Combinatorics of affine Lie algebras and loop groups

这个项目涉及李代数和循环群的组合、代数和拓扑方面，由三个相互关联的研究组成。第一部分探讨了因式分解现象，即无限维李群和无限维李群的表示之间的直接和意想不到的联系。第二部分试图将两个重要的组合工具Littelmann模型和京都路径模型与Bott和Mirkovic-Vilonen所研究的环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。第三部分应用Bott-Samels的几何方法推广和简化了组合学家最近的工作，计算了Loop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。Loop群是无限维对象的对称群，但与球面等有限维对象的对称性有密切的相似之处。在许多领域，如数论中的朗兰兹程序、粒子物理中的共形场论、可积系统中的孤子理论和统计物理中的可解晶格模型，在驯服(有限)和狂野(难处理)问题之间的圈群对称性是至关重要的。循环群预演的精确组合分析使所有这些领域的显式公式和解成为可能。此外，所得到的周期置换和Young表的组合在计算机科学问题中也有应用，例如在无限但周期的情况下的分类算法和网络设计。

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.