Representation theory of finite groups and association schemes

群是可用于理解对象的对称性的代数结构。群的表示理论是一个代数领域，它使我们能够用矩阵的形式来表示这些对称，这给了我们一种使用我们熟悉的线性代数工具来处理复杂结构的方法。对群表示理论的基本理解对于理解目前在计算机科学、化学、物理和信息论中所做的研究是必不可少的。 我在群表示理论方面的工作集中在当表示中使用的矩阵的条目被限制在有理数领域时可能出现的可能性。有了这个限制，可能会出现许多复杂的除法代数结构，当允许矩阵的项在实数或复数上自由取值时，这些结构不会出现。处理这些除法代数的理论工具之一是布劳尔群，它出现在数学的许多地方。我的建议的目标之一是寻找Brauer群的推广，它可以用于解决有限群表示理论中一些最重要的问题的新方法。 我的建议的另一个方面涉及联合方案，这是一种群的组合推广，大约50年前被发现在统计学和编码理论中有有用的应用。我的提议的另一个主要目标是沿着群的表示理论的路线，为联合方案的表示理论的发展做出贡献。我们所知道的关于团体的许多事情都是关于联合方案的公开问题。我和我的学生希望能够在这项工作的过程中解决其中的一些问题，并开辟协会方案的领域，以出现未来表征理论的应用。

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.