NORMALIZERS OF PARABOLIC SUBGROUPS OF COXETER GROUPS
NORMALIZERS OF PARABOLIC SUBGROUPS OF COXETER GROUPS
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Coxeter 群抛物线子群的标准化子
DOI：
10.2140/agt.2012.12.1137
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发表时间：
2011
影响因子：
0.7
通讯作者：
Daniel Allcock
中科院分区：
文献类型：
--
作者：
Daniel Allcock
我们改进了 Coxeter 群的抛物线子群的归一化器的非反射部分的虚拟上同调维数上的 Borcherds 的界。我们的界限是根据相应 Coxeter 子图的组件类型而不是节点数量。结果是反射扶正器的非反射部分是自由的布林克结果的扩展。即，D5 或 Am 奇数类型的抛物线子群的归一化子的非反射部分要么是自由的，要么具有索引为 2 的自由子群。假设是 Coxeter 图，J 是子图，WJ ⊆ Wi 是 Coxeter 群的相应包含。 Borcherds (3) 和 Brink-Howlett (5) 已详细描述了归一化器 NW� (WJ)。这种归一化器对于计算洛伦兹晶格和 K3 表面的自同构群有重要的应用；参见 (3) 及其参考文献。 NW� (WJ) 分为 3 部分：WJ 本身、另一个 Coxeter 群 W 和 W 的一组图自同构。最后两组称为归一化器的“反射”和“非反射”部分。 Borcherds 将 的虚上同调维数限制为|J|。我们的定理 1、3 和 4 根据 J 的分量类型而不是节点数量给出了更强的界限。 W 和 的定义涉及一些选择，并且无论如何做出这些选择，定理 3 中的界限都适用（定理 1 是一个特例）。当 W 为“最大”时，定理 4 改进了该界限。在这种情况下，当 J = D5 或 Amodd 时，结果要么是自由的，要么具有自由的索引 2 子群。这扩展了 Brink 的结果 (4)，当 J = A1 时该结果是自由的。作者感谢克莱数学研究所、日本学术振兴会、京都大学的支持和热情接待。
We improve a bound of Borcherds on the virtual co- homological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter sub- diagram rather than the number of nodes. A consequence is an extension of Brink's result that the non-reflection part of a re- flection centralizer is free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Am odd is either free or has a free subgroup of index 2. Supposeis a Coxeter diagram, J is a subdiagram and WJ ⊆ Wis the corresponding inclusion of Coxeter groups. The normalizer NW� (WJ) has been described in detail by Borcherds (3) and Brink- Howlett (5). Such normalizers have significant applications to working out the automorphism groups of Lorentzian lattices and K3 surfaces; see (3) and its references. NW� (WJ) falls into 3 pieces: WJ itself, another Coxeter group W, and a group of diagram automorphisms of W. The last two groups are called the "reflection" and "non-reflection" parts of the normalizer. Borcherds bounded the virtual cohomological dimension of by |J|. Our theorems 1, 3 and 4 give stronger bounds, in terms of the types of the components of J rather than the number of nodes. There are choices involved in the definition of W and , and our bound in theorem 3 applies regardless of how these choices are made (theorem 1 is a special case). Theorem 4 improves this bound when W is "maximal". In this case, when J = D5 or Amodd, turns out to either be free or have an index 2 subgroup that is free. This extends Brink's result (4) that is free when J = A1. The author is grateful to the Clay Mathematics Institute, the Japan Society for the Promotion of Science, and Kyoto University for their support and hospitality.