Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q, t)$-Gelfand-Tsetlin Graph

[摘要] We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [ Ann. Probab. 43 (2015), 3052-3132], and are expected to provide tools for the study of statistical mechanical models, representation theory and random matrices. As first application of our formulas, we characterize the boundary of the $(q,t)$-deformation of the Gelfand-Tsetlin graph when $t = q^{\theta}$ and $\theta$ is a positive integer.

[关键词] Branching graph;Macdonald polynomials;Gelfand–Tsetlin graph