Steiner bundles on Grassmannians and representations of wild Kronecker algebras

• Applicant: Dr. Daniel Bissinger
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 548677842

Let r > 2 and d < r be natural numbers, and let k be an algebraically closed field of arbitrary characteristic. The main goal of this project is to get a better understanding of the connection between Steiner bundles on the Grassmannian Gr_d(k^r) and finite dimensional r-Kronecker representations. The first considerations of Kronecker representations as a tool to the study vector bundles can be found in works by Hulek (1981) and Drézet-Le Potier (1985) for the special case of the full projective space over the field of complex numbers. A functorial connection between certain bundles of smooth projective varieties and abstractly described full subcategories of Kronecker representations goes back to Jardim and Prata (in characteristic 0).Together with Rolf Farnsteiner, the applicant of the project built on the approach of Jardim and Prata for the Grassmannian Gr_d(k^r) and proved the equivalence of the category of Steiner bundles on Gr_d(k^r) and a full subcategory of Kronecker representations for fields of arbitrary characteristic. This leads to applications concerning the connections between representations of finite group schemes and vector bundles à la Carlson, Friedlander, Pevtsova, Suslin. Furthermore, the Kronecker representations corresponding to Steiner bundles were identified as the category of so-called relatively projective representations. This new description enables the investigation of the category of Steiner bundles using combinatorial tools of modern representation theory. Among other things, one can prove that the category of relatively projective representations is closed under the inverse Auslander-Reiten translation, subrepresentations and extensions. As a consequence, Steiner bundles can be arranged in a directed graph (translation quiver) in which each vertex has a successor and exactly one vertex in each connected component has no predecessor, i.e. is a root. In this project, this new combinatorial description of the category of Steiner bundles and the category of relatively projective representations will be studied using methods of representation theory, algebraic geometry and algebraic groups.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Rolf Farnsteiner