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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675767
• 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.

我建议研究超组的表示理论。高组是具有杰出基础B = {b0，b1，，，Br-1}的有限差的联想代数A，b0 = 1 = 1在B中，并且B具有“伪内”属性：对于B中的每个BI，B0中的b0 bibi innon nonnonze ecor bibi is bibi is bi ecor中均具有唯一的bi*。因此，超级组将熟悉的组概念概括为组的逆属性被伪内替代。 *** My work will focus on how these structures can be represented as materies 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 core of every bk in a product of basis elements bibj is总是有一个层次结构：组代数为邻接代数，邻接代数是积分表代数。在过去的20年中，这些类型的超级组的大部分代表理论都是由群体和代数代表理论的思想所激发的，这导致了在图理论，设计理论和编码理论等领域的富有成果的应用。它为在保形场理论中出现的模块化数据提供了一个框架，偶尔在群体理论中发现了新的思想，这些想法被那些涉及超级组的代数属性的人发现了。超级组的表示理论是国际代数组合学的新兴领域。许多新捐款都在亚洲国家，欧洲和美国进行，这使其成为加拿大人的国际合作和交流机会的成熟领域。 ***我们的方法有一个实质性的计算代数组成部分，它与普通和积分代表理论，群体理论，环理论，代数图理论以及代数组合中的新兴思想相结合，以产生充满活力的研究和培训环境。该提案中的主要项目是关于找到超级组的不可约表示的最小实现领域的描述，发现用于构建超级组的不可还能表示的技术，描述可以在不交流性超级分组的基础基础上代表的有限顺序单位，并确定与整体相关性的相关性，并确定与之相关的相关性。群体代表理论中正在进行的协作项目与Zassenhaus集成群环的概念以及在有限的本地环的单一群体特征的无数性问题上，也是该提案的一部分。