The Combinatorics of Macdonald Polynomials and Related Objects

主要研究者：Haglund，James B.Proposal Number：DMS - 0901467 Institution：University of Pennsylvania题目：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 depends on several variables，and which have important applications. 在这个项目中，PI将研究与其中一些最有用的Macdonald多项式相关的组合学，Macdonald多项式是满足正交关系的多变量对称函数，并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。 他们最初的定义是相当困难和间接的，但在2004年，PI为他们找到了一个很好的组合公式，这在随后的海曼，Loehr和PI之间的联合工作中得到了证明。 PI和其他人已经发现麦克唐纳多项式的新组合学经常导致相关对象的新组合公式。 例如，在最近与Luoto、Mason和货车Willigenburg的联合工作中，PI一直在研究被称为Demazure字符和Demazure原子的非线性麦克唐纳多项式的某些极限情况，这些情况与表示论有关。 PI和他的合作者已经证明，重要的舒尔函数基所满足的一些基本关系有涉及Demazure字符和原子的改进版本，涉及麦克唐纳多项式的新组合学中的结构。 该项目的另一部分涉及PI最近形成的猜想，这些猜想是PI之前猜想的多变量版本，以及其他涉及车的零点和匹配多项式的猜想。 PI正在使用计算机结合当前的数学方法，以在这些方面取得进展。单变量多项式在科学和数学中发挥着基础作用，例如当用连续函数对离散数据建模时。 某些称为正交多项式的多项式族特别有用。 Macdonald多项式是多元正交多项式的一个主族，它包含了所有以前研究过的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。 PI在以前与Haiman和Loehr的合作中证明了它们的直接组合描述。 仍然有许多未解决的问题，涉及麦克唐纳多项式，在这个项目中，PI将继续发展理论的组合方面，应用于各种数学分支的各种多项式。 这个建议的另一个方面涉及使给定多项式为零的变量的值，PI正在使用计算机和实验方法进行调查。

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.