The Combinatorics of Higher Dimensional Catalan Polynomials

Macdonald对称函数理论是列举组合学中最活跃的领域之一。这一理论的核心是根据复平面上点族的代数几何解释麦克唐纳多项式，即希尔伯特格式。PI与格罗伊诺夫斯基联合，建议将这一点推广到更高的维度。为了避免所涉及的代数几何困难，他们研究了一小部分通常的二维类比理论，即二元加泰罗尼亚多项式理论。该项目将继续Fishel和Grojnowski定义加泰罗尼亚多项式的组合d维类似物的计划。它将解释高维代数几何的详细计算，并在此基础上，使用关联体的1-骨架上的偏序集结构族进行详细计算，关联体是一个顶点为Dyck路的单纯复形。PI期望对这些结构的进一步研究将导致几何量的组合描述，并导致麦克唐纳对称函数的许多理论的推广。麦克唐纳对称函数的理论已经是大量的工作，具有令人兴奋的应用和与表示理论、代数几何、统计学甚至物理的交互作用(例如，在研究弦理论的“拓扑顶点”方面)。PI预计，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。

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.