Quotienting by Quasisymmetrics: Combinatorics and Geometry

拟对称求商：组合学和几何

• 批准号: 2246961
• 负责人: Vasu Tewari
• 金额: $ 15.12万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246961&HistoricalAwards=false
• 关键词: Quotienting; Quasisymmetrics; Combinatorics; Geometry

Algebraic combinatorics is a field of mathematics originating from the dynamic interplay between discrete combinatorial objects and abstract algebraic notions. This interdisciplinary aspect makes for an extremely fertile field of investigation with applications to theoretical computer science, economics, statistics, computational biology, and other subjects of mathematics. A crucial component to answering questions in this area involves distilling hard-to-understand geometric or algebraic information into hands-on combinatorial data which has the added benefit of casting new light on the original context. This project applies tools from combinatorics and geometry to study classical algebraic constructions such as quotients of polynomial rings. The PI will develop combinatorial tools and techniques to gain insight, with the long term goal of developing a firmer grasp on positivity questions in algebraic combinatorics. Furthermore, this project provides several research training opportunities for graduate students.This project aims to advance our understanding of the quotient of the polynomial ring modulo the ideal of quasisymmetric polynomials. This quotient is an quasisymmetric analog of the coinvariant algebra, an object with a storied history going back to the work of A. Borel. A basis for the coinvariant algebra with deep geometric and combinatorial relevance is given by Schubert polynomials of Lascoux-Schützenberger. Motivated by questions pertaining to the permutahedral variety, the PI will study a new basis for the quasisymmetric quotient that is intimately tied with Schubert polynomials. The overarching goal of this project is to gain further insight into the long-standing open question of multiplying Schubert polynomials combinatorially. On the geometric side, the PI will draw upon connections between the permutahedral variety and the quasisymmetric quotient to bring forth novel aspects of lattice point enumeration in permutahedra.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.

代数组合学是一个数学领域，起源于离散组合对象和抽象代数概念之间的动态相互作用。这一跨学科的方面使得一个非常肥沃的调查领域，应用于理论计算机科学，经济学，统计学，计算生物学和其他数学学科。回答这一领域问题的一个关键组成部分是将难以理解的几何或代数信息提炼成动手组合数据，这具有在原始背景上投射新光线的额外好处。本计画应用组合学与几何学的工具来研究经典的代数结构，例如多项式环的代数。PI将开发组合工具和技术来获得洞察力，长期目标是更牢固地掌握代数组合学中的正性问题。此外，本计画亦提供研究生数个研究训练机会，旨在增进我们对拟对称多项式理想模多项式环之商之了解。这个商是共不变代数的一个准对称模拟，共不变代数是一个有着传奇历史的对象，可以追溯到A。博雷尔具有深刻的几何和组合相关性的协不变代数的基础由Lascoux-Schützenberger的舒伯特多项式给出。出于有关的问题permutahedral品种，PI将研究一个新的基础准对称商是密切联系在一起的舒伯特多项式。这个项目的首要目标是进一步深入了解舒伯特多项式组合相乘的长期未决问题。在几何方面，PI将利用置换面体多样性和准对称商之间的联系，提出置换面体中格点计数的新方面。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。