Collaborative Research: Special Functions for Diagonal Harmonics and Schubert Calculus

组合数学是一个数学领域，涉及分析，组织和优化离散数据。 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。 本计画将发展对称函数理论的组合方法，此领域应用于机率、统计力学与量子资讯理论。该项目包括进一步开发SAGE开源数学软件，这是这项调查的一个重要工具。研究生将作为该项目的一部分进行培训。该项目涉及与表示论，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（Macdonald polynomials）是一个著名的正交多项式族，它构成了对称函数环的基础。麦克唐纳多项式也与许多其他领域有联系，包括双仿射赫克代数，粒子物理学中的卡洛格罗-萨瑟兰模型和结不变量。麦克唐纳多项式的研究也导致了对角调和空间，对称群的分级表示，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起来。 这个项目将Macdonald理论与Catalan函数联系起来，在旗簇上对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研究加泰罗尼亚函数的工具，特别是与舒伯特微积分和量子群晶体的联系，以丰富椭圆霍尔代数及其与对角谐波的联系，并建立一个新的组合框架。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.