Combinatorics of Symmetric Functions

对称函数是多元多项式F(x1，x2，…，xn)，它在变量的任何排列下都是不变的。近年来，这类函数在数学的几个领域中已经变得非常重要，不同类型的对称函数的表示理论、解析、代数、几何和组合性质是当前研究的活跃领域。根据这一建议，PI将研究各种对称函数族以及与表示理论和代数相关的其他对象的组合性质。对于PI的研究项目来说，一个特别重要的对称函数族是麦克唐纳多项式族。这些是一组变量{x1，...，xn}中的对称函数，这些变量依赖于两个额外的参数q，t，也依赖于一个整数划分。到目前为止，麦克唐纳多项式已经发现了许多重要的应用，但也有一些与之相关的引人入胜的公开问题，特别是与它们的组合性质有关的问题。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Haiman，Cherednik和其他许多人的工作，Macdonald多项式和对角调和被联系到主流主题，如代数几何中的Hilbert格式和双仿射Hecke代数。麦克唐纳多项式和其他对称函数的组合学的新发展，以及相关的对象，如对角调和的希尔伯特级数，可能适用于数学和科学的许多领域。麦克唐纳多项式和对角调和空间的最新组合进展的显性和具体的性质使这些主题为广泛的研究人员和学生所接近。因此，与高级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的技术来解决，例如双射组合学。PI目前有三名博士生，并与主要是本科院校的教职员工保持合作。

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.