Representation Theory and Schubert Calculus: Combinatorics and Interactions

这个项目的一个统一的主题是组合学和计算的重点。在过去的几十年里，计算在数学研究中扮演着重要的角色。这刺激了组合数学的发展，因为它研究的各种离散结构（如图和偏序集）特别适合编码和操纵复杂的数学对象。PI将使用表示论中的组合技术（表示论是研究群对称性的基本工具，在数学及其他领域具有重要应用，例如，理论物理学），以及舒伯特微积分（其起源于枚举几何学，例如，对满足许多一般相交条件的线或平面进行计数，但目前与诸如量子上同调的现代领域有关）。通过开发和研究某些表示和相关代数簇的组合模型，PI将追求有效的相关计算;此外，这项工作预计将导致更好地理解上述数学对象及其之间的微妙联系。该项目中开发的一些模型将在开放源码计算机代数系统SAGE中实现。本研究计划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流形上的现代舒伯特演算。这项研究的一个重要途径是与壁龛模型有关，由PI与A。波斯特尼科夫;这是可对称化的Kac-Moody代数的可积最高权表示的组合模型，以及旗流形的K-理论中的某些乘法公式。在最近的PI研究中，给出了凹室模型的新应用。Naito，D. Sagaki，A. Schilling和M.下园这些突出有趣的连接Kirillov-Reshetikhin模的仿射李代数，麦克唐纳多项式，和量子K理论的旗帜流形。PI将进一步探索其中的一些联系。他还将追求应用壁龛模型的公式惠特克函数的p-adic组，这是一个基本的工具理论的自守形式。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），和组合方面的几何佐竹对应（这给出了一个几何建设的不可约李代数表示，基于仿射格拉斯曼）。

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).