Combinatorial and geometric aspects of the representation theory of finite group schemes

从最广泛的意义上讲，这个项目关注的是通过李理论和几何方法提供张量积的模范畴的研究。由有限群的群代数的模范畴给出的经典背景，是相当好理解的。更一般地说，我们可以考虑有限维霍普夫代数的表示，重点是有限群方案和量子群。几何通过上同调支撑簇和点的表示论支撑空间进入。后者产生了各种幂零算子，一方面提供了新的模不变量，同时定义了编码群方案结构特征的子范畴。组合方面出现在与Auslander-Reiten理论，其中不变量给出的不动点提供了新的见解，某些类别的不可分解的模块内的AR-不动点的分布。在约化代数群的经典背景下，婴儿Verma模的支撑的计算涉及幂零轨道和根系的组合学。关于块的表示类型的问题需要理解Ext-quivers，其结构与简单模块的张量积的分解有关。

Taken in its broadest sense, this project is concerned with the study of module categories affording tensor products via Lie-theoretic and geometric methods. The classical context, given by module categories of group algebras of finite groups, is fairly well understood. More generally, one can consider representations of finite-dimensional Hopf algebras, with emphasis on finite group schemes and quantum groups. Geometry enters via cohomological support varieties and representation-theoretic support spaces of Π-points. The latter give rise to varieties of nilpotent operators that provide new invariants of modules on the one hand, while defining subcategories that encode structural features of group schemes on the other. Combinatorial aspects arise in connection with Auslander-Reiten theory, where invariants given by Π-points provide new insight into the distribution of certain classes of indecomposable modules within the AR-quiver. In the classical context of reductive algebraic groups, the computation of supports of baby Verma modules involves the combinatorics of nilpotent orbits and root systems. Questions concerning the representation type of blocks require the understanding of Ext-quivers, whose structure is related to the decomposition of tensor products of simple modules.