Combinatorial Studies in Algebra, Geometry, and Topology

代数、几何和拓扑的组合研究

• 批准号: 0701044
• 负责人: Cristian Lenart
• 金额: $ 16.88万
• 依托单位: SUNY at Albany
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701044&HistoricalAwards=false
• 关键词: Combinatorial; Studies; Algebra; Geometry; Topology

This research plan is divided into three projects in algebraic combinatorics and its applications to other areas of mathematics. The first project is in combinatorial representation theory, and is mostly concerned with the development of the alcove path model; this is a simple combinatorial model (recently introduced by the investigator in collaboration with A. Postnikov) for the representation theory of complex semisimple Lie algebras and, more generally, of complex symmetrizable Kac-Moody algebras. One problem is to describe the way in which the alcove path model specializes to other models in this area, such as Kashiwara-Nakashima tableaux (in the classical types), and the Kyoto path model (in the affine Kac-Moody types). Other problems are related to the combinatorics of Kashiwara's crystals and an efficient construction of a monomial basis of an irreducible representation; the approach to these problems is based on the alcove path model. Unrelated to this model, the first part of the project also includes a combinatorial study of explicit constructions of irreducible representations based on lattices with a small number of covers. The second part of the project is concerned with modern Schubert calculus on generalized flag varieties. The main goal is to derive combinatorial multiplication formulas for Schubert classes (i.e., the natural basis elements) in the cohomology and K-theory of flag varieties. One such problem is a Chevalley-type multiplication formula (by a codimension 1 class) in the equivariant K-theory of flag varieties for Kac-Moody groups; this formula generalizes a similar formula in the finite case based on the alcove path model. More general such formulas will be investigated in cohomology, based on combinatorial structures such as certain monoids for the Bruhat order on the corresponding Weyl group. The third part of the project is concerned with combinatorial applications to algebraic topology. More precisely, it involves combinatorial formulas (based on trees) for the coefficients of certain formal group laws. There are applications to topological conjectures about classifying spaces of certain finite abelian groups.A unifying theme of the project outlined here 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 it became clear that combinatorial structures are particularly well suited for encoding complex mathematical objects, while combinatorial methods are well suited for related computations. This research plan is part of the ongoing effort to perform concrete computations, based on combinatorial structures. Such structures are used to study representations (i.e., actions on vector spaces) of complex Lie algebras. They are also used to study the geometry of certain classical algebraic varieties, namely flag varieties; related applications exist, for instance, to enumerative geometry (such as counting the lines or planes satisfying a number of generic intersection conditions, which is equivalent to performing certain cohomology calculations). The representations of Lie algebras and the geometry of flag varieties are related to each other, and they play a fundamental role in several areas of mathematics and theoretical physics. They display remarkable combinatorial complexity, which is investigated in this project.

该研究计划在代数组合学及其在其他数学领域的应用中分为三个项目。第一个项目是组合代表理论，主要与壁co路径模型的发展有关。这是一个简单的组合模型（由研究者与A. Postnikov合作引入），用于复杂半神经谎言代数的表示理论，更普遍地是相互对称性的KAC-MOODY代数。一个问题是描述Alcove路径模型专门针对该领域其他模型的方式，例如Kashiwara-Nakashima Tableaux（在经典类型中）和Kyoto Path模型（在Aggine Kac-Moody类型中）。其他问题与喀什瓦拉晶体的组合以及有效构建不可约形表示的基础有关。解决这些问题的方法是基于壁co路径模型。该项目的第一部分与该模型无关，还包括基于少量覆盖的晶格的明确构造的组合研究。该项目的第二部分与现代的舒伯特演算在广义标志品种上有关。主要目的是在旗品品种的共同体和K理论中得出舒伯特类（即自然基础元素）的组合乘法公式。一个问题之一是雪瓦利型乘法公式（通过codimension 1类）在kac-moody群体的旗品品种的等效性K理论中；该公式基于壁co路径模型在有限情况下概括了类似的公式。更通用的这种公式将基于组合结构（例如相应的Weyl组上的Bruhat顺序的某些单体）进行研究。该项目的第三部分与代数拓扑的组合应用有关。更确切地说，它涉及某些正式群体法律系数的组合公式（基于树木）。关于对某些有限阿贝尔组的空间进行分类的拓扑猜想，有应用于此处概述的项目的统一主题。在过去的几十年中，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，因为很明显组合结构特别适合编码复杂的数学对象，而组合方法非常适合相关计算。该研究计划是基于组合结构进行具体计算的持续努力的一部分。这种结构用于研究复杂谎言代数的表示表示（即在矢量空间上的作用）。它们还用于研究某些经典代数品种的几何形状，即标志品种。例如，存在相关的应用程序来列举几何形状（例如计算满足许多通用交叉条件的线或平面，这等同于执行某些共同体学计算）。 Lie代数和国旗品种的几何形状相互关联，它们在数学和理论物理学的多个领域中起着基本作用。他们表现出了显着的组合复杂性，该项目已在该项目中进行了研究。