Nilpotent Orbits, Representation Theory, and Combinatorics

主要的研究人员将研究表示论、代数几何和组合学中与李代数中的幂零轨道有关的几个问题。更具体地说，研究人员将继续研究正根偏序集中的理想及其与表示理论和簇代数理论的联系。他还将继续研究Lusztig-Vogan双射并寻找它的显式计算。他将试图完成确定例外李代数中哪些幂零轨道具有正规闭包，并证明(在所有李代数中)关于依赖于类似技巧的幂零轨道覆盖上的函数结构的结果。表示论是现代代数的一个分支，与理解对称性有关。一个中心思想是，复杂的代数或几何结构可以由一组比原始结构更容易理解的矩阵(数组)来表示。例如，正方形的对称性可以(以一种可能的方式)由一组8个2乘2的矩阵来表示。这种表象理论的观点在化学和物理中有很多应用。研究人员的工作将有助于理解李代数和李群的表示理论，这对于理解数学中出现的对象的对称性是必不可少的。

The principal investigator will study several questionsin representation theory, algebraic geometry, and combinatorics which are related to nilpotent orbits in Lie algebras.More specifically, the investigator will continue his work on ideals in the poset of positive rootsand their connection to representation theory and the theory of cluster algebras.He will also continue to study the Lusztig-Vogan bijection and to seek out its explicit computation. He will try to complete the determination of which nilpotent orbits in the exceptional Lie algebras have normal closure and prove results (in all Lie algebras) about the structure offunctions on covers of nilpotent orbits that rely on similar techniques.Representation theory is a branch of modern algebra that is concernedwith understanding symmetries. A central idea is that complicated algebraic or geometric structures can berepresented by a certain set of matrices (arrays of numbers), which are easier to understand than the original structure. For example, thesymmetries of the square can be represented (in one possible way) by a certain set of eight two-by-two matrices. This view of representation theory has many applications in chemistry and physics. The investigator's work will contribute tounderstanding the representation theory of Lie algebras and Lie groups, which are essential to understanding the symmetry of objects which arise in mathematics.