Generic points in representation varieties with applications to Lie theory and 0-Schur algebras

箭图的表示理论(RQT)和李理论(LT)是密切相关的，例如通过根系统。本文探索RTQ、LT、量化包络代数(QEA)和量化Schur代数(QSA)之间的新联系。申请者将首先研究表示簇中的一般现象，特别是开轨的存在，以及各种射影表示对中的通用点。然后，她将致力于解释LT、QEA和QSA环境中的一般现象。主要应用将证明某些类型的李代数，特别是海藻李代数的Richardson元的存在性。其他应用包括对单李代数的基本系统的组合性质的自然解释和新的结晶基的构造。

Representation theory of quivers (RQT)and Lie theory (LT) are closelyrelated, for instance via root systems.This proposalexplores new connectionsbetween RTQ, LT,quantised enveloping algebras (QEA) and quantised Schur algebras ( QSA).The applicant will first study generic phenomena inrepresentation varieties, in particular, the existence of open orbits, and genericpoints in varieties of pairs of projective representations. Then she will work oninterpreting the generic phenomena in the setting of LT, QEA and QSA.The main applications will be to show the existence of Richardson elementsfor some classes of Lie algebras, in particular for seaweed Lie algebras. Otherapplications include a natural interpretation of combinatorial properties ofroot systems of simple Lie algebras and a new construction of crystal bases.