Realization problems in Representation Theory and Algebraic Combinatorics

我建议研究超群的表示理论。超群是具有可区别基B={b0，b1，…的有限维结合代数A，br-1}，其乘法恒等式b0=1位于B中，且B具有“伪逆”性质：对于B中的任一bi，存在唯一的bi*，其b0在Bibi*中的系数为非零。因此，超群推广了常见的群概念，用伪逆代替了群的逆性。*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，除了群代数之外，主要涉及两种类型的超群：结合方案的邻接代数，其中基B的非单位元可以用图的集合来标识；以及积表代数，其中基元bibj的乘积中的每个bk的系数总是非负整数。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在很大程度上受到了群和代数表示理论的启发，这在图论、设计理论和编码理论等领域都产生了丰硕的应用。它为共形场理论中出现的模数据的研究提供了一个框架，研究超群的代数性质的人偶尔也会在群论中发现新的想法。超群的表示理论是国际上一个新兴的代数组合学研究领域。许多新的捐款发生在亚洲国家、欧洲和美国，这使它成为加拿大人拥有国际合作和交流机会的成熟地区。*我们的方法有一个重要的计算代数部分，它结合了普通和积分表示理论、群论、环论、代数图论和代数组合学中的新兴思想的技能和经验，以创造一个充满活力的研究和培训环境。该方案的主要工作是寻找超群的不可约表示的最小实现域的描述，发现构造超群的不可约表示的技巧，描述可在非交换超群的基础上积分表示的有限阶单位，以及确定可实现为结合方案的邻接代数的积分表代数。正在进行的关于整数群环的Zassenhaus猜想的群表示理论和关于有限局部环的酉群的Weil特征标的重数自由问题的合作项目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.