Barnes–Wall lattice

Barnes–Wall lattice

In mathematics, the Barnes–Wall lattice Λ16, discovered by harvtxt|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 harvtxt|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 harv|Conway|Sloane|1999|loc=section 4.10.

References

*citation|id=MR|0106893 |last=Barnes|first= E. S.|last2= Wall|first2= G. E.|title= Some extreme forms defined in terms of Abelian groups|journal= J. Austral. Math. Soc.|volume= 1 |year= 1959|issue=1|pages= 47-63
*Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | title=Sphere Packings, Lattices and Groups | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-98585-5 | id=MR|0920369 | year=1999 | volume=290
*citation|id=MR|1282375
last=Scharlau|first= Rudolf|last2= Venkov|first2= Boris B.
title=The genus of the Barnes-Wall lattice.
journal=Comment. Math. Helv.|volume= 69 |year=1994|issue= 2|pages= 322-333
url=http://retro.seals.ch/digbib/view?did=c1:421661&sdid=c1:422358

External links

* [http://www.research.att.com/~njas/lattices/BW16.html Barnes-Wall lattice] at Sloane's lattice catalogue.


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