Representation Theory and Schubert Calculus: Combinatorics and Interactions

这个项目的一个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，因为它研究的各种离散结构(如图和偏序集)特别适合于编码和处理复杂的数学对象。PI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，计数满足许多一般相交条件的直线或平面，但目前与量子上同调等现代领域相关)中使用组合技术。通过开发和研究某些表示和相关代数变体的组合模型，PI将追求有效的相关计算；此外，这项工作有望导致更好地理解所提到的数学对象以及它们之间的微妙联系。项目中开发的一些模型将在开源计算机代数系统SAGE中实现。这项研究项目使用组合结构和方法来解决两个主要领域的问题：李代数的表示理论和旗形上的现代Schubert演算。这项研究的一个重要途径是关于由PI和A.Postnikov合作开发的凹槽模型；这是一个组合模型，用于可对称化的Kac-Moody代数的可积最高权表示，以及旗形的K-理论中的某些乘法公式。在S.Naito，D.Sagaki，A.Schling和M.Shimozono最近的工作中，给出了凹穴模型的新应用。这些都突出了仿射李代数的Kirillov-Reshetikhin模、Macdonald多项式和旗形的量子K-理论之间有趣的联系。国际和平研究所将进一步探索其中一些联系。他还将继续将凹室模型应用于p-add群上的Whittaker函数的公式，这是自同构形式理论的基本工具。在现代的Schubert演算中，PI将致力于：统一和扩展普通上同调中Schubert结构常数的各种组合模型，超越K-理论的Schubert演算(例如，在椭圆上同调中)，以及几何Satake对应的组合方面(它给出了基于仿射Grassmanian的不可约李代数表示的几何构造)。

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 the various discrete structures it studies (such as graphs and partially ordered sets) are particularly well suited for encoding and manipulating complex mathematical objects. The PI 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, but is currently related to modern areas such as quantum cohomology). By developing and studying combinatorial models for certain representations and related algebraic varieties, the PI will pursue efficient related computations; furthermore, this work is expected to lead to a better understanding of the mentioned mathematical objects and of the subtle connections between them. Some of the models developed in the project will be implemented in the open source computer algebra system SAGE. This research project uses combinatorial structures and methods to solve problems in two main areas: the representation theory of Lie algebras and modern Schubert calculus on flag manifolds. An important avenue of this research is concerned with the alcove model, developed by the PI in collaboration with A. Postnikov; this is a combinatorial model for integrable highest weight representations of symmetrizable Kac-Moody algebras, as well as for certain multiplication formulas in the K-theory of flag manifolds. New applications of the alcove model were given in recent work of the PI with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono. These highlight interesting connections between Kirillov-Reshetikhin modules for affine Lie algebras, Macdonald polynomials, and the quantum K-theory of flag manifolds. The PI will further explore some of these connections. He will also pursue applications of the alcove model to formulas for Whittaker functions on p-adic groups, which are a basic tool in the theory of automorphic forms. In modern Schubert calculus, the PI will work on: unifying and extending various combinatorial models for the Schubert structure constants in ordinary cohomology, Schubert calculus beyond K-theory (e.g., in elliptic cohomology), and combinatorial aspects of the geometric Satake correspondence (which gives a geometric construction of irreducible Lie algebra representations, based on the affine Grassmannian).