Representation Theory

群是一个基本的数学概念，随着时间的推移，从相当具体的设置发展成为一个对象，现在纯粹根据公理抽象定义。在所有群的领域中，有大量的具体和自然的例子，它们比一般的抽象群更容易被研究和理解。线性变换群就是这种类型。群表示理论旨在理解群可以被实现为线性群的各种方式。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，相关性和内在美而被广泛研究。抽象的李代数可以用表示法具体化.它们以线性变换的李代数的形式产生了前者的镜像（并非所有镜像都是完全忠实的），这是李代数中最具体的类型。表示通常由不可分割或不可分解的成分组合而成。李代数表示论的目标之一（分别是）群）是构造，理解和分类所有不可分解的表示，为每一个和每一个不同类型的李代数（分别。组）。李代数中最好理解和表现最好的类型叫做半单。它们的内部结构和不可分解的表示是众所周知的。然而，大多数李代数以外的半单有一个野生的表示理论，是非常困难的，几乎乌托邦，分类所有不可分解的表示，他们，无论多么基本的李代数本身可能。尽管如此，我已经确定了一个大的，但合适的类非半单李代数，其中某些类型的不可分解模的分类是可行的，这是我计划做的。在这个方向上的重大进展将大大增加我们目前的知识表示理论的这些疯狂的行为李代数，并建立结构和基础的一个领域，我们的知识体系还没有很好地组织起来。在群的情况下，我关注的是R. Steinberg，这在一个重要的类群的表示论中起着突出的作用，称为有限群的李型。群表示分为两种类型，普通的和模的，后者是由前者通过一个归约过程得到的。我试图定位和识别模块斯坦伯格表示的不可分解的组件。由于这个表示在群论中的重要性，这个问题的解决方案可能会引起兴趣和关注。我的研究在数学界之外的一个更大的影响是我将在未来5年教的大约1000名学生。我的研究经费将使我能够保持一个合理的流动的高级本科生，研究生和博士后研究员，与谁分享和讨论我的研究项目。当我在教数百名刚开始的本科生时，他们将从一位不仅了解主题材料，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，这只能在那些在其专业领域最活跃的人的帮助下才能实现。加拿大将受益于分配资源，使其青年获得最高质量的教育。

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.