Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 001, 66 pages      arXiv:1704.02429      https://doi.org/10.3842/SIGMA.2018.001

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

Cesar Cuenca
Department of Mathematics, Massachusetts Institute of Technology, USA

Received April 21, 2017, in final form December 09, 2017; Published online January 02, 2018

Abstract
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.

Key words: branching graph; Macdonald polynomials; Gelfand-Tsetlin graph.

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