The Combinatorics of Macdonald Polynomials and Related Objects

主要研究人员：Haglund， James b .提案编号：DMS - 0901467机构：宾夕法尼亚大学题目：麦克唐纳多项式的组合及相关对象该奖项由2009年美国复苏与再投资法案（公法111-5）资助。在数学和科学中有许多多项式的例子，这些多项式依赖于几个变量，并且具有重要的应用。在这个项目中，PI将研究与麦克唐纳多项式相关的组合学，麦克唐纳多项式是一种多变量对称函数，满足正交关系，在代数组合学中发挥核心作用，应用于特殊函数、代数几何和统计力学。它们最初的定义是相当困难和间接的，但在2004年，PI为它们找到了一个很好的组合公式，这个公式在随后由海曼、Loehr和PI的联合研究中得到了证明。PI和其他人已经发现麦克唐纳多项式的新组合常常导致相关对象的新组合公式。例如，在最近与Luoto、Mason和Van Willigenburg的联合工作中，PI一直在研究非对称麦克唐纳多项式的某些极限情况，即与表示理论相关的demmazure字符和demmazure原子。PI和他的合作者已经证明，一些由重要的舒尔函数基所满足的基本关系有涉及Demazure特征和原子的精炼版本，涉及麦克唐纳多项式的新组合结构。该项目的另一部分涉及PI最近形成的猜想，这些猜想是PI先前猜想的多变量版本，其他猜想涉及到零和匹配多项式。PI正在使用计算机结合当前的数学方法在这些猜想上取得进展。单变量多项式在科学和数学中发挥着重要作用，例如用连续函数对离散数据建模。某些称为正交多项式的多项式族特别有用。麦克唐纳多项式是多变量正交多项式的一个主族，它包含了前人研究过的各种正交多项式族和其他有用的多项式作为特例。麦克唐纳证明了它们的存在，但没有给出特别简单的描述。在之前与海曼和勒尔的合作中，PI证明了对它们的直接组合描述。麦克唐纳多项式仍有许多未解决的问题，在这个项目中，PI将继续发展该理论的组合方面，应用于数学各个分支的各种多项式。该提议的另一个方面涉及使给定多项式为零的变量值，PI正在使用计算机和实验方法进行研究。

Principal Investigator: Haglund, James B.Proposal Number: DMS - 0901467 Institution: University of PennsylvaniaTitle: The Combinatorics of Macdonald Polynomials and Related ObjectsThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).There are many examples in mathematics and science of polynomials which depend on several variables, and which have important applications. In this project the PI will investigate the combinatorics associated with some of the most useful of these, Macdonald polynomials, which are multi-variate symmetric functions which satisfy an orthogonality relation and play a central role in algebraic combinatorics with applications to special functions, algebraic geometry, and statistical mechanics. Their original definition was rather difficult and indirect, but in 2004 the PI found a nice combinatorial formula for them, which was proved in subsequent joint work between Haiman, Loehr and the PI. The PI and others have been finding that the new combinatorics of Macdonald polynomials often leads to new combinatorial formulas for related objects. For example, in recent joint work with Luoto, Mason, and Van Willigenburg, the PI has been investigating certain limiting cases of nonsymmetricMacdonald polynomials known as Demazure characters and Demazure atoms, which are connected to representation theory. The PI and his collaborators have shown that some of the fundamental relations satisfied by the important Schur function basis have refined versions involving Demazure characters and atoms, involving constructs in the new combinatorics of Macdonald polynomials. Another part of the project involves conjectures the PI has recently formed which are multi-variate versions of previous conjectures of the PI and others involving the zeros of rook and matching polynomials. The PI is using computers combined with current mathematical methods to make progress on these conjectures.Polynomials in a single variable play a fundamental role in science and mathematics, for example when modeling discrete data by a continuous function. Certain families of polynomials known as orthogonal polynomials are especially useful. Macdonald polynomials are a master family of orthogonal polynomials in several variables which contain all sorts of previously studied families of orthogonal polynomials and other useful polynomials as special cases. Macdonald proved they exist, but gave no particularly simple description of them. The PI, in a previous collaboration with Haiman and Loehr, proved a direct combinatorial description of them. There are still many unsolved problems involving Macdonald polynomials, and in this projectthe PI will continue to develop the combinatorial side of the theory, with applications to a variety of polynomials from various branches of mathematics. Another aspect of this proposal involves values of the variables which make a given polynomial zero, which the PI is investigating using computers and experimental methods.