The Combinatorics of Higher Dimensional Catalan Polynomials

高维 Catalan 多项式的组合

• 批准号: 1200280
• 负责人: Susanna Fishel
• 金额: $ 13.4万
• 依托单位: Arizona State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200280&HistoricalAwards=false
• 关键词: Combinatorics; Higher; Dimensional; Catalan; Polynomials

The theory of Macdonald symmetric functions is one of the most active areas in enumerative combinatorics. At the heart of this theory is the interpretation, due to Haiman, of Macdonald polynomials in terms of the algebraic geometry of families of points in the complex plane, that is, of the Hilbert scheme. The PI, joint with Grojnowski, proposes to generalize this to higher dimensions. In order to avoid the algebraic geometric difficulties involved, they study an analog of a small piece of the usual 2-dimensional theory, the theory of the two variable Catalan polynomial. The project will continue Fishel and Grojnowski's program for defining the combinatorial d-dimensional analogs of the Catalan polynomial. It will explain detailed computations of the higher dimensional algebraic geometry, and motivated by this, detailed computations with a family of poset structures on the 1-skeleton of the associahedron, a simplicial complex whose vertices are Dyck paths. The PI expects that further study of these structures will lead to a combinatorial description of the geometric quantities, and to a generalization of much of the theory of Macdonald symmetric functions.The theory of Macdonald symmetric functions is already a vast body of work, with exciting applications and interactions with representation theory, algebraic geometry, statistics, and even physics (for example, in studying ``the topological vertex'' of string theory). The PI expects that, just as with the Macdonald polynomials, these higher dimensional symmetric functions will arise in diverse mathematical fields.

Macdonald对称函数理论是计数组合学中最活跃的领域之一。 在这个理论的核心是解释，由于海曼，麦克唐纳多项式的代数几何家庭的点在复杂的平面，即希尔伯特计划。PI与Grojnowski联合提出将其推广到更高的维度。为了避免代数几何的困难，他们研究了一个模拟的一小块通常的二维理论，理论的两个变量加泰罗尼亚多项式。 该项目将继续Fishel和Grojnowski的计划，用于定义加泰罗尼亚多项式的组合d维类似物。 它将解释高维代数几何的详细计算，并以此为动机，详细计算与一个家庭的偏序集结构上的1-骨架的associahedron，一个单纯的复杂的顶点是戴克路径。 PI希望对这些结构的进一步研究将导致几何量的组合描述，并推广麦克唐纳对称函数的大部分理论。麦克唐纳对称函数的理论已经是一个庞大的工作机构，具有令人兴奋的应用和与表示论，代数几何，统计学，甚至物理学的相互作用（例如，在研究弦理论的“拓扑顶点”时）。 PI期望，就像麦克唐纳多项式一样，这些高维对称函数将出现在不同的数学领域。