Representations and cohomology of algebras and categories

本提案的主要焦点是研究EI-范畴的表示和上同调--即所有自同构都是同构的范畴。这一建议的动机源于模表示理论中长期存在的猜想和p-局部群的同伦理论中的公开问题，p-局部群的同伦理论是由Broto，Levi和Oliver在工作中发展起来的一种p-完备拓扑空间。主要研究者最近证明了模表示理论中最著名的猜想之一Alperin权猜想(最初是涉及有限群的p-块的单模的个数的数值等式)具有关于适当的EI-范畴上的某个函子的上同调的结构重构形式。首席研究人员建议探索的方向之一是，应该有类似的结构结果-开发G的工作。R.Robinson-关于Dade的猜想(非常粗略地说，这是对Alperin的重量猜想的改进，考虑了更多的特征标不变量)。块论中的两个重要的公开问题可以完全一般地表示为某些函子在适当范畴上的2-上循环的粘合问题：是否可以将分类空间与每个p-块联系在一起，以及块的中心子群的自同构群上的K“ulshammer-Puig2-上循环是否受定义在块的融合系统上的2-上循环的限制。主要研究者的另一个最新结果是，存在一个谱序列，它把正则EI-范畴上的函子的上同调与块上的函子的上同调联系起来这类物体的同构类的偏序集。这个谱序列对于解决上述两个粘合问题应该是有用的；这是首席调查员提议采取的另一个方向。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，这些映射保持了该对象的结构(其“对称”群)，并从这些映射中推导出所考虑的对象的属性--这是一种非常简化的表象理论的一般原理。

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.