Some Questions in Algebraic Combinatorics

代数组合学的一些问题

• 批准号: 0757165
• 负责人: Pavlo Pylyavskyy
• 金额: $ 11.11万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757165&HistoricalAwards=false
• 关键词: Some; Questions; Algebraic; Combinatorics

John Stembridge provided a planar network interpretation for the pfaffian of a skew-symmetric matrix. This gave the original motivation for Thomas Lam and the PI to introduce and study the skew version of total positivity. Further investigation in this direction, and in particular the relation of skew total positivity to the theory of total positivity developed by George Lusztig, is the subject of the first project. The second project grew out of an attempt to reinterpret the often used standard bases of the coordinate ring of Grassmanians, and to find different kinds of bases occurring naturally. The question is closely related to the question of triangulating the cone of Gelfand-Tsetlin patterns, which motivated David Speyer and the PI to introduce the notion of driving rules as a way to locally determine the triangulation. Properties of the simplicial complexes and the bases which arise are the subject of the investigation. The third project deals with K-homology of Grassmanian, studied so far to a smaller extend than the dual notion of K-theory. The first step in this direction was taken by Thomas Lam and the PI who introduced dual stable Grothendieck polynomials as representatives of Schubert classes in the K-homology ring. Further questions to be studied include the K-theoretic version of the Robinson-Schensted correspondence, K-homology of complete flag varieties and more.The unifying theme of all three projects is the study of nice varieties, such as Grassmanians and more general flag varieties. A Grassmanian as the variety of all k-dimensional subspaces of an n-dimensional space. An old and yet active research area is Schubert calculus, which - very roughly - studies how subspaces of a space intersect. A simple example would be the following question: in a three-dimensional space four lines in generic position are chosen. What is the number of lines that intersect all four of them? Schubert calculus was recently enriched by the work of George Lusztig, who defined and studied a special part of flag varieties, called totally the positive part. Real numbers naturally come with a notion of positive and negative, and when we consider flag varieties over real numbers there is a way to identify a positive part in them. The study of the totally positive part has lead to a number of beautiful recent developments.

John Stembridge为Pfaffian提供了平面对称矩阵的平面网络解释。这给了Thomas Lam和Pi引入和研究完全阳性的偏斜版本的最初动机。在这个方向上进行进一步研究，尤其是偏斜的总阳性与乔治·卢斯蒂格（George Lusztig）开发的总积极性理论的关系，是第一个项目的主题。第二个项目的成长是从试图重新诠释格拉马尼亚人坐标环的常用标准基础的尝试，并发现自然发生的不同种类的基础。这个问题与三角测量的问题密切相关，该问题激发了戴维·斯皮尔（David Speyer）和PI介绍驾驶规则的概念，以此作为本地确定三角调节的一种方式。简单复合物的特性和出现的基础是研究的主题。第三个项目涉及格拉马尼亚人的K-词素学，其研究范围比K理论的双重概念较小。托马斯·兰（Thomas Lam）和PI迈出了这一方向的第一步，他们引入了双重稳定的Grothendieck多项式，作为K-同学环中Schubert类的代表。要研究的其他问题包括Robinson-Schensted对应的K理论版本，完整国旗品种的K-词素等。这三个项目的统一主题是研究好品种的研究，例如格拉斯马人和更多的一般国旗品种。 Grassmanian作为N维空间的所有K维子空间的种类。舒伯特演算是一个古老而活跃的研究领域，它非常粗略地研究了空间的子空间是如何相交的。一个简单的示例是以下问题：在三维空间中，选择了四行在通用位置。与所有四个相交的线数是多少？舒伯特演算物最近被乔治·卢斯蒂格（George Lusztig）的作品所丰富，后者定义并研究了国旗品种的特殊部分，称为完全是积极的部分。实际数字自然会带有正面和负面的概念，当我们考虑旗帜品种在实数上时，有一种方法可以识别它们中的积极部分。对完全积极的部分的研究导致了许多美丽的发展。