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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711617
• 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论。 超群是一个有限维关联代数 A，具有显着基 B={b0, b1,..., br-1}，其中乘法恒等式 b0 = 1 位于 B 中，并且 B 具有“伪逆”属性：对于 B 中的每个 bi，B 中存在唯一的 bi*，其中 bibi* 中 b0 的系数不为零。 因此，超群概括了熟悉的群概念，用伪逆来代替群的逆性质。 我的工作将集中于如何将这些结构表示为尽可能小的域或环上的矩阵，主要处理除群代数之外的两种类型的超群：关联方案的邻接代数，其中基 B 的非同一性元素可以通过图的集合来标识，以及积分表代数，其中每个 bk 的系数是基元素 bibj 的乘积 始终是非负整数。 这里有一个层次结构：群代数是邻接代数，邻接代数是积分表代数。 在过去的 20 年里，这类超群的表示论大部分都是受到群和代数表示论思想的启发，这在图论、设计理论和编码理论等领域取得了富有成效的应用。 它为共形场论中出现的模数据的研究提供了一个框架，并且偶尔那些研究超群代数性质的人也会发现群论中的新思想。 超群表示论是国际代数组合学的一个新兴研究领域。 许多新的贡献发生在亚洲国家、欧洲和美国，这使得加拿大成为一个拥有成熟国际合作和交流机会的地区。 我们的方法中有一个重要的计算代数组成部分，它与普通和积分表示论、群论、环理论、代数图论以及代数组合学中的新兴思想的技能和经验相结合，以产生一个充满活力的研究和培训环境。 该提案的主要项目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，描述可以在非交换超群的基础上积分表示的有限阶单元，以及确定可以实现为关联方案的邻接代数的积分表代数。 正在进行的群表示论合作项目，涉及整数群环的 Zassenhaus 猜想和有限局部环酉群的 Weil 特性的无多重性问题，也是该提案的一部分。