Representation Theory

表征论

• 批准号: RGPIN-2014-06255
• 负责人: Szechtman, Fernando
• 金额: $ 0.8万
• 依托单位: University of Regina
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567693
• 关键词: Representation; Theory

A group is a fundamental mathematical notion that developed over time from fairly concrete settings into an object that is now abstractly defined purely in terms of axioms. In the realm of all groups, there are large reservoirs of concrete and natural examples that can be studied and understood far more easily than general abstract groups. Linear groups of transformations are of this kind. Group representation theory aims at understanding the various ways in which groups can be realized as linear groups. An analogous situation occurs for algebras, which are basic mathematical objects as well. There are various types of them, and I am interested in algebras of Lie type, which are among the most widely studied because of their applications, relevance and inner beauty. Abstract Lie algebras can be made concrete by means representations. These produce mirror images (not all of them entirely faithful) of the former in the shape of Lie algebras of linear transformations, which are among the most concrete types of Lie algebras. Representations are usually made up by combining unbreakable or indecomposable components. One of the goals of the representation theory of Lie algebras (resp. groups) is to construct, understand and classify all indecomposable representations for each and every one the various types of Lie algebras (resp. groups). The most well understood and better behaved types of Lie algebras are called semisimple. Their inner structure and indecomposable representations are well known. However, most Lie algebras other than semisimple have a wild representation theory and is exceedingly difficult, almost utopian, to classify all indecomposable representations for them, regardless of how elementary the Lie algebra itself might be. Nevertheless, I have identified a large but suitable class of non-semisimple Lie algebras for which the classification of certain types of indecomposable modules is feasible, which is what I plan to do. Significant progress in this direction will greatly augment our present knowledge of the representation theory of these wildly behaved Lie algebras, and build structure and foundation to an area where our body of knowledge is not yet well organized. In the case of groups, I am focused on a distinguished kind of representation, discovered by R. Steinberg, that plays a prominent role in the representation theory of an important class groups called finite groups of Lie type. Group representations come into two types, ordinary and modular, the latter obtained from the former by means of a reduction process. I am trying to locate and identify the indecomposable components of the modular Steinberg representation. Due to the importance of this representation in group theory, the solution to this problem is likely to attract interest and attention. An even greater impact of my research, outside of the mathematical community, is to the roughly 1000 students that I will teach over the next 5 years. Funding for my research will allow me to maintain a reasonable flow of advanced undergraduate students, graduate students and postdoctoral fellows, with whom to share and discuss my research projects. While I am teaching the hundreds of beginning undergraduate students, they will greatly benefit from having a professor that not only knows the subject material, but is actively engaged in research. The young minds that come to our classroom need to be nurtured, challenged and led to grow to their full potential, and it is clear to me that this can only be achieved with the help of those that are most active in their field of expertise. Canada will benefit by allocating resources so that its youth receive top quality education.

群是一个基本的数学概念，随着时间的推移，它从相当具体的环境发展成为现在纯粹根据公理抽象定义的对象。在所有群体的领域中，都有大量具体和自然的例子，这些例子比一般的抽象群体更容易研究和理解。线性变换群就是这种类型的。群体表征理论旨在理解群体作为线性群体实现的各种方式。类似的情况也发生在代数上，代数也是基本的数学对象。它们有各种各样的类型，我对Lie型代数很感兴趣，因为它们的应用、相关性和内在美，它们是研究最广泛的代数之一。抽象李代数可以通过表示来具体化。它们以线性变换的李代数的形式产生前者的镜像（并非所有的镜像都完全忠实），这是李代数最具体的类型之一。表示通常由不可破坏或不可分解的组件组合而成。李代数的表示理论的目标之一。群)是构造，理解和分类所有不可分解的表示为每一个不同类型的李代数(见。组)。最容易理解和表现较好的李代数类型被称为半简单李代数。它们的内部结构和不可分解的表示是众所周知的。然而，除了半简单李代数之外，大多数李代数都有一个疯狂的表示理论，无论李代数本身是多么初等，对它们的所有不可分解的表示进行分类都是非常困难的，几乎是乌托邦式的。尽管如此，我已经确定了一个大而合适的非半简单李代数类，其中某些类型的不可分解模块的分类是可行的，这就是我打算做的。在这个方向上的重大进展将极大地增加我们目前对这些行为狂野的李代数的表示理论的知识，并为我们的知识体系尚未组织良好的领域建立结构和基础。就群而言，我关注的是R. Steinberg发现的一种特殊的表示，它在李氏有限群这一重要类群的表示理论中起着重要作用。群表示有普通表示和模表示两种类型，模表示是由普通表示通过约简过程得到的。我试图找到和识别模块化斯坦伯格表示的不可分解的组成部分。由于这种表示在群论中的重要性，这个问题的解决方案很可能会引起人们的兴趣和关注。在数学界之外，我的研究还有一个更大的影响，那就是在接下来的5年里，我将教大约1000名学生。我的研究经费将允许我保持一个合理流动的高级本科生，研究生和博士后，与他们分享和讨论我的研究项目。在我教授数百名本科新生的过程中，他们将从一位不仅了解学科材料，而且积极从事研究的教授中受益匪浅。来到我们教室的年轻人需要得到培养、挑战和引导，以充分发挥他们的潜力，我很清楚，这只能在那些在他们的专业领域最活跃的人的帮助下实现。加拿大将通过分配资源使其青年接受高质量的教育而受益。