Barnes–Wall lattice

In mathematics, the Barnes–Wall lattice Λ₁₆, discovered by Eric Stephen Barnes and G. E. (Tim) Wall ((Barnes Wall)), 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) 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).

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 
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, https://archive.org/details/spherepackingsla0000conw_b8u0 
• Scharlau, Rudolf; Venkov, Boris B. (1994), "The genus of the Barnes–Wall lattice.", Comment. Math. Helv. 69 (2): 322–333, doi:10.1007/BF02564490 

External links

• Barnes–Wall lattice at Sloane's lattice catalogue.