Higher Schur-Weyl dualities and gradings

• Applicant: Professorin Dr. Catharina Stroppel
• Subject Area: Mathematics
• Term: from 2009 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 124909482

The objective of the proposed project is a construction of graded versions of a family of algebras which arise, apart from representation theory, in invariant theory, topology and statistical mechanics. The focus is on the cyclotomic Wenzl algebras with the ultimate goal of a graded representation theory for nonsemisimple Brauer algebras. The approach is a type BCD analogue of higher Schur-Weyl duality; a newly developed method with big impact on the modular representation theory of the symmetric group, higher representation theory and categorification. While some of the techniques needed are already established in the case of the general linear Lie algebra, many of these fail when moving away from the type A setup, and we focus on developing suitable substitutes. The most fundamental one is a construction of quiver Hecke (KLR) algebras for cyclotomicWenzl algebras, both combinatorially as well as geometrically. As an application we expect to obtain character formulas and combinatorial models for tensor product decompositions of rational representations for Lie super groups, in particular in the ortho-symplectic cases. Related topics: Lie (super)algebras, quiver Hecke algebras, Deligne's category Rep GL(δ), twisted Yangians, perverse sheaves on isotropic Grassmannians.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1388: Representation Theory