Morita equivalence classes of blocks

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，并且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，即那些在素特征p的域上的模表示。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，相当于对各个块的模范畴的依次研究。与块相关联的不变量是其缺陷群。在具有平凡缺陷群的块中，所有表示基本上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以非常复杂。我们比较模块类别的块使用森田等价。森田等价模范畴在某种意义上是“相同的”。一个基本的问题是解决多诺万的猜想，提出了在20世纪70年代，它预测，对于一个固定的缺陷组只有1000多个森田等价类的块。多诺万猜想对于我们理解这个问题来说既简单又基本。如果它是真的，那么在理论上我们可以分类模块类别的块，而如果它是假的，那么这个主题甚至比我们想象的更怀尔德和更不可预测。PI，在最近的一篇论文中与Kessar，Külshammer和Sambale，给出了一个分类高达森田等价块的拟单群与阿贝尔亏损群时，素数p是2。这是一个巨大的工具，不仅用于验证多诺万猜想的情况下，但进一步分类的森田等价类块与给定的缺陷组。在极少数有意义的情况下这样做了，这样做将在一个目前非常黑暗的地区点亮一盏灯。这一建议的很大一部分是利用上述文件给精确的描述森田等价类在一系列的情况下，以及证明多诺万猜想在更广泛的情况下。这也是调查的一般现象，森田等价块之间的有限群，特别是在伽罗瓦共轭块的情况下，考虑由Kessar。这里的情况是非常神秘的，因为存在伽罗瓦共轭块（这是几乎无法区分的），不是森田等价。这将涉及一些代数数论，对于我们理解多诺万猜想至关重要。该项目的主要成果将是有关森田等价类的详细信息，为未来的研究提供宝贵的资源。将建立一个网站，提供这些详细信息，并记录多诺万猜想和森田等价类分类的进展。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. 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.