, doi:10.1017/S1446788700025064, MR 0106893
- Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, MR 0920369
- Scharlau, Rudolf; Venkov, Boris B. (1994), "The genus of the Barnes–Wall lattice.", Comment. Math. Helv., 69 (2): 322–333, CiteSeerX 10.1.1.29.9284, doi:10.1007/BF02564490, MR 1282375