Polytopes in Combinatorics and Algebra

组合学和代数中的多面体

基本信息

批准号：
1501059
负责人：
Karola Meszaros
金额：
$ 17万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-07-15 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501059&HistoricalAwards=false
关键词：
Polytopes Combinatorics Algebra

项目摘要

Polytopes are geometric objects with flat sides: polygons in two dimensions, polyhedra in three dimensions, and higher-dimensional generalizations. Polytopes are ubiquitous in mathematics and play important roles throughout science and engineering. The simplest n-dimensional polytopes have only n+1 vertices and are called simplices: triangles, tetrahedra, and their higher-dimensional generalizations. While it is straightforward to determine the volume of a simplex in any dimension, the volume of a general polyhedron in high dimension can be challenging to compute. One way to calculate volume is to dissect a polytope into simplices. This research project studies dissections, volumes, and integer points of special families of polytopes known as flow and root polytopes. Flow and root polytopes naturally appear in problems in several areas of mathematics, such as representation theory and algebraic geometry. Results of this project will have impact in these areas of mathematics as well as in scientific and engineering applications.A number of important conjectures and questions in algebraic combinatorics have flow polytopes and root polytopes at their core. These special polytopes can be systematically dissected into simplices. The project will investigate a conjecture of Haglund about the bigraded Hilbert series of the space of diagonal harmonics that can be stated in terms of a sum of a certain weight function over the integer points of a flow polytope. The research will also study the subword complexes introduced by Knutson and Miller, conceived to illustrate the combinatorics of Schubert polynomials and determinantal ideals, by relating them to root polytopes.

多面体是具有平坦侧面的几何对象：二维多边形、三维多面体以及更高维的概括。多胞体在数学中无处不在，并在整个科学和工程中发挥着重要作用。最简单的 n 维多面体只有 n+1 个顶点，称为单纯形：三角形、四面体及其更高维的推广。虽然确定任何维度中单纯形的体积都很简单，但高维中一般多面体的体积计算起来却很困难。计算体积的一种方法是将多面体剖成单纯形。该研究项目研究特殊多胞体家族（称为流多胞体和根多胞体）的解剖、体积和整数点。流和根多胞体自然地出现在多个数学领域的问题中，例如表示论和代数几何。该项目的结果将对这些数学领域以及科学和工程应用产生影响。代数组合学中的许多重要猜想和问题都以流多胞形和根多胞形为核心。这些特殊的多胞体可以系统地分解为单纯形。该项目将研究哈格伦德关于对角调和空间的二级希尔伯特级数的猜想，该级数可以用流多胞形整数点上的某个权重函数之和来表示。该研究还将研究克纳森和米勒引入的子词复合体，旨在通过将它们与根多胞体联系起来来说明舒伯特多项式和行列式理想的组合学。