Finite Groups (Sporadic Simple Groups) and Related Topics

有限群（零星简单群）及相关主题

• 批准号: 09440006
• 负责人: KITAZUME Masaaki
• 金额: $ 5.25万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440006/
• 关键词: Finite Groups; Simple Groups; Designs; Codes; Lattices; Graphs; Vertex Operator Algebras; Monster simple group; 散在型単純群; 符号(code); 格子(lattice); 自己同型群

We have studied finite simple groups and related graphs, designs, codes, lattices and vertex operator algebras. Main results are as follows :1. The 2- and 3-radical subgroups of Fischer's simple groups FィイD222ィエD2, FィイD223ィエD2, F'ィイD224ィエD2 have been classified.2. Two kinds of non-split extensions of OィイD27ィエD2(3) have been constructed explicitly by using Dickson's trilinear form and some special Moufang loop.3. By using a lattice VOA, a new proof of the Borwein identity has been given.4. New VOAs related to ternary codes have been introduced, and the irreducible representations of L(ィイD74(/)5ィエD7, 0) 【symmetry】 L(ィイD74(/)5ィエD7, 3) have been determined.5. By using an embedding ィイD82ィエD8AィイD312(/)2ィエD3 into the Leech lattice and a ZィイD22ィエD2 × ZィイD22ィエD2-code, some decomposition of the Moonshine VOA has been given.6. Some 5-designs invariant by PSL(2, 23) and related to S(5, 8, 24) have been classified.7. A simple characterization of S(5, 8, 24) (or Golay code) has been given.8. The Niemeier lattices have been constructed form ZィイD24ィエD2-codes, and their embeddings of them into 1/ィイD82ィエD8 times the Leech lattice have been given.9. The exceptional graphs embedded into the root system EィイD28ィエD2 have been classified.10. Even unimodular Gaussian (resp. Quaternionic) lattices of dimension 12 (resp. 6) have been classified.

我们已经研究了有限单群和相关的图，设计，码，格和顶点算子代数。主要研究结果如下：1.对Fischer单群F D222 D2，F D223 D2，F 'D224 D2的2-和3-根式子群进行了分类.利用Dickson的三线性形式和某些特殊的Moufang环，构造了O_n_D_27_n_D_2（3）的两类非分裂扩张.利用格VOA给出了Borwein恒等式的一个新的证明.引入了与三进制码相关的新的VOA，并确定了L（D 74（/）5 D 7，0）[对称性] L（D 74（/）5 D 7，3）的不可约表示.通过在Leech格上嵌入一个Z_（82）N_（8A）N_（312（/）2）N_（312）D_（3对PSL（2，23）不变量和S（5，8，24）相关的5-设计进行了分类.给出了S（5，8，24）（或Golay码）的一个简单特征.从Z_（24）× D_（24）× D_2-码构造了Niemeier格，并给出了它们嵌入到1/1 × D_（82）× D_（8）× Leech格中的嵌入.嵌入到根系统E D28 D2中的例外图已经分类。偶数单模高斯（分别为四元数）12维晶格（分别为6)都是机密

项目成果

M.Kitazume,M.Miyamoto,H.Yamada: "Borwein Identity and Vertex Operator Algebras"Journal of Number Theory. (発表予定). (2000)

M.Kitazume、M.Miyamoto、H.Yamada：《Borwein 恒等式和顶点算子代数》《数论杂志》（即将出版）。

A.Bonnecaze,P.Gaborit,M.Harada,M.Kitazume,P.Sole: "Niemeier Lattices and Type II Codes over Z_4"Discrete Mathematics. 205. 1-21 (1999)

A.Bonnecaze、P.Gaborit、M.Harada、M.Kitazume、P.Sole：“尼迈尔格和 Z_4 上的 II 型代码”离散数学。

K.Yamauchi: "On a Theorem of J.A.Green" Journal of Algebra. 209. 708-712 (1998)

K.Yamauchi：“关于 J.A.Green 定理”代数杂志。

M.Kitazume: "Some Non-split Extensions of the Orthogonal Group Ω(7,3)"Journal of Algebra. 224. 59-76 (2000)

M.Kitazume：“正交群 Ω(7,3) 的一些非分裂扩展”代数杂志 224. 59-76 (2000)

M. Kitazume, K. Koike and H. Shiga: "Notes on a certain integral orthogonal group and its quotient by the principal congruence subgroup of level 2"Kyushu Journal of Mathematics. (to appear). (2000)

M. Kitazume、K. Koike 和 H. Shiga：“关于某个积分正交群及其由 2 级主同余子群的商的注释”《九州数学杂志》。