Refined symmetric functions and affine analogs in combinatorics

这一建议是为了探索A型仿射Weyl群和对称函数之间的联系，这些对称函数称为k-Schur函数。K-Schur函数起源于对Macdonald多项式基的研究，Macdonald多项式基是对称函数空间的基，在几何、表示理论和物理学中扮演着重要的角色。随着对k-Schur函数的深入研究，发现k-Schur函数不仅为麦克唐纳多项式理论中的难题提供了一种途径，而且为对称空间的子空间提供了一个基本的基础。在对k-Schur函数的研究中，产生了大量美丽的组合猜想，扩展和提炼了对称函数论中的经典思想，本文就是在这些猜想的基础上，进一步研究k-Schur函数。例如，最近关于k-SchurFunctions的工作表明了Macdonald多项式和仿射对称群之间的联系。组合学是一个广阔的数学领域，大致被描述为对满足特定标准的对象集合进行计数的研究。因此，组合数学在许多领域的发展中扮演着不可或缺的角色，组合方法被从生物学家到理论物理学家的科学家所使用。PI试图理解对称函数理论中出现的美丽组合数学的自然精炼-对称函数理论是数学的一个经典部分，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.