Combinatorics of affine Lie algebras and loop groups

仿射李代数和循环群的组合

• 批准号: 0703524
• 负责人: Peter Magyar
• 金额: $ 10.48万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0703524&HistoricalAwards=false
• 关键词: Combinatorics; affine; Lie; algebras; loop

This project relates combinatorial, algebraic, and topological aspects ofaffine Lie algebras and loop groups, and consists of three connectedinvestigations. The first part explores the factorization phenomenon, adirect and unexpected link between the representations offinite-dimensional Lie groups and infinite-dimensional loop groups. Thesecond part seeks to relate two important combinatorial tools, theLittelmann and Kyoto path models, to the geometric topology of loop groupsas studied by Bott and Mirkovic-Vilonen. The aim is to give a naturalgeometric framework for these tools, which are originally defined in purelycombinatorial terms, in order to better understand their combinatorialstructure. The third part seeks to compute explicit bases for the(co)homology and K-theory of loop groups, so-called affine Schubert andGrothendieck polynomials, applying the geometric approach of Bott-Samelsonto extend and simplify the recent work of combinatorists.Loop groups are symmetry groups of infinite-dimensional objects, but areclosely analogous to symmetries of finite-dimensional objects such asspheres. Loop group symmetry, at the boundary between tame (finite) andwild (intractable) problems, is crucial in many areas, such as theLanglands program in number theory, conformal field theory in particlephysics, soliton theory in integrable systems, and solvable lattice modelsin statistical physics. Precise combinatorial analysis of loop-grouprepresentations makes possible explicit formulas and solutions in all theseareas. Furthermore, the resulting combinatorics of periodic permutationsand Young tableaux has applications to problems of computer science such assorting algorithms and network design in cases which are infinite, butperiodic.

本项目涉及仿射李代数和环群的组合、代数和拓扑方面，由三个相关的研究组成。第一部分探讨了有限维李群表示与无限维环群表示之间的因式分解现象和直接的意想不到的联系。第二部分试图将两个重要的组合工具，littelmann和Kyoto路径模型，与Bott和Mirkovic-Vilonen研究的环路群的几何拓扑联系起来。其目的是为这些工具提供一个自然的几何框架，这些工具最初是用纯组合术语定义的，以便更好地理解它们的组合结构。第三部分试图计算环群的（co）同调和k理论的显式基，即所谓的仿射Schubert和grothendieck多项式，应用bot - samelson的几何方法来扩展和简化组合学家最近的工作。环群是无限维物体的对称群，但与有限维物体（如球体）的对称性非常相似。环群对称，在驯服（有限）和狂野（棘手）问题之间的边界，在许多领域都是至关重要的，比如数论中的兰兰兹程序，粒子物理中的共形场论，可积系统中的孤子理论，以及统计物理中的可解晶格模型。对环群表示的精确组合分析使所有这些领域的显式公式和解决方案成为可能。此外，周期排列和Young表的组合可以应用于计算机科学的问题，例如在无限但周期性的情况下的分类算法和网络设计。