Morita equivalence classes of blocks

群是一个抽象的结构，它几乎可以出现在数学或物理的任何领域。因此，它是普遍的，可以成为弥合不同领域的一种手段。群的一些例子是整数(带加法)、多面体的对称性(具有对称性的合成)或曲面上的基本路群。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，就像在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的一个表示，而且需要研究该群的所有表示。捕获此信息的对象是模块类别。我们感兴趣的是群的模表示，也就是素数特征为p的域上的模表示。在这里，细化我们的模范畴是有意义的。我们不是研究集团本身，而是研究它的区块。研究一个群的模范畴相当于依次研究每个块的模范畴。与块相关的不变量是它的缺陷组。在具有平凡缺陷组的块中，所有表示本质上都是单个表示的副本的和，但是具有大缺陷的块的表示的结构可能非常复杂。我们使用Morita等价来比较块的模范畴。Morita等价模范畴在某种意义上是“相同的”。一个基本问题是由1970年S提出的Donovan猜想提出的，该猜想预言对于固定的缺陷群，只有有限多个Morita等价类的块。多诺万的猜想对我们对这一主题的理解既简单又基本。如果它是真的，那么在理论上我们可以对块的模范畴进行分类，而如果它是假的，那么这个主题比我们想象的更广泛和更不可预测。在最近与Kessar，Külshammer和Sambale的一篇论文中，PI给出了当素数p为2时具有交换亏群的拟单群的块的Morita等价的分类。这是一个巨大的工具，不仅可以验证Donovan猜想的情况，而且可以进一步分类具有给定缺陷群的块的Morita等价类。这在很少有意义的情况下完成，这样做会在目前非常黑暗的区域照亮。这一建议的很大一部分是利用上述论文在一系列情况下给出森田等价类的精确描述，并在更广泛的情况下证明Donovan猜想。本文还研究了有限群的块之间Morita等价的一般现象，特别是在Kessar所考虑的Galois共轭块的情况下。这里的情况非常神秘，因为存在不是森田等价的伽罗瓦共轭块(几乎无法区分)。这将涉及到一些代数数论，对我们理解Donovan猜想是至关重要的。该项目的主要成果将是关于Morita等价类的详细信息，为未来的研究提供宝贵的资源。将建立一个网站，以提供这一详细信息，并记录多诺万猜想和森田等价类分类的一般进展。该项目涉及李型有限群、同调代数、数论、群论和表示论的知识，并将受益于与英国和国外强大的代数界的合作。

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. Study of the module category of a group amounts to study of the module category of each block in turn. An invariant associated to a block is its defect group. In a block with trivial defect group all representations are essentially sums of copies of a single representation, but the structure of representations of a block of large defect can be very complex. We compare module categories of blocks using Morita equivalence. Morita equivalent module categories are in a sense 'the same'. A fundamental question is addressed by Donovan's conjecture, posed in the 1970's, which predicts that for a fixed defect group there are only finitely many Morita equivalence classes of blocks. Donovan's conjecture is both simple and fundamental to our perception of the subject. If it is true, then in theory we could classify module categories of blocks, whilst if it is false, then the subject is even wilder and more unpredictable than we imagined.The PI, in a recent paper with Kessar, Külshammer and Sambale, gave a classification up to Morita equivalence of blocks of quasisimple groups with abelian defect groups when the prime p is 2. This is a tremendous tool not only for verifying cases of Donovan's conjecture, but for going further and classifying the Morita equivalence classes of blocks with a given defect group. This has been done in very few meaningful cases and doing so would shine a light in an area which is currently very dark. A large part of this proposal is to exploit the above paper to give precise descriptions of the Morita equivalence classes in a range of cases, as well as to prove Donovan's conjecture in an even wider range of cases. It is also to investigate the general phenomenon of Morita equivalence between blocks of finite groups, particularly in the situation of Galois conjugate blocks as considered by Kessar. Here the situation is very mysterious, in that there exist Galois conjugate blocks (which are almost indistinguishable) that are not Morita equivalent. This will involve some algebraic number theory, and is crucial to our understanding of Donovan's conjecture.The principal outcome of the project will be detailed information on Morita equivalence classes, providing an invaluable resource for future research. A website will be constructed to make this detailed information available and to record progress on Donovan's conjecture and the classification of Morita equivalence classes in general. The project involves knowledge of finite groups of Lie type, of homological algebra, number theory, group theory and representation theory, and will benefit from collaborations with the strong algebra community both in the UK and outside.