Niemeier lattice
In mathematics, a Niemeier lattice is one of the 24
positive definite even unimodular lattices of rank 24,
which were classified by Niemeier. The best known example is the Leech lattice.
Classification Niemeier lattices are usually labeled by the Dynkin diagram of their
root systems. These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number. (The Coxeter number, at least in these cases, is
the number of roots divided by the dimension.) There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier
lattice for each of these Dynkin diagrams. The complete list of Niemeier lattices is The Leech lattice (empty root system), Coxeter number 0. A124, Coxeter number 2.A212, Coxeter number 3.A38, Coxeter number 4.A46, Coxeter number 5.A54D4, Coxeter number 6.D46, Coxeter number 6.A64, Coxeter number 7.A72D52, Coxeter number 8.A83, Coxeter number 9.A92D6, Coxeter number 10.D64, Coxeter number 10.E64, Coxeter number 12.A11D7E6, Coxeter number 12.A122, Coxeter number 13.D83, Coxeter number 14.A15D9, Coxeter number 16.A17E7, Coxeter number 18.D10E72, Coxeter number 18.D122, Coxeter number 22.A24, Coxeter number 25.D16E8, Coxeter number 30.E83, Coxeter number 30.D24, Coxeter number 46.
Properties Some of the Niemeier lattices are related to sporadic simple groups.
The Leech lattice is acted on by a double cover of the Conway group,
and the lattices A124 and A212
are acted on by the Mathieu groups M24 and M12. The Niemeier lattices, other than the Leech lattice, correspond to
the deep holes of the Leech lattice. This implies that the affine Dynkin diagrams of the Niemeier lattices can be seen inside the Leech lattice, when
two points of the Leech lattice are joined by no lines when they have distance
, by 1 line if they have distance ,
and by a double line if they have distance .
Further reading
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