Combinatorics of Multivariate Orthogonal Polynomials

多元正交多项式的组合

基本信息

批准号：
2054482
负责人：
Sylvie Corteel
金额：
$ 30万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054482&HistoricalAwards=false
关键词：
Combinatorics Multivariate Orthogonal Polynomials

项目摘要

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.

正交多项式族的研究旨在概括对经典多项式族的理解，例如勒让德多项式，这些多项式在微分方程研究中出现，在物理、工程、数值逼近和其他领域有着广泛的应用。该研究项目解决了有关多元正交多项式组合的几个问题。目标是开发枚举组合学中通用且有效的技术，并将其应用于来自组合学、代数和物理学的问题。研究主题包括 Askey-Wilson 多项式系数的组合解释及其多元泛化、排除过程和 MacDonald (Koornwinder) 多项式之间的相互作用、q-Jacobi 多项式和演讲厅画面的组合以及 Rogers 之间的关系-拉马努金恒等式和圆柱分区。该项目将让研究生参与研究。更具体地说，该项目涉及围绕 Askey-Wilson 多项式组合及其多元泛化的几个相互关联的问题。第一个研究方向涉及 Schur 基中系数的正性和这些多项式的展开。该项目将探索格路径组合学、表格组合学、代数和概率的相互作用来解决这些问题。第二个方向旨在采用多物种不对称简单排除过程（ASEP）和广义版本来理解不同类型的麦克唐纳多项式的组合以及对拟对称类似物的推广。 PI 旨在使用来自统计物理学、组合代数和多行队列/顶点模型组合学的技术来开发这种组合理论。第三个方向是研究 Lecture Hall Schur 函数及其偏斜类似物、它们与 q-Jacobi 多元类似物和概括的联系，以及来自 Lecture Hall 对象的平铺模型及其渐近属性。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 Hall Littlewood 多项式的联系。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。