Combinatorics of Skew Tableaux and Flow Polytopes

倾斜表格和流动多面体的组合

• 批准号: 1855536
• 负责人: Alejandro Morales
• 金额: $ 15万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855536&HistoricalAwards=false
• 关键词: Combinatorics; Skew; Tableaux; Flow; Polytopes

Enumerative combinatorics is an area of mathematics that studies counting finite objects and has powerful applications in numerous areas of math and science including representation theory, geometry, probability, statistical mechanics, and theoretical computer science. Two important questions in combinatorics that are of interest in computer science, algebra and optimization are how many ways are there to order objects/tasks with certain constraints and to transport goods through a network. This research studies special cases of these two problems using two fascinating mathematical objects called partially ordered sets and polytopes. The topics of this project are accessible to students and the project includes collaboration with students. Also, the objects studied in the project can be visualized and will be disseminated through videos, student lectures, and STEM outreach activities. Partially ordered sets are fundamental objects in combinatorics and computer science. A measure of the complexity of a finite poset with is its number of linear extensions: the number of orderings of its elements compatible with the order of the poset. The first part of this project studies linear extensions for certain families of partial orders that appear in algebraic combinatorics and can be computed efficiently like posets of (skew) Young diagrams and trees (not necessarily rooted). The tools used include new positive formulas coming from geometry to count skew Young tableaux and determinantal identities for counting linear extensions of trees. The second part of the project studies flow polytopes of graphs or networks and the volumes, the number of lattice points and triangulations of these polytopes. This project has two main goals. The first one is to compute the Ehrhart series of flow polytopes using new objects that encode the volume of these polytopes. The second goal is to study a phenomenon that flow polytopes share with another family of polytopes called generalized permutahedra relating the volume of a polytope with the number of lattice points of a related polytope.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

枚举组合学是一个研究有限对象计数的数学领域，在数学和科学的许多领域都有强大的应用，包括表示论，几何，概率，统计力学和理论计算机科学。在计算机科学、代数和优化中感兴趣的组合数学中的两个重要问题是，有多少种方法可以在某些约束下订购对象/任务，以及通过网络运输货物。本研究使用两个有趣的数学对象（称为偏序集和多面体）来研究这两个问题的特殊情况。该项目的主题是学生可以访问的，该项目包括与学生的合作。此外，该项目中研究的对象可以可视化，并将通过视频，学生讲座和STEM外展活动进行传播。偏序集是组合数学和计算机科学中的基本对象。一个有限偏序集的复杂性的度量是它的线性扩张的数量：它的元素的排序与偏序集的顺序相容的数量。该项目的第一部分研究了某些偏序族的线性扩展，这些偏序出现在代数组合学中，并且可以像（斜）杨图和树（不一定有根）的偏序集那样有效地计算。所使用的工具包括新的正公式来自几何计算斜杨tableaux和行列式的身份，计算线性扩展的树木。第二部分研究了图或网络的流多面体及其体积、格点数和三角剖分。该项目有两个主要目标。第一个是使用编码这些多面体的体积的新对象来计算流多面体的Ehrhart系列。第二个目标是研究流动多面体与另一个多面体家族（称为广义置换面）共享的现象，该多面体家族将多面体的体积与相关多面体的格点数量联系起来。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Generalized Pitman–Stanley flow polytopes

广义 Pitman—Stanley 流多面体

• 发表时间: 2023
• 期刊: Séminaire lotharingien de combinatoire
• 作者: Dugan, William T.;Hegarty, Maura;Morales, Alejandro H.;Raymond, Annie
• 通讯作者: Raymond, Annie

Factorization problems in complex reflection groups

• DOI: 10.4153/s0008414x2000022x
• 发表时间: 2019-06
• 期刊: Canadian Journal of Mathematics
• 作者: J. Lewis;A. Morales

Hook Formulas for Skew Shapes IV. Increasing Tableaux and Factorial Grothendieck Polynomials

• DOI: 10.1007/s10958-022-05777-0
• 发表时间: 2021-08
• 期刊: Journal of Mathematical Sciences
• 作者: A. Morales;I. Pak;G. Panova

Bijecting hidden symmetries for skew staircase shapes

斜楼梯形状的双射隐藏对称性

• DOI: 10.5802/alco.285
• 发表时间: 2023
• 期刊: Algebraic Combinatorics
• 作者: Hamaker, Zachary;Morales, Alejandro H.;Pak, Igor;Serrano, Luis;Williams, Nathan
• 通讯作者: Williams, Nathan

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

流多胞体体积的 Morris 恒等式的改进和对称性

• DOI: 10.5802/crmath.218
• 发表时间: 2021
• 期刊: Comptes Rendus. Mathématique
• 作者: Morales, Alejandro H.;Shi, William
• 通讯作者: Shi, William