Imprimitive Geometries and Groups

原始几何和群

• 批准号: 0601621
• 负责人: Jonathan Hall
• 金额: $ 16.5万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601621&HistoricalAwards=false
• 关键词: Imprimitive; Geometries; Groups

A result of Cayley implies that every group has a faithfulrepresentations as a transitive permutation group. Many have primitivefaithful representations, for instance, all finite simplegroups. Indeed the classical simple groups act primitively on thepoints of a projective or polar space. However many groups have noprimitive faithful permutation representation, so imprimitive faithfulrepresentations must be considered. In this project Hall studiesfinite imprimitive groups that act as automorphism groups ofgeometries. He also considers the converse---description andcharacterization of geometries that come provided with a natural andnontrivial equivalence relation. Of particular interest areequivalence relations whose classes are the orbits of some nontrivialnormal subgroup, the group then naturally embedded in thecorresponding wreath product. The geometries studied andcharacterized are highly homogeneous partial linear spaces,particularly those associated with isometry groups of forms, whereimprimitivity for the group can come from degeneracy of the form. Forforms of degree greater than two, this leads to questions aboutpossible generalizations of Witt's theorem on bilinear and relatedforms.Group theory is often described as the mathematics of symmetry. Thatis, many groups are best described as collections of symmetries ofgeometric objects. Conversely many geometries are best characterizedby their associated groups, a point of view that goes back to Hilbert.Finite groups appear prominently via symmetry in many fields otherthan mathematics, for example, physics, chemistry, and computerscience. The building blocks of finite symmetry groups are the finitesimple groups and are realized as primitive permutationgroups---permutation groups that in an appropriate sense areefficient. A general finite group is then glued together from simplegroups, and its symmetry realization is imprimitive---constructed fromvarious primitive constituents. There has been great success recentlyin the description of simple groups and their primitive permuationrepresentations. Understanding of and facility with imprimitivegroups is the important next step. The project is aimed at givingelementary geometric descriptions of certain imprimitive groups andconversely characterizing and recognizing geometries starting fromproperties of an imprimitive symmetry group.

Cayley的一个结果意味着每个群作为传递置换群都有一个忠实的表示。许多群有连续的忠实表示，例如，所有有限的单群。实际上，经典单群在射影空间或极空间的点上起着互补作用。 然而，许多群没有本原的忠实置换表示，因此必须考虑非本原的忠实置换表示。 在这个项目霍尔研究有限非本原群作为自同构群的几何。 他还考虑了几何的匡威描述和表征，这些描述和表征提供了一个自然的和非平凡的等价关系。 特别感兴趣的是等价关系，其类是某些非平凡正规子群的轨道，然后该群自然嵌入相应的圈积中。 研究和表征的几何是高度齐性的部分线性空间，特别是那些与等距群的形式，其中immigrativity的组可以来自退化的形式。 对于次数大于2的形式，这就引出了关于维特定理在双线性和相关形式上的可能推广的问题。群论通常被描述为对称性的数学。 也就是说，许多组最好被描述为几何对象的对称性的集合。相反，许多几何学的最佳特征是它们的相关群，这一观点可以追溯到希尔伯特。有限群通过对称性在数学以外的许多领域中表现突出，例如物理学、化学和计算机科学。 有限对称群的构造块是有限单群，并且被实现为本原置换群-在适当意义上是有效的置换群。一般有限群是由简单群粘合在一起的，它的对称实现是非本原的--由各种本原成分构造而成。 最近在单群及其本原置换表示的刻画方面取得了巨大的成功。 了解和处理重要的群体是下一步的重要工作。 该项目旨在给出某些非本原群的初等几何描述，并从非本原对称群的性质出发，反过来刻画和认识几何。