A characterisation of Leech's lattice
A characterisation of Leech's lattice
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Leech 晶格的表征
DOI：
10.1007/bf01389796
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发表时间：
1969
影响因子：
3.1
通讯作者：
J. Conway
中科院分区：
文献类型：
--
作者：
J. Conway
[-5, 6] 有望成为许多调查的主题。我们在这里给出一个简短的证明，证明该晶格具有一些最简单的属性。尽管我们必须引用两个定理来打开和结束证明，但读者如果愿意的话可以相信它们，并且他会发现证明在其他方面是完全独立的。特别是，Leech 晶格本身将在其表征过程中被定义，因此不需要熟悉它，尽管为了某些读者的方便，我们坚持使用 [1, 2] 的符号。本文的方法立即给出了群的顺序，这是我们两篇论文 [1, 2] 的主题，并且可以用来给出有关群和格的其他信息。但本文的主要兴趣可能在于该论证可以扩展到其他有趣格子的构造的可能性。然而，我们推测，任何这样的扩展论证一定要复杂得多，因为读者在我们的证明中经常观察到的极端“严格性”是否会再次出现是非常值得怀疑的。我们立即进行证明。我们将证明，Leech 的格子是小于 32 维的唯一格，每单位体积有一个点，并且其中每个非零距离的平方都是大于 2 的偶整数。每单位体积有一个点的格子称为单模格，即使每个平方距离都是偶数整数。在维度为 d 的偶数单模晶格中，我们使用 u 来表示平方长度为 n 的向量的数量。然后我们的定理断言 Leech 的晶格是唯一一个 d < 32 且 u 2 = 0 的偶模晶格。
[-5, 6] promises to be the subject of many investigations. We give here a short proof that this lattice is characterised by some of its simplest properties. Although we must quote two theorems to open and close the proof, the reader can take these on trust if he wishes, and he will find that the proof is otherwise completely self-contained. In particular the Leech lattice will itself be defined in the course of its characterisation, so that no acquaintance with it is presupposed, although for the convenience of certain readers we have adhered to the notation of [1, 2]. The methods of this paper immediately give the order of the group which is the theme of our two papers [1, 2], and can be used to give other information about the group and the lattice. But the main interest of this paper probably lies in the possibility that the argument can be extended to yield constructions for other interesting lattices. We conjecture, however, that any such extended argument must be considerably more complicated, since it is very doubtful that the extreme "tightness" which the reader will observe so often in our proof can ever occur again. We proceed at once to the proof. We shall show that Leech's is the only lattice in fewer than 32 dimensions which has one point per unit volume and in which the square of every non-zero distance is an even integer greater than 2. A lattice with one point per unit volume is called unimodular, and then also even if every squared distance is an even integer. In an even unimodular lattice of dimension d we use u, for the number of vectors of squared length n. Our theorem then asserts that Leech's is the only even unimodular lattice with d < 32 and u 2 = 0.