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

在最广泛的意义上，这个项目关注于通过李论和几何方法来研究提供张量积的模范畴。由有限群的群代数的模范畴给出的经典上下文是相当好理解的。更一般地，我们可以考虑有限维Hopf代数的表示，重点是有限群方案和量子群。几何是通过Π点的上同调支集和表示论支集空间进入的。后者产生了各种幂零算子，一方面提供了新的模不变量，另一方面定义了编码群方案结构特征的子范畴。组合方面与Auslander-Reiten理论有关，其中由Π点给出的不变量提供了对AR-箭图中某些类不可分解模的分布的新见解。在经典约化代数群的背景下，Baby Verma模的支撑度的计算涉及到幂零轨道和根系的组合。关于块的表示类型的问题需要理解外箭图，它的结构与单模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.