Vector stabilizers and subgroups of Leech lattice groups
Vector stabilizers and subgroups of Leech lattice groups
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矢量稳定剂和 Leech 晶格群的子群
DOI：
10.1016/0021-8693(89)90260-3
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发表时间：
1989
影响因子：
0.9
通讯作者：
R. Wilson
中科院分区：
文献类型：
--
作者：
R. Wilson
Leech 格子是一个二维格子，其自模群是 Conway 群 Co 的双覆盖 2’Co，（参见 [3, p. 1801）。由此，可以通过固定某些向量集来获得其他康威群 Co 和 Co，、麦克劳林群 McL 和 Higman-Sims 群 HS。格在确定 Co，（参见[10））和 Co L（参见[1 I]）的最大子群中发挥了核心作用。关键的观察是，如果一个群在有理表示中固定一个非零向量，那么它会在该表示模 2 的任何约简中固定一个非零向量。本文的目的之一是推广这个结果，并将其放在模表示理论中适当的背景中。另一个主要目标是将这个结果应用于其他Leech格群，从而给出McL和Co（最初由Finkelstein在[S]中确定）和HS（最初由Magliveras在[S]中确定）和Co（参见[lo]）的最大非局部子群的统一分类，以及自同构群McL：McL的2和HS：HS的2（最初在[127中确定）。
The Leech lattice is a 2bdimensional lattice whose automo~ hism group is the double cover 2 ‘Co, of Conway’s group Co,(see [3, p. 1801). From it the other Conway groups Co, and Co,, McLaughlin’s group McL, and the Higman-Sims group HS, may be obtained by fixing certain sets of vectors.The lattice played a central role in the determination of the maximal subgroups of Co,(see [10)) and Co L (see [1 I]). The crucial observation is that if a group fixes a non-zero vector in a rational representation then it fixes a non-zero vector in any reduction of this representation modulo 2. One of the aims of the present paper is to generalise this result, and put it in its proper context in modular representation theory. The other main aim is to apply this result to other Leech lattice groups, and so give a uniform classification of the maximal non-local subgroups of McL and Co,(originally determined by Finkelstein in [S]) and HS (originally determined by Magliveras in [S]) and Co,(see [lo]), as well as the automorphism groups McL: 2 of McL and HS: 2 of HS (originally determined in [127).