Combinatorics of Crystals, Macdonald Polynomials, and Schubert Calculus

这个研究项目使用组合结构和方法来解决两个领域的问题：李代数的表示理论和舒伯特演算。在表象理论中，研究人员将继续从麦克唐纳多项式的Ram-Yip公式及其乘积的Yip公式中推导出经典Lie类型的显式Tableau公式；这些公式是基于凹槽行走模型的，该模型是由Gaussen-Littelmann和研究人员与A.Postnikov合作提出的，然后由其他数学家发展。其中一个应用是能量函数的有效计算，它定义了Kirillov-Reshetikhin晶体张量积的仿射分级。这个应用是基于麦克唐纳多项式、量子上同调和仿射晶体之间的有趣联系。为了更好地理解它，研究者提出了“量子=仿射”现象的晶体理论对应(基于凹室模型)，该现象与FLAG簇的量子上同调和仿射Grassmanian的同调有关。表示论中的其他项目涉及某些表示的显式构造，以及晶体的拓扑(作为偏序集)和编码Mirkovic-Vilonen环结构的偏序集的拓扑。在Schubert演算中，研究者有与广义FLAG簇的上同调、量子上同调和量子K-理论相关的项目。这些项目中的大多数都涉及到舒伯特结构常数的正组合公式的新方法，舒伯特结构常数在舒伯特类的基础上表示舒伯特类的乘积。这个项目的统一主题是强调组合学和计算。在过去的几十年里，计算在数学研究中扮演了重要的角色。这刺激了组合学的发展，因为很明显，组合结构特别适合于对复杂的数学对象进行编码，而组合方法很适合进行相关的计算。研究人员将在表示理论(这是研究群对称性的基本工具，在数学和其他领域有重要应用，例如，理论物理)和舒伯特微积分(其起源于计数几何，例如，计算满足一些一般相交条件的直线或平面)中使用组合技术。某些表示和相关的代数簇由图或偏序集来建模。通过研究这些显示出显著复杂性的离散物体的结构，研究人员将能够获得重要的代数和几何信息。

This research project uses combinatorial structures and methods to solve problems in two areas: the representation theory of Lie algebras and Schubert calculus. In representation theory, the investigator will continue his work on deriving explicit tableau formulas in classical Lie types from the Ram-Yip formula for Macdonald polynomials and the Yip formula for their product; these formulas are in terms of the alcove walk model, which was introduced by Gaussent-Littelmann and the investigator in collaboration with A. Postnikov, and was then developed by other mathematicians. One of the applications is an efficient computation of the energy function, which defines the affine grading on a tensor product of Kirillov-Reshetikhin crystals. This application is based on an interesting connection between Macdonald polynomials, quantum cohomology, and affine crystals. In order to better understand it, the investigator proposes a crystal-theoretic counterpart (based on the alcove model) of the "quantum=affine" phenomenon relating the quantum cohomology of flag varieties and the homology of the affine Grassmannian. Other projects in representation theory involve explicit constructions of certain representations, as well as the topology of crystals (as posets) and of a poset which encodes the structure of Mirkovic-Vilonen cycles. In Schubert calculus, the investigator has projects related to the cohomology, quantum cohomology, and quantum K-theory of generalized flag varieties. Most of these projects involve new approaches to positive combinatorial formulas for the Schubert structure constants, which express the product of Schubert classes in the basis of Schubert classes.A unifying theme of this project is the emphasis on combinatorics and computation. During the last decades, computation has gained an important role in mathematical research. This stimulated the development of combinatorics, as it became clear that combinatorial structures are particularly well suited for encoding complex mathematical objects, while combinatorial methods are well suited for related computations. The investigator will use combinatorial techniques in representation theory (which is a fundamental tool for studying group symmetry, and which has important applications in mathematics and beyond, e.g., to theoretical physics), and in Schubert calculus (which has its origins in enumerative geometry, e.g., counting the lines or planes satisfying a number of generic intersection conditions). Certain representations and the related algebraic varieties are modeled by graphs or partially ordered sets. By studying the structure of these discrete objects, which displays remarkable complexity, the investigator will be able to derive important algebraic and geometric information.