Some Questions in Algebraic Combinatorics

John Stembridge为斜对称矩阵的pfaffian提供了一个平面网络解释。这给了林郑月娥和公安部引入和研究完全积极的歪斜版本的最初动机。在这个方向上的进一步研究，特别是由George Lusztig发展的倾斜全正性与全正性理论的关系，是第一个项目的主题。第二个项目是试图重新解释格拉斯马尼亚坐标环中经常使用的标准碱基，并寻找自然发生的不同类型的碱基。这个问题与Gelfand-Tsetlin模式的锥体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。单纯复形的性质和由此产生的基是研究的主题。第三个项目涉及Grassmanian的K-同调，到目前为止研究的范围比K-理论的对偶概念要小。这个方向的第一步是Thomas Lam和PI在K-同调环上引入了对偶稳定的Grothendieck多项式作为Schubert类的代表。需要进一步研究的问题包括Robinson-Schensted对应的K-理论版本、完全旗帜品种的K-同调以及更多。所有三个项目的统一主题都是研究更好的品种，如Grassmanian和更一般的旗帜品种。作为n维空间中所有k维子空间的簇的Grassmanian。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，选择了四条处于通用位置的线。这四条线都相交的线数是多少？最近，George Lusztig的工作丰富了舒伯特微积分，他定义并研究了旗帜变种的一个特殊部分，完全称为正部分。实数自然伴随着正负的概念，当我们考虑旗帜的变化而不是实数时，就有一种方法来识别它们中的积极部分。对完全积极的部分的研究已经导致了一些美丽的最近的发展。

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.