The genus of the Coxeter-Todd lattice
by Rudolf Scharlau and Boris B. Venkov.
Abstract:
The classification of all 3-elementary even lattices of determinant
$3^6$ and dimension 12 is given. This genus of lattices consists of
10 isometry classes. Like in the cases of 24-dimensional, even
unimodular lattices, or 16-dimensional totally even 2-elementary
lattices of determinant $2^8$ (genus of $4D_4$), all but one of them
are reflective in the sense that the root system has maximal rank
$12$. Following earlier work (R. Scharlau and B. Venkov, `The genus
of the Barnes Wall lattice', Comment. Math. Helv. 69 (1994)), a list
of `possible' root systems is derived which are subject to two
essential restrictions. It turns out that for each root system in this
list, there exists a unique lattice.
Key words and phrases: integral lattice, integral quadratic form,
modular lattice, extremal lattice, reflective lattice, Coxeter-Todd
lattice,
class number.
1991 Mathematics Subject Classification: Primary 11H55; Secondary 11E41.
Available as : PostScript File
Contact: Rudolf.Scharlau@Mathematik.Uni-Dortmund.DE