Presentations, Cohomology and Representations of Finite Groups and Coverings of Curves

有限群和曲线覆盖的表示、上同调和表示

• 批准号: 1001962
• 负责人: Robert Guralnick
• 金额: $ 22.09万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001962&HistoricalAwards=false
• 关键词: Presentations; Cohomology; Representations; Finite; Groups

The PI plans to study some basic problems about finite and algebraic groups related to presentations, linear and permutation representations, and cohomology with applications to the problems in arithmetic algebraic geometry--particularly questions related to polynomials, rational functions, and automorphisms and coverings of curves. Using the work of Tiep and the PI, small representations of simple groups will be studied. This work has already had applications to various questions of Katz, Kollar, and Larsen in algebraic geometry. The most critical open case is to classify all closed subgroups which are irreducible on exterior powers of a module. This is also closely related to questions about the classification of maximal subgroups of the finite simple groups. This latter questions leads to many basic problems. In particular the PI will look at problems about bounding the number of irreducible representations of bounded dimension for various families of simple groups.The PI and his collaborators have shown recently that with the possible exception of one family, every finite simple group has a presentation with at most fifty relations. In many cases, one can do much better (for example, an infinite number of alternating groups can be presented with two generators and four relations). Allowing the number of relations to increase a bit, one can even produce short presentations (essentially best possible). The most fundamental problem is the relationship between discrete presentations and profinite presentations and cohomology (one way to think of a profinite presentation is that if one is given by generators and relations and one already knows the group is finite, then one can identify the group). This leads one to try to produce very good bounds on the size of the first and second cohomology groups of finite and algebraic groups with coefficients in a simple module. One goal is to prove that every finite simple group has a profinite presentation with two generators and at most four relations (one cannot do better). This may be even be true without the profiniteness condition but no one has any idea of how to approach this in general. Another major problem is to complete the classification of exceptional polynomials over finite fields. Exceptional polynomials are precisely the bijective polynomials assuming the field size is sufficiently large compared to the degree and have been studied seriously since the thesis of Dickson in the 1890's, as well as by Schur, Fried and others. The classification of indecomposable exceptional polynomials whose degree is not a power of the characteristic used a combination of deep group theory together with methods arithmetic algebraic geometry. The PI intends to use these methods and use new results to study these problems. This should have applications to cryptography.

PI计划研究有关有限和代数群体与演示，线性和置换表示相关的一些基本问题，以及在算术代数几何学中的问题上应用的，以及与多项式，理性功能以及自动形态和曲线覆盖的问题有关的问题。使用TIEP和PI的工作，将研究简单组的少量表示。 这项工作已经在代数几何形状中对Katz，Kollar和Larsen的各种问题提出了应用。 最关键的开放式案例是对模块外部功能上不可还原的所有闭合子组进行分类。 这也与有限简单组的最大亚组分类的问题密切相关。后一个问题导致许多基本问题。特别是，PI将研究有关简单群体各个家庭的界限不可约定表示的数量的问题。PI及其合作者最近表明，除了一个家庭外，每个有限的简单集团都可能具有五十个关系的介绍。在许多情况下，一个人可以做得更好（例如，可以提供两个发电机和四个关系的无限交替组）。 允许关系的数量增加一点，甚至可以产生简短的演示文稿（本质上最好）。最根本的问题是离散演示文稿与涂鸦演示文稿与共同体之间的关系（考虑一种概述的演示方法是，如果一个人由生成器和关系给出，并且已经知道该组是有限的，那么一个人可以识别该组）。这导致人们试图在一个简单模块中具有系数的有限和代数组的第一和第二个共同体组的大小产生良好的界限。 一个目标是证明每个有限的简单组都有与两个发电机和最多四个关系（一个做得更好）的大量演示。没有熟练条件，这甚至可能是正确的，但是没有人对如何处理这种情况有任何想法。 另一个主要问题是完成对有限领域的特殊多项式的分类。出色的多项式恰恰是射精的多项式，假设田间大小与该程度相比足够大，并且自1890年代迪克森（Dickson）以及Schur，Fried和其他人以来就经过了认真的研究。不可分解的异常多项式的分类，其程度不是特征的力量，将深组理论的结合与方法算术代数几何结合在一起。 PI打算使用这些方法并使用新的结果来研究这些问题。 这应该在密码学上应用。