Realization problems in Representation Theory and Algebraic Combinatorics

表示论和代数组合学中的实现问题

• 批准号: RGPIN-2017-05331
• 负责人: Herman, Allen
• 金额: $ 1.17万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758992
• 关键词: Realization; problems; Representation; Theory; Algebraic

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.

我建议研究超群的表示理论。超群是具有可分辨基B={b0, b1,， br-1}的有限维结合代数A，它的乘法单位b0 = 1存在于B中，并且B具有“伪逆”性质：对于B中的每一个bi， B中存在一个唯一的bi*，该bi*中b0的系数非零。因此超群推广了群的概念，群的逆性质被伪逆所代替。我的工作将集中在如何将这些结构可以表示成矩阵在尽可能小的字段或环,主要处理两种类型的超群除了组代数:邻接代数协会的计划,不同一性的基础元素B可以确定图的集合,和表代数积分,超群的系数每bk的产品基础元素bibj始终是一个非负整数。这里有一个层次：群代数是邻接代数，而邻接代数是整表代数。在过去的20年里，这类超群的许多表示理论都是受到群和代数的表示理论的启发，这在图论、设计理论和编码理论等领域产生了丰硕的应用。它为共形场论中出现的模数据的研究提供了一个框架，并且那些研究超群的代数性质的人偶尔也发现了群论中的新思想。超群表示理论是国际上代数组合学研究的一个新兴领域。许多新的贡献发生在亚洲国家、欧洲和美国，这使它成为加拿大人进行国际合作和交流机会的成熟领域。我们的方法中有大量的计算代数成分，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，创造了一个充满活力的研究和培训环境。本文的主要工作是寻找超群不可约表示的最小实现域的描述，发现构造超群不可约表示的技术，描述可在非交换超群的基础上积分表示的有限阶单位，确定可作为关联方案邻接代数实现的积分表代数。正在进行的关于整群环的Zassenhaus猜想的群的表示理论和有限局部环的酉群的Weil特征的无多重性问题的合作项目也是提案的一部分。