New Applications of Combinatorics to Representation Theory and Schubert Calculus

该研究的主要目标是使用组合结构和方法，以便在代数（更具体地说，表示论），几何，拓扑和数论中执行计算;在这个过程中，还揭示了各个领域之间的隐藏联系。组合学研究各种离散结构（如排列、偏序集和图），它们非常适合编码复杂的数学对象和相关的计算。表示论是研究对称性的基本工具，通过将抽象群/代数的元素实现为（某些向量空间的）线性变换。PI研究李代数和量子群的表示，这些表示在物理学中有许多应用，例如计算粒子系统在特定时间处于给定状态的概率。在几何学中，PI专注于舒伯特微积分，它起源于枚举几何学（例如，计算满足多个一般相交条件的线或平面），但是目前与诸如量子上同调的现代领域有关。PI开发的一些模型已经或将在开放源码计算机代数系统SAGE中实现。拟议的研究包括以下主要项目，将与几个合作者一起进行。(1)PI将扩展他的工作均匀组合模型的Kirillov-Reshetikhin（KR）晶体的仿射李代数从单列KR晶体的任意。(2)统一的组合公式，寻求（非metaplectic和metaplectic）Iwahori惠特克函数，这是一个基本的工具，在理论的自守形式。与舒伯特微积分的连接也将被调查。(3)几何佐竹对应的组合学（实现几何上的不可约表示的约化群）研究通过组合分解Lusztig的q-模拟的Weyl字符。(4)在舒伯特演算中，PI有几个与广义旗簇的各种上同调相关的项目。特别地，基于所谓的Kazhdan-Lusztig Schubert类的双曲上同调，我们将Schubert演算推广到K-理论之外（它给出了Ginzburg-Kapranov-Vasserot椭圆上同调的一个茎版本）;这些类别是由PI在以前的联合工作中定义的。该奖项反映了NSF的法定使命，并通过利用基金会的智力价值进行评估，更广泛的影响审查标准。

The main goal of the research is to use combinatorial structures and methods in order to perform computations in algebra (more specifically, representation theory), geometry, topology, and number theory; hidden connections between various areas are also revealed in this process. Combinatorics studies various discrete structures (such as permutations, partially ordered sets, and graphs), which are well suited for encoding complex mathematical objects and for the related computations. Representation theory is a fundamental tool for studying symmetry, by realizing the elements of abstract groups/algebras as linear transformations (of some vector spaces). The PI studies representations of Lie algebras and quantum groups, which have many applications to physics, such as calculating the probability of a particle system being in a given state at a particular time. In geometry, the PI focuses on Schubert calculus, which has its origins in enumerative geometry (e.g., counting the lines or planes satisfying a number of generic intersection conditions), but is currently related to modern areas such as quantum cohomology. Some of the models developed by the PI have been or will be implemented in the open source computer algebra system SAGE.The proposed research consists of the following main projects, to be pursued with several collaborators. (1) The PI will extend his work on uniform combinatorial models for Kirillov-Reshetikhin (KR) crystals of affine Lie algebras from the single column KR crystals to the arbitrary ones. (2) Uniform combinatorial formulas are sought for the (non-metaplectic and metaplectic) Iwahori Whittaker functions, which are a basic tool in the theory of automorphic forms. Connections with Schubert calculus will also be investigated. (3) The combinatorics of the geometric Satake correspondence (realizing geometrically the irreducible representations of reductive groups) is studied via a combinatorial decomposition of Lusztig's q-analogue of the Weyl character. (4) In Schubert calculus, the PI has several projects related to various cohomologies of generalized flag varieties. In particular, an extension of Schubert calculus beyond K-theory is pursued based on the so-called Kazhdan-Lusztig Schubert classes in hyperbolic cohomology (which gives a stalk version of the elliptic cohomology of Ginzburg-Kapranov-Vasserot); these classes were defined by the PI in previous joint work.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.