Combinatorics of Multivariate Orthogonal Polynomials

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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.