Some Questions in Algebraic Combinatorics

John Stembridge为斜对称矩阵的pfuncan提供了一个平面网络解释。这给了托马斯林和PI引入和研究总积极性的倾斜版本的最初动机。在这个方向上的进一步调查，特别是倾斜的总积极性的关系，以理论的总积极性发展的乔治Lusztig，是主题的第一个项目。第二个项目是试图重新解释经常使用的格拉斯曼坐标环的标准基，并寻找自然发生的不同类型的基。这个问题与三角化Gelfand-Tsetlin模式的锥的问题密切相关，这促使大卫斯派尔和PI引入驾驶规则的概念，作为局部确定三角化的一种方式。性质的单纯复形和基地出现的主题的调查。第三个项目涉及格拉斯曼的K-同调，到目前为止，研究的范围比K-理论的对偶概念小。在这个方向上的第一步是由托马斯林和PI谁介绍了双稳定的Grothendieck多项式的代表舒伯特类的K-同调环。进一步研究的问题包括K理论版本的罗宾逊-申斯特对应，K-同源的完整的旗品种和更多。所有三个项目的统一主题是研究好品种，如Grassmanians和更一般的旗品种。一个格拉斯曼作为一个n维空间的所有k维子空间的变种。舒伯特微积分是一个古老而活跃的研究领域，它粗略地研究了空间的子空间如何相交。一个简单的例子是下面的问题：在三维空间中，选择四条处于通用位置的线。与这四条线相交的直线的个数是多少？舒伯特微积分最近被乔治Lusztig的工作所丰富，他定义并研究了旗簇的一个特殊部分，称为完全正部分。真实的数自然带有正和负的概念，当我们考虑真实的数上的旗形变量时，有一种方法可以识别其中的正部分。对完全积极的部分的研究导致了一些美丽的最新发展。

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.