New Applications of Combinatorics to Representation Theory and Schubert Calculus

组合数学在表示论和舒伯特微积分中的新应用

基本信息

批准号：
1855592
负责人：
Cristian Lenart
金额：
$ 20万
依托单位：
SUNY at Albany
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855592&HistoricalAwards=false
关键词：
New Applications Combinatorics Representation Theory

项目摘要

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.

该研究的主要目的是使用组合结构和方法，以便在代数（更具体地说是表示理论），几何，拓扑和数理论中执行计算；在此过程中还揭示了各个区域之间的隐藏连接。组合学研究各种离散结构（例如排列，部分有序集和图），非常适合编码复杂的数学对象和相关计算。表示理论是研究对称性的基本工具，通过将抽象组/代数的元素视为线性变换（某些向量空间）。 PI研究代数和量子组的表示表示，它们在物理上有许多应用，例如计算特定时间在给定状态处的粒子系统的概率。在几何形状中，PI专注于Schubert演算，其起源于枚举几何形状（例如，计算满足许多通用交叉条件的线或平面），但目前与诸如量子共同体学之类的现代领域有关。 PI开发的一些模型已经或将在开源计算机代数系统Sage中实施。拟议的研究包括以下主要项目，将与几个合作者一起追求。（1）PI将把他的作品扩展到仿射代数的Kirillov-Reshetikhin（KR）晶体的均匀组合模型上，从单列KR晶体到任意晶体。（2）寻求均匀的组合公式的（非属于元素和元素）iwahori whittaker函数，这是自动形式理论中的基本工具。与舒伯特演算的联系也将进行研究。（3）通过Lusztig的Weyl特征的Q-Analeogue的组合分解研究了几何萨克对应关系的组合（从几何实现还原性群体的不可减至表示）。（4）在舒伯特演算中，PI有几个与广义标志品种的各种共同体有关的项目。特别是，基于所谓的Kazhdan-Lusztig Schubert在双曲线共同体中的所谓的Kazhdan-Lusztig Schubert类（提供了Ginzburg-Kapranov-vasserot的椭圆形共同体的茎版本）；这些班级是由PI在先前的联合工作中定义的。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来支持。