Combinatorics of Symmetric Functions

对称函数是多变量多项式F（x1，x2，...，xn），它在变量的任何排列下都是不变的。 近年来，这样的功能已成为相当重要的几个领域的数学，和代表性的理论，分析，代数，几何和组合性质的不同家庭的对称函数形式的活跃领域，目前的研究。 根据这个提议，PI将研究各种对称函数家族的组合性质以及与表示论和代数相关的其他对象。 一个特别重要的对称函数家族是麦克唐纳多项式家族。 这些是在一组变量{x1，...，xn}，其取决于两个附加参数q、t并且还取决于整数划分。 麦克唐纳多项式已经发现了许多重要的应用日期，但也有几个迷人的开放问题与他们有关，特别是与他们的组合性质。 它们与对角调和空间密切相关，对角调和空间是代数组合学和代数几何中的一个重要课题。 对称函数作为数学和数学物理研究人员的工具正变得越来越重要。 通过海曼，切雷德尼克和许多其他人的工作，麦克唐纳多项式和对角谐波被链接到主流主题，如希尔伯特计划从代数几何和双仿射Hecke代数。 麦克唐纳多项式和其他对称函数的组合数学的新发展，以及相关的对象，如希尔伯特系列的对角谐波，可能适用于许多数学和科学领域。 麦克唐纳多项式和对角谐波空间最近的组合进展的明确和具体的性质，使这些主题可以访问广泛的研究人员和学生。 因此，与高等理论数学有关的问题可以使用更具体和更容易理解的领域（如双射组合学）的技术来解决。PI目前有三个博士学位。学生，并保持与教师的合作，主要是本科院校。

A symmetric function is a multi-variate polynomial F(x1,x2,...,xn) which is invariant under any permutation of the variables. In recent years such functions have become quite important to several fields of mathematics, and the representation theoretic, analytic, algebraic, geometric, and combinatorial properties of different families of symmetric functions form active areas of current research. Under this proposal the PI will investigate the combinatorial properties of various families of symmetric functions and other objects of relevance to representation theory and algebra. One particularly important family of symmetric functions central to the PI's research program is the family of Macdonald polynomials. These are symmetric functions in a set of variables {x1,...,xn} which depend on two additional parameters q,t and also on an integer partition. Macdonald polynomials have found many significant applications to date, but there are also several captivating open problems associated with them, particularly with their combinatorial properties. They are closely related to the space of diagonal harmonics, an important topic in algebraic combinatorics and algebraic geometry. Symmetric functions are becoming increasingly important as tools for researchers in mathematics and mathematical physics. Through the work of Haiman, Cherednik and many others, Macdonald polynomials and diagonal harmonics are linked to mainstream topics such as the Hilbert Scheme from algebraic geometry and the double affine Hecke algebra. New developments in the combinatorics of Macdonald polynomials and other symmetric functions, and related objects such as the Hilbert series of diagonal harmonics, are potentially applicable to many areas of mathematics and science. The explicit and concrete nature of recent combinatorial advances in Macdonald polynomials and the space of diagonal harmonics makes these subjects accessible to a broad range of researchers and students. Thus problems with connections to advanced theoretical mathematics can be attacked using techniques from more concrete and accessible areas such as bijective combinatorics. The PI currently has three Ph. D. students, and maintains collaborations with faculty at primarily undergraduate institutions.