Collaborative Research: Special Functions for Diagonal Harmonics and Schubert Calculus

组合学是一个涉及分析、组织和优化离散数据的数学领域。它是许多科学领域的基本工具，如基因组学、计算机科学、统计学和物理学。这个项目将开发对称函数理论的组合方法，这是一个应用于概率、统计力学和量子信息理论的领域。该项目包括进一步开发SAGE开源数学软件，这是本次调查的关键工具。研究生将作为这个项目的一部分接受培训。这个项目解决与表示理论、代数几何和物理相关的组合问题，重点是麦克唐纳多项式和舒伯特微积分。麦克唐纳多项式是构成对称函数环的基的一族重要的正交多项式。麦克唐纳多项式还与许多其他领域有联系，包括双仿射Hecke代数、粒子物理中的Calogero-Sutherland模型和纽结不变量。对麦克唐纳多项式的研究还导致了对角调和空间，这是对称群的一种分级表示，其特征与麦克唐纳多项式密切相关。在过去的二十年里，出现了一幅美丽的图画，将这个故事与椭圆曲线的霍尔代数联系在一起。这个项目将Macdonald理论与加泰罗尼亚函数联系起来，对旗簇上的某些向量丛的欧拉特征进行分级。研究人员的目标是应用以前加泰罗尼亚函数研究中的工具，特别是与舒伯特微积分和量子群晶体的联系，通过一个新的组合框架来丰富椭圆霍尔代数及其与对角调和的联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Combinatorics is an area of mathematics concerned with analyzing, organizing, and optimizing over discrete data. It is a fundamental tool in many scientific areas such as genomics, computer science, statistics, and physics. This project will develop combinatorial methods in symmetric function theory, an area with applications to probability, statistical mechanics, and quantum information theory. The project includes further development of the SAGE open-source mathematics software, a crucial tool for this investigation. Graduate students will be trained as part of this project.This project addresses combinatorial questions tied to representation theory, algebraic geometry, and physics, with a focus on Macdonald polynomials and Schubert calculus. Macdonald polynomials are a remarkable family of orthogonal polynomials that form a basis for the ring of symmetric functions. Macdonald polynomials also have connections with many other areas, including double affine Hecke algebras, the Calogero-Sutherland model in particle physics, and knot invariants. Studies of Macdonald polynomials also led to the space of diagonal harmonics, a graded representation of the symmetric group whose character is closely tied to Macdonald polynomials. Over the last two decades a beautiful picture has emerged connecting this story to the Hall algebra of an elliptic curve. This project ties Macdonald theory to Catalan functions, graded Euler characteristics of certain vector bundles on the flag variety. The investigators aim to apply tools from previous study of Catalan functions, particularly connections with Schubert calculus and crystals of quantum groups, to enrich the elliptic Hall algebra and its ties to diagonal harmonics with a new combinatorial framework.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.