Realization problems in Representation Theory and Algebraic Combinatorics

我打算研究超群的表示论。 超群是一个有限维结合代数A，它有一个特殊的基B={b 0，b1，，br-1}，其中乘法单位元b 0 = 1位于B中，并且B具有“伪逆”性质：对于B中的每个bi，在B中存在唯一的bi*，其中b 0在bibi* 中的系数不为零。 因此，一个超群推广了熟悉的群概念，用伪逆代替了群的逆性质。 我的工作将集中在如何将这些结构表示为尽可能小的域或环上的矩阵，主要处理除了群代数之外的两种类型的超群：结合概型的邻接代数，其中基B的非单位元素可以用图的集合来标识，以及整表代数，它们是其中基元素bibj的乘积中的每个bk的系数总是非负整数的超群。 这里有一个层次：群代数是邻接代数，邻接代数是整表代数。 在过去的20年里，这类超群的表示理论的大部分都是由群和代数的表示理论的思想所激发的，这导致了在图论，设计理论和编码理论等领域的富有成效的应用。 它提供了一个框架的研究模块数据出现在共形场理论，偶尔新的想法在群论已被发现的那些工作的代数性质的超群。 超群表示论是国际上代数组合学的一个新兴研究领域。 许多新的捐款发生在亚洲国家、欧洲和美国，这使它成为一个为加拿大人提供国际合作和交流机会的地区。 有大量的计算代数组件，我们的方法，它与普通和积分表示理论，群论，环理论，代数图论的技能和经验相结合，并在代数组合学的新兴思想，以产生一个充满活力的研究和培训环境。 这个建议的主要项目是关于寻找描述的最小域实现的不可约表示的超群，发现技术，建设不可约表示的超群，描述单位的有限阶，可以表示为一体的基础上一个非交换的超群，并确定积分表代数，可以实现为邻接代数的一个协会计划。 正在进行的合作项目表示理论的团体有关的Zassenhaus猜想的整群环和多重性的问题的Weil字符的酉群的一个有限的本地环也是该提案的一部分。

I propose to study the representation theory of hypergroups. A hypergroup is a finite-dimensional associative algebra A with a distinguished basis B={b0, b1, , br-1} for which the multiplicative identity b0 = 1 lies in B, and B has the “pseudo-inverse” property: for every bi in B, there is a unique bi* in B for which the coefficient of b0 in bibi* is nonzero. So a hypergroup generalizes the familiar group concept with the group's inverse property replaced by the pseudo-inverse. My work will focus on how these structures can be represented as matrices over as small a field or ring as possible, dealing mainly with two types of hypergroup in addition to group algebras: adjacency algebras of association schemes, in which the nonidentity elements of the basis B can be identified with a collection of graphs, and integral table algebras, which are hypergroups in which the coefficient of every bk in a product of basis elements bibj is always a nonnegative integer. There is a hierarchy here: group algebras are adjacency algebras, and adjacency algebras are integral table algebras. Over the last 20 years, much of the representation theory of these kinds of hypergroups has been motivated by ideas from the representation theory of groups and algebras, and this has resulted in fruitful applications in areas such as graph theory, design theory, and coding theory. It has provided a framework for studies of modular data appearing in conformal field theory, and occasionally new ideas in group theory have been uncovered by those working out the algebraic properties of hypergroups. Representation theory of hypergroups is an emerging area of research in algebraic combinatorics internationally. Many of the new contributions are taking place in Asian nations, Europe, and the U.S., which makes it an area ripe with international collaborative and exchange opportunities for Canadians. There is a substantial computational algebra component to our approach, which mixes with skills and experience in ordinary and integral representation theory, group theory, ring theory, algebraic graph theory, and emerging ideas in algebraic combinatorics to produce a vibrant research and training environment. The main projects in this proposal are about finding descriptions of the smallest field of realization of irreducible representations of hypergroups, discovering techniques for constructing irreducible representations of hypergroups, describing the units of finite order that can be represented integrally in the basis of a noncommutative hypergroup, and determining the integral table algebras that can be realized as the adjacency algebra of an association scheme. Ongoing collaborative projects in the representation theory of groups concerning the Zassenhaus conjecture for integral group rings and on the multiplicity-free question for the Weil character of a unitary group of a finite local ring are also part of the proposal.