Representation theory over local rings

群是一种抽象的结构，几乎可以出现在任何数学或物理领域。因此，它具有普遍性，可以成为弥合不同领域的一种手段。群的一些例子是整数（有加法），多面体的对称性（有对称性的合成）或曲面上路径的基本群。为了理解这些抽象对象，我们需要以某种方式表示一个组。我们这样做是因为把它看作是空间变换的集合。这个群可能已经有了自然的表示，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，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，我们不仅需要研究群的一个表示，而且需要研究该群的所有表示。捕获这些信息的对象是模块类别。我们感兴趣的是一个群的模表示，也就是说，那些在一个领域的主要特征p。在这里，它是有意义的细化我们的模范畴。我们不研究群体本身，而是研究它的区块。对一个组的模范畴的研究，相当于对每个块的模范畴的依次研究。人们早就意识到，而不是仅仅研究关于一个域的表示，使用局部环作为连接特征零（经典表示论）和特征p（模表示论）的表示的桥梁是有益的。这种方法已经如此成功，我们越来越多地研究表示论方面的本地环本身的权利。这个项目的首要主题是以新的方式开发这种方法，开发三个相互关联的理论，旨在阐明模块化表示理论的一些重大问题。一个理论，很少被探索，是采取某些块的替代品（即，较小的对象），这些对象足够大，可以包含我们感兴趣的关于我们正在研究的任何问题的信息。这通常只能在本地环的上下文中完成。这个项目的很大一部分将奠定这种方法的基础，以及计算的例子需要看到的模式，我们可以根据理论。著名的Alperin-McKay猜想从20世纪70年代是一个例子，这种方法将被使用。另一个理论是研究块的Picard群，这与块的自相似性有关。定义在局部环上的皮卡德群特别适合于研究，正如最近Boltje，Kessar和Linkelmann所表明的那样，伊顿已经使用它来非常精确地分析模范畴。这个项目的一个主题是发展我们对Picard群的理解，并回答一些关于它们的大小和结构的突出问题，以及开发它们的应用。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。第三个理论涉及实现模块和代数的小领域和相关的本地环和它们之间的关系。这个项目的主要成果一方面是新的理论和技术，这将促进进一步的研究，另一方面是关于块的数据，他们的皮卡德群和他们的商对象，这将被纳入伊顿的网站编目有限群的块。该项目涉及表示论，群论，同调代数，数论，并将受益于与强大的代数社区在英国和国外的合作。

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.