Representation Theory

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，都有大量的具体和自然的例子，它们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。群表示理论旨在理解群可以实现为线性群的各种方式。类似的情况也发生在代数上，它也是基本的数学对象。它们有各种各样的类型，我对李型代数很感兴趣，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的方式具体化。它们以线性变换的李代数的形式产生了前者的镜像(并不是所有的镜像都是完全忠实的)，线性变换的李代数是最具体的李代数类型之一。表示通常由不可拆分或不可分解的组件组合而成。李代数表示论的目标之一。群)是构造、理解和分类各种类型李代数的所有不可分解表示。组)。李代数中最广为人知、表现最好的类型称为半单李代数。它们的内部结构和不可分解的表示是众所周知的。然而，除了半单李代数以外的大多数李代数都有一个狂野的表示理论，并且非常困难，几乎是乌托邦的，要对它们的所有不可分解表示进行分类，无论李代数本身是多么初等。然而，我已经确定了一大类但合适的非半单李代数，对于它，某些类型的不可分解模的分类是可行的，这也是我计划要做的。在这个方向上的重大进展将极大地增加我们对这些疯狂行为的李代数的表示理论的现有知识，并为我们的知识体系尚未很好地组织的领域建立结构和基础。在群的情况下，我专注于由R.Steinberg发现的一种特殊的表示，它在称为李型有限群的一类重要群的表示理论中起着突出的作用。群的表示分为普通和模数两种类型，后者是通过约化过程从前者获得的。我试图找到和识别模块化斯坦伯格表示法中不可分解的组成部分。由于这一表象在群论中的重要性，这一问题的解决很可能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，那就是影响到我将在未来5年内教授的大约1000名学生。为我的研究提供资金将使我能够保持高级本科生、研究生和博士后研究员的合理流动，与他们分享和讨论我的研究项目。当我教数百名本科生时，他们将受益于有一位教授，他不仅了解学科材料，而且积极参与研究。来到我们课堂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，只有在他们的专业领域最活跃的人的帮助下，才能实现这一点。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。

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.