Representations and cohomology of algebras and categories

本建议的主要焦点是EI-范畴的表示和上同调的研究-也就是说，所有自同态都是同构的范畴。这一提议的动机源于模表示论中长期存在的理论以及p-局部群同伦理论中的公开问题，p-局部群是一种p-完备拓扑空间，其同伦理论由Broto，Levi和奥利弗在工作中发展。主要研究者最近证明了模表示论中最著名的理论之一，Alperin的重量猜想，（它最初是一个涉及有限群的p-块的单模个数的数值等式），在一个适当的EI-范畴上，用某个函子的上同调表示，它有一个结构上的重新表述。 主要研究者提议探索的方向之一是应该有类似的结构结果-利用G. R.罗宾逊-关于戴德的猜想（这是，非常粗略，细化Alperin的重量猜想考虑到更微妙的不变量的字符）。块理论中两个重要的公开问题可以用完全一般的方式表示为某些函子在适当范畴上的2-上圈的“胶合问题”：问题，以及块的中心子群的自同构群上的K“ulshammer-Puig ~ 2-上圈是否是一个2-主要研究者最近的另一个结果是，伊萨谱序列将正则EI-范畴上函子的上同调与该范畴对象的同构类偏序集上函子的上同调联系起来，这个谱序列对于解决上述两个胶合问题应该是有用的，这是主要研究者提出的另一个方向。 看待一个对象（甚至不一定是数学对象）的一种方法是考虑该对象到其自身的所有映射，这些映射保持其结构（其“对称性”群），并从这些映射中推导出所考虑对象的属性--这是以一种非常简化的方式，表示论的一般原理。

The main focus of the present proposal is the study of representationsand cohomology of EI-categories - that is, categories all of whoseendomorphisms are isomorphisms. The motivations for this proposal havetheir roots in long standing conjectures in modular representation theory aswell as open problems in the homotopy theory of p-local groups, a certaintype of p-complete topological spaces whose homotopy theory has beendeveloped in work by Broto, Levi and Oliver.The principal investigator has recently shown that one of the mostwell-known conjectures in modular representation theory, Alperin's weightconjecture, (which is originally a numerical equality involving the number of simple modules of a p-block of a finite group), has a structuralreformulation in terms of the cohomology of a certain functor on a suitableEI-category. One of the directions the principal investigator proposes to explore isthat there should be similar structural results - exploiting work ofG. R. Robinson - regarding Dade's conjectures (which are, very roughly,refinements of Alperin's weight conjecture taking into account moresubtle invariants of characters). The problem is to find the ``right"EI-category.Two important open problems in block theory can be formulated in acompletely general way as ``gluing problems" of 2-cocycles of certain functors onappropriate categories: the question whether one can associate aclassifying space with each p-block and the question whether the K"ulshammer-Puig2-cocycles on automorphism groups of centric subgroups of a block arethe restrictions of a 2-cocycle defined on the fusion system of theblock.Another recent result of the principal investigator is that there isa spectral sequence relating the cohomology of a functoron a regular EI-category to the cohomology of functors on the poset ofisomorphism classes of objects of that category.This spectral sequence should be useful for addressingthe two mentioned gluing problems; thisis another direction the principal investigator proposes to take. One way to look at a - not even necessarily mathematical - object is toconsider all maps of that object to itself which preserve itsstructure (the group of its "symmetries") and deduce from those maps properties of the considered object -this is, in a very simplified way, the general principle ofrepresentationtheory.