DFG-RSF: Geometry and representation theory at the interface of Lie algebras and quivers

• Applicant: Professor Dr. Markus Reineke
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 308831127

• Final Report Year: 2020

Final Report Abstract

This project, in the framework of a joint call of DFG and Russian Science Foundation in the fields of mathematics and physics, grew out of a long-lasting collaboration between members of the project groups on degenerate structures in Algebraic Lie Theory. A beautiful interplay between Lie-theoretic methods and quiver-theoretic methods for the study of the geometry of degenerate flag varieties emerged, from which a wealth of potential research directions was derived, which was therefore chosen as the central theme of the project. Among the main achievements of the project are: • the construction of cell decompositions for arbitrary quiver Grassmannians of Dynkin quivers, • the derivation of a surprising new relation between filtrations on quantum groups, the K-theory of Dynkin quivers and the tropical flaf variety, • the analysis of the geometry of degenerate versions of affine flag varieties and their finitedimensional approximations via quiver Grassmannians, • the discovery of a wealth of new flat (so-called linear) degeneration of flag varieties, • the study of the local and global algebraic geometry of linear degenerations, • the study of the global geometry of the family of linear degenerations, culminating in the calculation of their support sheaves with quantum group methods. The project consolidated and strengthened the collaboration between the project groups in Germany and Russia considerably.

Publications

• Linear degenerations of flag varieties. Math. Z. 287 (2017), no. 1-2, 615–654
  G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke
  
• Cell decompositions and algebraicity of cohomology for quiver Grassmannians. Preprint 2018
  G. Cerulli Irelli, F. Esposito, H. Franzen, M. Reineke
  
• Supports for linear degenerations of flag varieties. Preprint 2018
  X. Fang, M. Reineke
  
• Cones from quantum groups to tropical flag varieties. Preprint 2019
  X. Fang, G. Fourier, M. Reineke
  
• Linear degenerations of flag varieties: partial flags, defining equations, and group actions. Math. Z. (2019)
  G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke