Barnes–Wall lattice

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In mathematics, the Barnes–Wall lattice Λ₁₆, discovered by 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 similar 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).

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).

[edit] 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, MR0106893 
• Conway, John Horton & Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, vol. 290 (3rd ed.), Grundlehren der Mathematischen Wissenschaften, Berlin, New York: Springer-Verlag, MR0920369, ISBN 978-0-387-98585-5 
• Scharlau, Rudolf & Venkov, Boris B. (1994), “The genus of the Barnes-Wall lattice.”, Comment. Math. Helv. 69 (2): 322-333, MR1282375, <http://retro.seals.ch/digbib/view?did=c1:421661&sdid=c1:422358> 

[edit] External links

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