Combinatorics and Geometry of Symmetric Group Representations

代数组合学是一个使用有限和离散结构来研究更复杂的代数和几何结构的数学领域。组合数学的思想和技术越来越多地被用于纯数学的其他领域，如代数几何和表示论，以及应用领域，如数学物理和复杂性理论。这个项目的中心主题是改编代数组合学中经典对象类型的定义，通常只为划分或完整图定义，以应用于一般的图或图。通常，研究这种广义环境会揭示几何和代数之间的联系，而这些联系在原始环境中是可见的。这个项目包括几个研究方向。一个研究方向是研究一般图的Speht模、图Schubert簇的上同调类和匹配系综多面体的几何之间的结构关系。另一个主题是研究图的流多面体上的一类加权格点和，在完全图的情况下，它与停车函数的组合和对角调和空间的Hilbert级数有关。第三个主题是从几何和表示论的角度研究舒伯特多项式的各种推广。

Algebraic combinatorics is an area of mathematics that uses finite and discrete structures to study more complex algebraic and geometric structures. Ideas and techniques from combinatorics are increasingly being used in other areas of pure mathematics such as algebraic geometry and representation theory, as well as applied areas such as mathematical physics and complexity theory. The central theme of this project is to adapt definitions of classical types of objects in algebraic combinatorics, usually defined perhaps only for partitions or the complete graph, to apply to general diagrams or graphs. Often studying this generalized setting reveals geometric and algebraic connections of which only a shadow is visible in the original setting.This project includes several directions for research. One direction of study is to investigate the structural relationships between Specht modules for general diagrams, the cohomology classes of diagram Schubert varieties, and the geometry of matching ensemble polytopes. Another topic is to investigate certain weighted lattice point sums on flow polytopes of graphs, which in the case of the complete graph are related to the combinatorics of parking functions and the Hilbert series of the space of diagonal harmonics. A third topic is to study various generalizations of Schubert polynomials from a geometric and representation-theoretic perspective.