Combinatorial Studies in Algebra, Geometry, and Topology

这项研究计划分为三个项目，分别是代数组合学及其在其他数学领域的应用。第一个项目是组合表示理论，主要涉及凹路模型的发展；这是一个简单的组合模型(最近由研究者与A.Postnikov合作引入)，用于复半单李代数的表示理论，更一般地，复对称Kac-Moody代数的表示理论。一个问题是描述凹室路径模型在这一领域专门用于其他模型的方式，例如柏原-中岛型(古典类型)和京都路径模型(仿射Kac-Moody类型)。其他问题涉及柏原晶体的组合学和不可约表示的单项基的有效构造；这些问题的解决方法是基于凹槽路径模型。与这个模型无关，该项目的第一部分还包括对基于具有少量覆盖的格的不可约表示的显式构造的组合研究。该项目的第二部分是关于广义旗簇的现代舒伯特演算。主要目的是得到旗簇的上同调和K-理论中Schubert类(即自然基元)的组合乘法公式。一个这样的问题是Kac-Moody群的旗簇的等变K-理论中的Chvalley型乘法公式(余维1类)；该公式推广了基于凹路模型的有限情况下的类似公式。更一般的这样的公式将在上同调中进行研究，基于组合结构，例如相应Weyl群上Bruhat阶的某些么半群。该项目的第三部分是关于代数拓扑学的组合应用。更准确地说，它涉及某些形式群律的系数的组合公式(基于树)。关于某些有限阿贝尔群的分类空间的拓扑猜想有一定的应用。这里概述的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，计算在数学研究中扮演了重要的角色。这刺激了组合学的发展，因为很明显，组合结构特别适合于对复杂的数学对象进行编码，而组合方法很适合进行相关的计算。这项研究计划是正在进行的基于组合结构进行具体计算的努力的一部分。这种结构被用来研究复李代数的表示(即，在向量空间上的作用)。它们也被用来研究某些经典代数簇的几何，即旗簇；相关的应用存在，例如，对于计数几何(例如对满足一些通用交集条件的直线或平面进行计数，这等同于执行某些上同调计算)。李代数的表示和旗簇的几何是相互联系的，它们在数学和理论物理的几个领域中起着基本的作用。它们表现出显著的组合复杂性，这一点在本项目中进行了研究。

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.