|
In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by . gave a simplified proof of the classification. has a sentence mentioning that he found more than 10 such lattices, but gives no further details. One example of a Niemeier lattice 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 given in the following table. In the table, :''G''0 is the order of the group generated by reflections :''G''1 is the order of the group of automorphisms fixing all components of the Dynkin diagram :''G''2 is the order of the group of automorphisms of permutations of components of the Dynkin diagram :''G''∞ is the index of the root lattice in the Niemeier lattice, in other words the order of the "glue code". It is the square root of the discriminant of the root lattice. :''G''0×''G''1×''G''2 is the order of the automorphism group of the lattice :''G''∞×''G''1×''G''2 is the order of the automorphism group of the corresponding deep hole. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Niemeier lattice」の詳細全文を読む スポンサード リンク
|