Representation theory over local rings

局部环的表示论

• 批准号: EP/T004592/1
• 负责人: Radha Kessar
• 金额: $ 49.76万
• 依托单位: City, University of London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT004592%2F1
• 关键词: Representation; theory; over; local; rings

A group is an abstract structure which can arise in almost any area of mathematics or in physics. As such it is universal and can be a means of bridging disparate areas. Some examples of groups are the integers (with addition), the symmetries of a polyhedron (with composition of symmetries) or the fundamental group of paths on a surface. To understand these abstract objects, we need to represent a group in some way. We do this by considering it as a collection of transformations of space. The group may already have natural representations, as happens often in physics, e.g., orthogonal groups, or they may be obscure and involve transformations of very high dimensional spaces (for example the 'monster' sporadic group requires a 196,883 dimensional space). Further we need to study not just one representation of a group, but the entirety of the representations of that group. An object capturing this information is a module category. Our interest is in the modular representations of a group, that is, those over a field of prime characteristic p. Here it makes sense to refine our module category. Instead of studying the group itself, we study its blocks. The study of the module category of a group amounts to study of the module category of each block in turn. It has long been realised that rather than just study representations with respect to a field, it is beneficial to use a local ring as a bridge to connect representations in characteristic zero (classical representation theory) to those in characteristic p (modular representation theory). This approach has been so successful that we are increasingly studying representation theory with respect to local rings in its own right. The overarching theme of this project is the exploitation of this approach in new ways, developing three interrelated bodies of theory aimed at shedding light on some of the big problems of modular representation theory.One theory, which has been little explored, is to take certain quotients of blocks (i.e., smaller objects) which are just large enough to contain information that we are interested in with respect to whichever problem we are looking at. This can usually only be done in the context of local rings. A large part of this project will be laying the foundations of this approach, together with the calculations of examples needed to see patterns on which we can base theory. The famous Alperin-McKay conjecture from the 1970's is an example where this approach will be used. Another theory is the study of the Picard group of a block, which is related to the block's self-similarities. The Picard group defined over a local ring is particularly amenable to study, as shown recently by Boltje, Kessar and Linckelmann, and has been used by Eaton to great effect to analyse module categories very precisely. A main theme of this project is to develop our understanding of Picard groups, and answer some outstanding question regarding their size and structure, as well as developing their application. The study of Picard groups of the quotient objects described above will further bring together the themes of the project.The third theory concerns the realisation of modules and algebras of small fields and associated local rings and the relationships between them. This promises to be a powerful viewpoint for examining existing conjectures and Picard groups.The main outcomes of the project will be on the one hand new theory and techniques which will spur further research, and on the other data about blocks, their Picard groups and their quotient objects, which will be incorporated into Eaton's website cataloguing blocks of finite groups.The project involves knowledge of representation theory, group theory, homological algebra, and number theory, and will benefit from collaborations with the strong algebra community both in the UK and outside.

群是一种抽象结构，几乎可以出现在数学或物理学的任何领域。因此，它是通用的，可以成为连接不同领域的一种手段。群的一些例子是整数（带有加法）、多面体的对称性（带有对称性的组合）或表面上的基本路径群。为了理解这些抽象对象，我们需要以某种方式表示一个组。我们通过将其视为空间变换的集合来做到这一点。该群可能已经具有自然表示，就像物理学中经常发生的那样，例如正交群，或者它们可能是模糊的并且涉及非常高维空间的变换（例如“怪物”零星群需要 196,883 维空间）。此外，我们不仅需要研究一个群体的一种表征，而且需要研究该群体的全部表征。捕获此信息的对象是模块类别。我们的兴趣在于群的模块化表示，即素数特征 p 的域上的模块化表示。在这里，完善我们的模块类别是有意义的。我们不是研究群体本身，而是研究它的区块。对一个组的模块类别的研究，相当于依次对每个块的模块类别的研究。人们很早就认识到，使用局部环作为桥梁来连接特征零（经典表示理论）中的表示与特征 p（模表示理论）中的表示，而不是仅仅研究关于域的表示，这是有益的。这种方法非常成功，以至于我们越来越多地研究关于局部环本身的表示理论。该项目的首要主题是以新的方式利用这种方法，发展三个相互关联的理论体系，旨在阐明模块化表示理论的一些大问题。其中一个很少被探索的理论是采取某些块（即较小的对象）的商，这些商足够大以包含我们对我们正在研究的任何问题感兴趣的信息。这通常只能在本地环的上下文中完成。该项目的很大一部分将奠定这种方法的基础，以及所需示例的计算，以了解我们可以作为理论基础的模式。 1970 年代著名的 Alperin-McKay 猜想就是使用这种方法的一个例子。另一种理论是块的皮卡德群的研究，这与块的自相似性有关。正如 Boltje、Kessar 和 Linckelmann 最近所表明的那样，在局部环上定义的皮卡德群特别适合研究，并且伊顿使用它来非常精确地分析模块类别，效果非常好。该项目的一个主题是加深我们对皮卡德群的理解，并回答有关其规模和结构的一些突出问题，以及开发其应用程序。对上述商对象的皮卡德群的研究将进一步汇集该项目的主题。第三个理论涉及小域和相关局部环的模和代数的实现以及它们之间的关系。这有望成为检验现有猜想和皮卡德群的有力观点。该项目的主要成果一方面是促进进一步研究的新理论和技术，另一方面是关于块、其皮卡德群及其商对象，将被纳入伊顿有限群块编目网站中。该项目涉及表示论、群论、同调代​​数和数论知识，并将受益于与英国和英国强大的代数社区的合作 外部。

项目成果

On the BV structure of the Hochschild cohomology of finite group algebras

• DOI: 10.2140/pjm.2021.313.1
• 发表时间: 2020-05
• 期刊: Pacific Journal of Mathematics
• 影响因子: 0.6
• 作者: D. Benson;R. Kessar;M. Linckelmann

Structure of blocks with normal defect and abelian inertial quotient

具有正态缺陷和阿贝尔惯性商的块的结构

• DOI: 10.1017/fms.2023.13
• 发表时间: 2023
• 期刊: Forum of Mathematics, Sigma
• 作者: Benson D

Arbitrarily large Morita Frobenius numbers

任意大的 Morita Frobenius 数

• DOI: 10.2140/ant.2022.16.1889
• 发表时间: 2022
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Eisele F

Bijections of silting complexes and derived Picard groups

淤积复合体和派生皮卡德群的双射

• DOI: 10.1112/jlms.12591
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Eisele F

Hochschild cohomology of symmetric groups and generating functions, II

对称群和生成函数的 Hochschild 上同调，II

• DOI: 10.1007/s40687-023-00382-2
• 发表时间: 2023
• 期刊: Research in the Mathematical Sciences
• 影响因子: 1.2
• 作者: Benson D