Some Questions in Algebraic Combinatorics

John Stembridge提供了一种斜对称矩阵的平面网络解释。这给了Thomas Lam和PI引入和研究总正性的倾斜版本的最初动机。第一个项目的主题是在这个方向上进一步研究，特别是研究偏总正性与乔治·吕茨蒂格（George Lusztig）提出的总正性理论的关系。第二个项目源于重新解释格拉斯曼人坐标环中常用的标准碱基，并寻找自然发生的不同碱基。这个问题与Gelfand-Tsetlin模式锥体的三角测量问题密切相关，这促使David Speyer和PI引入了驾驶规则的概念，作为局部确定三角测量的一种方式。简单配合物的性质和由此产生的碱是研究的主题。第三个项目涉及格拉斯曼的k -同调，迄今为止研究的范围比k理论的对偶概念要小。Thomas Lam和PI在这个方向上迈出了第一步，他们引入了对偶稳定的Grothendieck多项式作为k -同调环中Schubert类的代表。进一步研究的问题包括Robinson-Schensted对应的k -理论版本，完全旗变体的k -同源性等。这三个项目的统一主题是研究优良品种，如格拉斯曼和更一般的旗品种。格拉斯曼函数是一个n维空间的所有k维子空间的变化。舒伯特微积分是一个古老而活跃的研究领域，它非常粗略地研究一个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一般位置的线。与这四条直线相交的直线数是多少？最近，George Lusztig的工作丰富了Schubert微积分，他定义并研究了旗变体的一个特殊部分，称为完全正部分。实数自然带有正负的概念，当我们考虑与实数不同的标志变量时，有一种方法可以识别它们中的正部分。对完全积极的部分的研究导致了许多美丽的最新发展。

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.