Refined symmetric functions and affine analogs in combinatorics

本研究旨在探讨a型仿射Weyl群与对称函数k-Schur函数之间的联系。k-Schur函数是在对Macdonald多项式基的研究中产生的，Macdonald多项式基是对称函数空间的基础，在几何、表示理论和物理中起着重要作用。随着对k-Schur函数的深入研究，这些函数不仅可以为麦克唐纳多项式理论中的难题提供一种方法，而且可以为对称空间的子空间提供一个基本的基础。大量美丽的组合猜想扩展和完善了对称函数理论中的经典思想，这些猜想来自于对k-舒尔函数的研究。这个建议是对这些猜想进行研究，并进一步研究k-舒尔函数。例如，最近关于k-Schurfunctions的研究表明Macdonald多项式和仿射对称群之间存在联系。组合学是一个广阔的数学领域，可以粗略地描述为对满足特定标准的对象集合进行计数的研究。因此，组合学在许多领域的发展中起着不可或缺的作用，从生物学家到理论物理学家，许多科学家都使用组合方法。PI试图理解对称函数理论中出现的美丽组合的自然细化-这是数学的经典部分，在物理，工程和计算机科学等领域有着广泛的应用。

This proposal is to explore a connection between the type-A affine Weyl group and symmetric functions called k-Schur functions. The k-Schur functions arose in a study of the Macdonald polynomial basis, a basis for the symmetric function space that plays an important role in geometry, representation theory, and physics. As the k-Schur functions were investigated more deeply, it came to light that these functions may not only offer an approach to difficult problems in the theory of Macdonald polynomials, but also provide a fundamental basis for a subspace of the symmetric space. A wealth of beautiful combinatorial conjectures that extend and refine classical ideas in symmetric function theory came out of the study of k-Schur functions.This proposal is to work on these conjectures and to further investigatethe k-Schur functions. For example, recent work with the k-Schurfunctions suggests a connection between the Macdonald polynomials andthe affine symmetric group. Combinatorics is a vast area of mathematics loosely described as the study of counting collections of objects that satisfy specified criteria. As such, combinatorics plays an integral role in the development of many fields, and combinatorial methods are employed by scientists ranging from biologists to theoretical physicists.The PI seeks to understand a natural refinement for the beautiful combinatorics that arises in the theory of symmetric functions - a classical part of mathematics with a wide variety of applications in fields including physics, engineering, and computer science.