Combinatorics of Multivariate Orthogonal Polynomials

正交多项式族的研究旨在推广对经典多项式族的理解，例如在微分方程研究中出现的勒让德多项式，并在物理，工程，数值逼近和其他领域中有广泛的应用。本研究计画针对多元正交多项式组合数学的几个问题。目标是发展一般和有效的技术，在枚举组合与应用的问题，来自组合，代数和物理。正在研究的主题包括组合解释的系数的阿斯基威尔逊多项式和他们的多元推广，排斥过程和麦克唐纳（Koornwinder）多项式之间的相互作用，组合的q-雅可比多项式和演讲厅tableaux，和罗杰斯-拉马努金身份和圆柱分区之间的关系。该项目将涉及研究生的研究。更具体地说，该项目涉及围绕Askey-Wilson多项式及其多元推广的组合学的几个相关问题。第一个研究方向涉及系数的正性和这些多项式在Schur基中的展开。该项目将探讨格路径组合，tableaux组合，代数和概率的相互作用，以解决这些问题。第二个方向的目的是采用多物种不对称简单排斥过程（ASEP）和广义版本来理解不同类型的Macdonald多项式的组合学和准对称类似物的推广。PI的目标是使用来自统计物理，组合代数和多线队列/顶点模型组合数学的技术来发展这种组合理论。第三个方向是研究演讲厅舒尔函数及其斜向类似物，它们与q-Jacobi多元类似物和推广的联系，以及来自演讲厅对象的平铺模型及其渐近性质。最后一个方向涉及罗杰斯-拉马努金恒等式及其与圆柱分区和霍尔-利特尔伍德多项式的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The study of families of orthogonal polynomials aims to generalize understanding of the classical families of polynomials, such as the Legendre polynomials, that arise in the study of differential equations and have wide application in physics, engineering, numerical approximation, and other fields. This research project addresses several questions on the combinatorics of multivariate orthogonal polynomials. The goal is to develop general and efficient techniques in enumerative combinatorics with application to questions coming from combinatorics, algebra, and physics. The topics under study include the combinatorial interpretation of the coefficients of Askey-Wilson polynomials and their multivariate generalization, the interplay between exclusion processes and MacDonald (Koornwinder) polynomials, the combinatorics of q-Jacobi polynomials and Lecture Hall tableaux, and the relations between Rogers-Ramanujan identities and cylindric partitions. The project will involve graduate students in research.More specifically, this project concerns several interrelated questions surrounding the combinatorics of Askey-Wilson polynomials and their multivariate generalization. The first research direction concerns the positivity of the coefficients and the expansion of these polynomials in the Schur basis. The project will explore the interplay of lattice paths combinatorics, tableaux combinatorics, algebra, and probability to address these questions. The second direction aims to employ multispecies asymmetric simple exclusion process (ASEP) and generalized versions to understand the combinatorics of Macdonald polynomials of different types and generalization to quasisymmetric analogues. The PI aims to use techniques coming from statistical physics, combinatorial algebras, and multiline queue/vertex model combinatorics to develop this combinatorial theory. A third direction is to study the Lecture Hall Schur functions and their skew analogues, their connections with q-Jacobi multivariate analogues and generalizations, and tiling models coming from Lecture Hall objects and their asymptotic properties. A final direction concerns the Rogers-Ramanujan identities and their connection to cylindric partitions and Hall Littlewood polynomials.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.