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

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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.