Representation theory of finite groups and association schemes

有限群表示论和关联格式

• 批准号: 194195-2012
• 负责人: Herman, Allen
• 金额: $ 0.87万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569748
• 关键词: Representation; theory; finite; groups; association

Groups are algebraic structures that can be used to understand symmetries of objects. The representation theory of groups is the area of algebra that makes it possible for us to represent these symmetries in a matrix form, which gives us a means to work with complicated structure using the familiar tools of linear algebra. A basic understanding of group representation theory is essential for an understanding of research being done now in computer science, chemistry, physics, and information theory. My work in group representation theory focuses on the possibilities that can arise when the entries of the matrices being used in the representations are restricted to the field of rational numbers. With this restriction, many complicated division algebra structures can be present that do not occur when the entries of the matrices are allowed to range freely over the real or complex numbers. One of the theoretical tools available for dealing with these division algebras is the Brauer group, something that appears in many places in mathematics. One of the goals of my proposal is look for a generalization of the Brauer group that could be of use in new approaches to some of the most important problems in the representation theory of finite groups. The other aspect of my proposal concerns association schemes, a combinatorial generalization of groups that was discovered to have useful applications in statistics and coding theory about 50 years ago. Another main goal of my proposal is to contribute to the development of a representation theory for association schemes along the lines of the representation theory for groups. Many of the things that we know for groups are open questions for association schemes. My students and I hope to be able to settle some of these issues in the course of this work, and open up the area of association schemes to appear future applications of representation theory.

组是代数结构，可用于理解对象的对称性。 组的表示理论是代数的区域，使我们有可能以矩阵形式表示这些对称性，这使我们使用熟悉的线性代数工具来使用复杂结构的手段。 对小组代表理论的基本理解对于对现已在计算机科学，化学，物理学和信息理论中进行的研究的理解至关重要。 我在小组表示理论中的工作着重于当表示中使用的矩阵的条目仅限于理性数字领域时可能出现的可能性。 通过这种限制，可以存在许多复杂的分区代数结构，而当矩阵的条目允许在真实或复数上自由范围时，就不会发生。 Brauer群体是用于处理这些分区代数的理论工具之一，它在数学中出现的许多地方出现。 我的提案的目标之一是寻找Brauer群体的概括，该群体可以在有限群体表示理论中最重要的问题中使用新方法。 我的提案的另一个方面涉及关联方案，这是对大约50年前在统计和编码理论中具有有用应用的组的组合概括。 我的提议的另一个主要目标是为沿群体代表理论的界限的代表理论做出贡献。 我们知道的小组中知道的许多事情都是关联计划的开放问题。 我和我的学生希望能够在这项工作过程中解决其中一些问题，并开放协会方案领域，以表现出代表理论的未来应用。