Combinatorics and Geometry of Symmetric Group Representations

对称群表示的组合学和几何

• 批准号: 2204415
• 负责人: Ricky Liu
• 金额: $ 18.52万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2022-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204415&HistoricalAwards=false
• 关键词: Combinatorics; Geometry; Symmetric; Group; Representations

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.

代数组合学是数学领域，使用有限和离散结构来研究更复杂的代数和几何结构。组合学的思想和技术越来越多地用于纯数学的其他领域，例如代数几何学和表示理论，以及数学物理和复杂性理论等应用领域。该项目的中心主题是在代数组合物中调整经典对象类型的定义，通常仅适用于分区或完整图，以应用于一般图或图形。经常研究这种广义设置会揭示几何和代数连接，在原始设置中只能看到阴影。该项目包括多个研究方向。一个研究方向是研究通用图，图表的共同图类别的SCHUBERT品种的共同图和匹配集合多型的几何形状之间的SpecHT模块之间的结构关系。另一个主题是研究图表流程上的某些加权晶格点总和，在完整图的情况下，这与停车功能的组合和对角谐波空间的Hilbert系列有关。第三个主题是从几何和表示理论的角度研究舒伯特多项式的各种概括。

项目成果

Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes

作为 Gelfand-Tsetlin 多胞形 Minkowski 和的投影的舒伯特多项式

• DOI: 10.5070/c62359152
• 发表时间: 2022
• 期刊: Combinatorial Theory
• 作者: Liu, Ricky Ini;Mészáros, Karola;Dizier, Avery St.
• 通讯作者: Dizier, Avery St.

Twisted Schubert polynomials

• DOI: 10.1007/s00029-022-00802-1
• 发表时间: 2019-05
• 期刊: Selecta Mathematica
• 作者: Ricky Ini Liu

Up- and Down-Operators on Young's Lattice

杨氏格上的向上和向下算子

• DOI: 10.37236/10099
• 发表时间: 2021
• 期刊: The Electronic Journal of Combinatorics
• 作者: Liu, Ricky;Smith, Christian
• 通讯作者: Smith, Christian

Determinantal Formulas for SEM Expansions of Schubert Polynomials

舒伯特多项式 SEM 展开式的行列式

• DOI: 10.1007/s00026-021-00558-z
• 发表时间: 2021
• 期刊: Annals of Combinatorics
• 影响因子: 0.5
• 作者: Hatam, Hassan;Johnson, Joseph;Liu, Ricky Ini;Macaulay, Maria
• 通讯作者: Macaulay, Maria