Quotienting by Quasisymmetrics: Combinatorics and Geometry

代数组合学是数学的一个领域，起源于离散组合对象与抽象代数概念之间的动态相互作用。这种跨学科的方面使得研究领域极其丰富，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。要回答这个领域的问题，一个关键的组成部分包括将难以理解的几何或代数信息提炼成可操作的组合数据，这有一个额外的好处，那就是对原始环境有了新的认识。本项目应用组合学和几何学的工具来研究经典的代数结构，如多项式环的商。PI将开发组合工具和技术来获得洞察力，其长期目标是在代数组合学中更牢固地掌握正性问题。此外，本项目还为研究生提供了多个研究训练机会。本项目旨在促进我们对拟对称多项式理想中多项式环模的商的理解。这个商是协不变代数的准对称类比，这个对象的历史可以追溯到a . Borel的工作。lascoux - sch<s:1>岑伯格的舒伯特多项式给出了具有深刻几何和组合意义的协不变代数的一个基础。在有关复面体变化的问题的激励下，PI将研究与舒伯特多项式密切相关的准对称商的新基础。该项目的总体目标是进一步深入了解长期存在的舒伯特多项式组合相乘的开放问题。在几何方面，PI将利用多面体变化和准对称商之间的联系，提出多面体中点阵枚举的新方面。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.