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，计算在数学研究中发挥了重要作用。这刺激了组合学的发展，因为它研究的各种离散结构（如图和部分有序集）特别适合于编码和操作复杂的数学对象。PI将在表示理论（这是研究群对称的基本工具，在数学和理论物理等领域有重要应用）和舒伯特微积分（起源于枚举几何，例如，计算满足若干一般相交条件的线或面，但目前与量子上同调等现代领域有关）中使用组合技术。通过开发和研究某些表示和相关代数变量的组合模型，PI将追求高效的相关计算；此外，这项工作有望导致更好地理解所提到的数学对象和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。本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现代舒伯特在旗流形上的微积分。这项研究的一个重要途径是与凹室模型有关，由PI与A. Postnikov合作开发；这是可对称Kac-Moody代数的可积最高权表示的组合模型，以及标志流形k理论中的某些乘法公式。凹室模型的新应用是在PI最近的工作与S.内藤，D. Sagaki， A. Schilling，和M. Shimozono。这些突出了仿射李代数的Kirillov-Reshetikhin模块、Macdonald多项式和flag流形的量子k理论之间的有趣联系。PI将进一步探讨其中的一些联系。他还将研究凹形模型在p进群上的Whittaker函数公式中的应用，这是自同构形式理论中的一个基本工具。在现代舒伯特微积分中，PI将致力于：统一和扩展普通上同调中舒伯特结构常数的各种组合模型，超越k理论的舒伯特微积分（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Grassmannian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).