Higher Schur-Weyl dualities and gradings

该项目的目标是构建除表示论外，在不变理论、拓扑学和统计力学中出现的一族代数族的分级版本。重点研究了分圆Wenzl代数，最终目的是建立非半单Brauer代数的分次表示理论。该方法是一种具有较高Schur-Weyl对偶性的BCD模拟，是对对称群的模表示理论、高表示理论和范畴化产生重大影响的一种新方法。虽然在一般线性李代数的情况下已经建立了一些所需的技术，但当脱离A型设置时，许多技术都失败了，我们专注于开发合适的替代品。最基本的是构造分圆Wenzl代数的箭图Hecke(KLR)代数，包括组合和几何两个方面。作为应用，我们期望得到李超群的有理表示的张量积分解的特征标公式和组合模型，特别是在正交辛的情况下。相关主题：Lie(超)代数，箭图Hecke代数，Deligne范畴RepGL(δ)，扭曲的延吉安，各向同性Grassmannians上的倒叶。

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.