Combinatorics of Multivariate Orthogonal Polynomials

对正交多项式族的研究旨在推广对经典多项式族的理解，如勒让德多项式，这些经典多项式族是在研究微分方程式时产生的，在物理、工程、数值逼近等领域有着广泛的应用。本研究项目研究了多元正交多项式组合的几个问题。其目标是开发通用和有效的技术在列举组合学与应用问题来自组合数学，代数和物理。所研究的课题包括Askey-Wilson多项式系数的组合解释及其多元推广、排除过程与MacDonald(Koornwinder)多项式之间的相互作用、Q-Jacobi多项式与演讲厅表的组合以及Rogers-Ramanujan恒等式与柱面划分之间的关系。这个项目将包括研究生参与研究。更具体地说，这个项目涉及围绕Askey-Wilson多项式的组合学及其多元推广的几个相关问题。第一个研究方向是系数的正性和这些多项式在Schur基上的展开式。该项目将探索格路组合学、表格组合学、代数和概率之间的相互作用，以解决这些问题。第二个方向旨在利用多物种非对称简单排除过程(ASEP)和广义形式来理解不同类型的Macdonald多项式的组合学，并将其推广到准对称类似物。PI旨在使用统计物理、组合代数和多行队列/顶点模型组合数学中的技术来发展这一组合理论。第三个方向是研究Lesson Hall Schur函数及其斜类比，它们与q-Jacobi多元类比和推广的关系，以及来自Lesson Hall对象的平铺模型及其渐近性质。最后一个方向涉及罗杰斯-拉马努扬身份及其与圆柱形隔断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，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.